Classical and Non-Classical Limit Theorems for the Ising Model on Random Graphs

Classical and non-classical limit theorems for the Ising model on random graphs

Citation for published version (APA): Prioriello, M. L. (2016). Classical and non-classical limit theorems for the Ising model on random graphs. Technische Universiteit Eindhoven.

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Download date: 25. Sep. 2021 CLASSICALANDNON-CLASSICALLIMITTHEOREMS FORTHEISINGMODELONRANDOMGRAPHS

Maria Luisa Prioriello This work was ﬁnancially supported by The Netherlands Organization for Scientiﬁc Research (NWO) through the VICI grant of Remco van der Hofstad.

Classical and non-classical limit theorems for the Ising model on random graphs by M. L. Prioriello - Eindhoven University of Technology, 2016.

A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-4004-4

Cover design: Caroline Ashwood, “Copper tree”.

Printed by Gildeprint Drukkerijen, Enschede. Classical and non-classical limit theorems for the Ising model on random graphs

proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magniﬁcus, prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 27 januari 2016 om 16.00 uur

door

Maria Luisa Prioriello

geboren te Isernia, Italië

iii Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt:

voorzitter: prof. dr. ir. B. Koren 1e promotor: prof. dr. R. W. van der Hofstad 2e promotor: prof. C. Giardinà (Università di Modena e Reggio Emilia) copromotor: prof. C. Giberti (Università di Modena e Reggio Emilia) leden: prof. dr. A. C. D. van Enter (University of Groningen) prof. dr. J. S. H. van Leeuwaarden dr. F. R. Nardi prof. S. Polidoro (Università di Modena e Reggio Emilia)

Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in overeen- stemming met de TU/e Gedragscode Wetenschapsbeoefening.

iv ACKNOWLEDGMENTS

This thesis is the result of four years of research carried out between the Universities of Modena and Eindhoven. There are many people who deserve my gratitude for having supported me along this path. First of all, my special thanks go to my three supervisors. Thanks for having been such an edifying example of excellent researchers, thanks for the wonderful synergy among you that allowed me to work and grow up in a comfortable environment and thanks to your perseverance that made this double doctoral degree project possible. I want to thank Claudio for his patience and kindness: when I had a question, the door of your oﬃce was always open. Thank you for having helped me to pursue my research, putting beside your strong mathematical rigor a good dose of humor. Thank you Cristian, for having driven me to accomplish my PhD with great positivity and pragmatism. I admire your eagerness and passion for the research. Remco, thank you for having given me the opportunity to work with you. You are one of the most brilliant mind that I ever met, always smiling and polite, with a contagious enthusiasm for mathematics. I am indebted to each of you. I would like to express my gratitude to Aernout van Enter, Johan van Leeuwaar- den, Francesca Nardi and Sergio Polidoro for agreeing to be part of my doctoral committee, for reading my thesis and for their useful comments. I am also grateful to my master thesis supervisor Silvana De Lillo for having encouraged me to enroll the PhD program and for having continued our pleasant collaboration during these years. I am indebted with my co-author Sander Dommers for his signiﬁcant contribute on the subjects of Chapter 6 and for the fruitful discussions. I would like to thank my delightful housemates Alessandro G. e Melania. Thanks for making the house such a cheerful and loving place. I enjoyed a lot the amazing time spent together and the great harmony among us. I could not meet better housemates than you are. A big thank you also to my housemates in Bologna, especially to the people of Casa Molise for being such a family, for the great feasts and the endless laughs. I wish to thank Alessandro Z. e Carlo, the people who welcomed me here, who have always supported me giving me thousands of advices and suggestions on any matter. I would like to show my gratitude to all my colleagues of the Stochastics section for the awesome atmosphere and their great company. In particular, thanks to the Italian group: Enrico, Alessandro G., Lorenzo, Marta, Gianmarco, Alessandro Z., Fabio and Carlo for the wonderful time we have had together enjoying dinners, barbecues and trips.

v I also had a pleasant time in the oﬃce with Angelos, Jaron, Rick, Serban and Thembile. Thanks for tolerating my chitchat and my video calls. In particular, thanks to Jaron for the useful conversations and for the style tips about my thesis. Many thanks also to my colleagues in Modena, especially Carlos and Gioia, for their cheerful company during our travels and conferences. A kind thought is devoted to my graduate fellows Giacomo e Irma for carrying forward our friendship despite the distance. Thanks for coming to visit me and for showing always your support. I am immensely grateful to my best friends Fiorella, Giovanna and Noemi. Each one of you knows how much I owe to you and how much you are indispensable to me. There is no need to list all the reasons for which I should like to thank you. One of my biggest thanks is for my family. Thanks to my parents Mario and Patrizia, my brother Antonio e my sister Serena to be always present in my life. You continuously encourage and support me, expressing your interest and curiosity and making sure that everything goes well with your help. Without your love I would never have arrived this far. It is wonderful to see you so proud of me. I would also to express my gratitude to zia Angela and zio Pietro for their aﬀection and care. Finally, my last acknowledgment is dedicated to you, Dino. Thanks for being so close to me with your love, especially when we are far away. Thanks for your constant reassurance, for your sweet carefulness and for managing always to get a smile out of me. You are my strength and I am extremely lucky to have you by my side.

vi CONTENTS

1 Introduction and motivations 1 2 Ising model on random graphs 5 2.1 Random graph models ...... 5 2.1.1 Locally tree-like random graph ...... 5 2.1.2 Conﬁguration model ...... 7 2.1.3 Generalized random graph ...... 8

2.1.4 Local tree-likeness of CMN (d) and GRGN (w) ...... 9 2.1.5 Phase transition in random graphs ...... 9 2.1.6 Why these graphs? ...... 10 2.1.7 Other random graph models ...... 10 2.2 Ising model ...... 11 2.3 Measures and thermodynamic quantities for the Ising model on random graphs ...... 12 2.4 Phase transition and critical behavior ...... 15 2.5 Preliminaries ...... 16 2.5.1 State of the art ...... 17 2.5.2 Correlation inequalities ...... 17 2.5.3 Thermodynamic limits for the random quenched law . . . 18 2.5.4 Critical exponents for the random quenched law . . . . . 20 2.6 Other spin models ...... 21 2.6.1 The inhomogeneous Curie–Weiss model ...... 21 2.6.2 The Potts model ...... 22 2.6.3 The XY model ...... 22 2.7 Overview ...... 23 2.7.1 Discussion on the measures ...... 23 2.7.2 Overview of results ...... 25 2.7.3 CLT proof strategy ...... 26 2.7.4 Diﬀerences between random quenched and averaged quenched setting ...... 28 2.7.5 Properties of annealing ...... 28 2.7.6 Universality classes for power-law random graphs . . . . . 29 2.7.7 Violation of CLT at βc ...... 29 2.7.8 Low-temperature region ...... 30 2.8 Organization of the thesis ...... 30 3 Random quenched measure 33 3.1 LLN and CLT for locally tree-like random graphs ...... 33 3.2 Proofs ...... 34 3.2.1 Exponential convergence ...... 34 3.2.2 Random Quenched SLLN: Proof of Theorem 3.1.1 . . . . 35 3.2.3 Random Quenched CLT: Proof of Theorem 3.1.2 . . . . . 36

vii Contents

4 Averaged quenched measure 39 4.1 Thermodynamic limits and LLN for locally tree-like random graphs 39

4.2 CLT for CMN (2) ...... 40

4.3 CLT for CMN (1, 2) ...... 41 4.4 Proofs for locally tree-like random graphs ...... 41 4.4.1 Averaged quenched thermodynamic limits: Proof of Theo- rem 4.1.1 ...... 41 4.4.2 Averaged quenched WLLN: Proof of Theorem 4.1.2 . . . . 42

4.5 Proofs for CMN (2) ...... 43 4.5.1 Partition functions for the one-dimensional Ising model . 43 4.5.2 Quenched pressure ...... 44 4.5.3 Cumulant generating functions ...... 46 4.5.4 Averaged quenched CLT: Proof of Theorem 4.2.1 . . . . . 47

4.6 Proofs for CMN (1, 2) ...... 51 4.6.1 Quenched pressure ...... 52 4.6.2 Averaged quenched CLT: Proof of Theorem: 4.3.1 . . . . . 54 5 Annealed measure: uniqueness regime 65

5.1 Thermodynamic limits, SLLN and CLT for GRGN (w) ...... 65

5.2 Thermodynamic limits, SLLN and CLT for CMN (2) ...... 67

5.3 Thermodynamic limits, SLLN and CLT for CMN (1, 2) ...... 68

5.4 Proofs for GRGN (w) ...... 70 5.4.1 Annealed thermodynamic limits: Proof of Theorem 5.1.1 . 71 5.4.2 Annealed SLLN: Proof of Theorem 5.1.2 ...... 76 5.4.3 Annealed CLT: Proof of Theorem 5.1.3 ...... 76

5.5 Proofs for CMN (2) ...... 77 5.5.1 Annealed thermodynamic limits: Proof of Theorem 5.2.1 . 77 5.5.2 Annealed SLLN: Proof of Theorem 5.2.2 ...... 78 5.5.3 Annealed CLT: Proof of Theorem 5.2.3 ...... 78

5.6 Proofs for CMN (1, 2) ...... 83 5.6.1 Annealed partition function ...... 84 5.6.2 Annealed thermodynamic limits: Proof of Theorem 5.3.1 . 99 5.6.3 Annealed SLLN: Proof of Theorem 5.3.3 ...... 100 5.6.4 Annealed CLT: Proof of Theorem 5.3.4 ...... 100 5.6.5 Annealed pressure (way 2): Proof of Theorem 5.3.2 . . . . 101 6 Annealed measure: at criticality 107 6.1 Weight sequences conditions ...... 107

6.2 Critical exponents for GRGN (w) ...... 108

6.3 Non-classical limit theorem for GRGN (w) ...... 109 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents ...... 111 6.4.1 Magnetization: critical exponents β and δ ...... 112 6.4.2 Susceptibility: critical exponents γ and γ0 ...... 120 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem ...... 129 6.5.1 Rewrite of the moment generating function ...... 130 6.5.2 Convergence for E[W 4] < ∞ ...... 131

viii Contents

6.5.3 Convergence for τ ∈ (3, 5) ...... 135 6.5.4 Proof of Theorem 6.3.1 ...... 139 6.5.5 Scaling window ...... 144 7 Zero temperature recursion 147 7.1 Analytical solution ...... 147 7.1.1 Fixed point at zero temperature and zero external field . 148 7.1.2 Fixed point at zero temperature and positive external field 149 7.1.3 Fixed point approaching zero temperature ...... 151 7.2 Numerical analysis ...... 153 7.2.1 Zero temperature and zero external field case ...... 154 7.2.2 Zero temperature and positive external field case . . . . . 155 7.2.3 Non-zero temperature case ...... 159

Bibliography 161

INTRODUCTIONANDMOTIVATIONS 1

The Ising model, named after Ernst Ising, is a mathematical model introduced to understand ferromagnetism. The model was suggested to Ising for his Ph.D thesis [41] by his adviser Lenz in the 1920s [57, 58, 59]. The idea is to represent the interaction between atoms of a magnetic material. Each atom has a magnetic moment, typically represented by an arrow, which is allowed to point either up or down and flips between the two orientations. The interactions among atoms decay rapidly with distance, so in general, only the influence on neighboring elements is taken into account. In addition an external magnetic field, which tends to align the moments of the atoms in its direction, may be present. The classical mathematical model lives on a lattice and a particle is assigned to each lattice site. The magnetic moment (spin) of a particle is restricted to take only two values: +1 or −1.

Ising phase transition. An interesting feature of the Ising model is the presence of a phase transition, which in general occurs when a small change in parameters, such as temperature or pressure, causes a qualitative change in the state of the system. Phase transitions are common in nature and in everyday life, for example, a liquid becomes a gas above a ﬁxed temperature, or freezes when the temperature goes down. In ferromagnetic materials the phase transition can appear when, below a certain temperature, the system becomes magnetized because a fraction of atoms larger than 1/2 are magnets pointing in a particular direction, while this magnetization is lost at suﬃciently high temperatures. In his study, Ising discovered that the one-dimensional model does not exhibit a phase transition and he asserted, erroneously, that there was no phase transition in any dimension.

Ising models and cooperative phenomena. The interactions between the spin random variables are crucially determined by the spatial structure where the spin variables are sitting. Besides regular lattices, in recent years much attention has been devoted to the setting in which the spins variables are placed on the vertices of random graphs. These models can be studied as a paradigmatic model for cooperative phenomena living on complex networks. The behavior of the whole system can be understood as the consequence of the cooperation between units of the system. This turns out to be useful for analyzing problems in ﬁelds as diverse as biology, economy, neuroscience, social science, opinion formation. We can think, for example, at the mapping between the dynamics of the Ising model and some processes on brain networks [30, 40, 61, 63]. Our brain consists of approximately 1010 neurons connected in a very complex network, called

1 Introduction and motivations neural network. Neurons communicate with other neurons via electric signals. At a fixed time, a neuron either can or cannot generate an electric signal, i.e. we say that it has two states: it is active or it is at rest. It turns out that this system can be described by an Ising model, where the external field is replaced by the average of emitting electric signal for each cell. An other instance is the dynamic of the opinion formation in a social system [24, 25, 33, 31, 32]. We consider a population of N individuals and we associate at each individual a variable which can take only two values. This variable represents the opinion, being in favor or against, about a certain topic, i.e. vote yes/no in a referendum. The individuals may change their opinions after the interaction with others. The strength of this interaction could be positive or negative, favors agreement or disagreement and it could be stronger among particular individuals. We can also take an additional external force that favors one of the two opinions into account, like the influence of media, government, existing culture. This kind of model has been implemented to study phenomena of interaction between two different cultures [11, 10]. In this case the whole population is divided into two homogeneous groups with interactions that are stronger and imitative between the individuals of each group, whereas they could be even counter-imitative between individuals belonging to different groups. Processes like this one do not live on lattices, but more likely they take place on networks that could be formed, for example, by a set of people (the vertices of the network) connected by links that represent friendships, or some other form of social interaction, between individuals. Networks of this type are called social networks.

Random graphs as network models. In the real world a large number of examples of complex networks exist [55, 56]. Beyond the social networks, we can find information networks (e.g., World Wide Web, citation networks between academic papers), technological networks (e.g., Internet, transportation networks such as airline routes, road and rail networks) and biological networks (e.g., protein interaction network, neural networks). Many of these real-world networks share features. First of all, these networks are large and they grow in size as time proceeds. Nevertheless, a small-world behavior appears, meaning that most pairs of vertices are connected by relatively short chains of links. Another interesting characteristic is that the networks are scale free, meaning that the majority of vertices have “few” connections, whereas the number of vertices with high connections decays slowly. We model networks using random graphs. We choose random graph models that possess particular properties of interest, such as specified degree distributions, to describe a given network. A simple example of a random graph is obtained by fixing a number of vertices and adding connections between them with a certain probability. Alternatively we can get a random graph growing in size adding vertices successively and attaching or removing connections. There exist

2 Introduction and motivations diﬀerent ways to build a random graph, each of which leads to a graph with diﬀerent properties [3, 43, 64, 65].

Ising models on random graphs. Ising models on random graphs are also an ideal probabilistic model to study dependent random variables in the presence of two sources of randomness. Indeed they possess a dependence structure between the Ising random variables and an extra level of spatial disorder given by the random graph. Taking this double randomness into account in various ways, three different measures arise. For a given assignment of the graph the Random Quenched measure coincides with the random Boltzmann–Gibbs distribution on that random graph. This means that we just look at the evolution of the dynamic of the spin towards an equilibrium state whereas the environment is frozen. If we think of the example of opinion formation, the network of the social interactions is fixed and the individuals just change their opinion according to that of their neighbors. The Averaged Quenched and the Annealed measures are obtained by averaging over all possible random graphs in two different ways. These two measures arise in the presence of a dynamical evolution of the environment. In particular, in the annealed setting the dynamics of the graph is much faster than the dynamics of the spins, i.e. the friendship network changes much more quickly than the opinion of the individuals. Another interesting feature that emerges from the study of the Ising model on random graphs is the prominent role of the inhomogeneity of the graph models. This inhomogeneity, that concerns the variable numbers of connections of vertices, is not present in the analysis on lattice and it has to be taken in account in our investigations.

Contribution of this thesis. The aim of the thesis is to study asymptotic theorems for spin variables of the Ising model on random graphs. In particular, we look at the behavior of the sum of the spins under the three different measures mentioned in the previous paragraph. Asymptotic theorems are relevant to understand properties of the system when its size becomes large, as for the models under consideration. We highlight that the study of these theorems in this setting is particularly challenging because we are dealing with dependent random variables. Our analysis is inspired by the results proved by Ellis for the Ising model on Zd (see [26]). The first step is to prove the law of large numbers. We will show the convergence of the average of the sum of the spins to their expected value. The rate of convergence will be different for the different measures. Then, by analyzing the fluctuations around the expected value, we prove the central limit theorem. For the random quenched setting, the result holds for a generic class of random graphs. In the averaged quenched and annealed case, instead, we have to limit our analysis to particular random graph models in

3 Introduction and motivations which explicit calculations are possible. Indeed, in these two settings we have to take into account the contribution of the graph and we can not replicate the general strategy used with the random quenched measure. We emphasize that we restrict these theorems to the set of the parameters such that there exists a unique Gibbs measure, i.e., the uniqueness regime, outside the critical region. Finally we analyze what happens at the critical point. Here the central limit theorem breaks down and a diﬀerent scaling of the total spin is needed to obtain a non-classical limiting distribution.

Organization. This thesis is structured in the following way. In the next chapter we introduce the spin models, the random graphs, the main deﬁnitions and we provide a general overview on this work. Chapters 3, 4 and 5 are organized according to the three diﬀerent measures and they collect results proved in [37, 36]. For each measure we present the asymptotic theorems for the random graphs under consideration outside the critical region. Chapter 6, based on [19], concerns the study at criticality: we show results about critical exponents and a non-classical limit theorem. Chapter 7 investigates a recursion equation that arises for the Ising model on random trees. For a more extensive description of the organization of the thesis see Section 2.8.

4 ISINGMODELONRANDOMGRAPHS 2

2.1 Random graph models

In this section we present the spatial structures, i. e., the random graphs, on which our spin model is defined. We first introduce the general class of locally tree-like random graphs, where the local structure around any vertex is given by a random tree. Then we define two specific models, namely the configuration model and the generalized random graph.

We denote by GN = (VN , EN ) a random graph with vertex set VN = [N] and edge set EN ⊂ VN × VN and by GN the set of all possible graphs with N vertices. Here and in the rest of this thesis [N] = {1, ... , N} will be used to denote the vertex set of GN .

2.1.1 Locally tree-like random graph

The class of models that we consider here is the same as deﬁned in [16, 20].

Namely, we study random graph sequences (GN )N≥1 that are assumed to be locally like a homogeneous random tree, uniformly sparse and whose degree distribution has a strongly ﬁnite mean. In order to formally state these assumptions, we need to introduce some notation.

Let the integer-valued random variable D have distribution P = (pk)k≥1 , i.e., P (D = k) = pk, for k = 1, 2, ... . We deﬁne the size-biased law ρ = (ρk)k≥0 of D by (k + 1) p ρ = k+1 , (2.1.1) k E [D] where the expected value of D is supposed to be ﬁnite. Let K be a random variable with P (K = k) = ρk and let ν be the average value of K, i.e.,

X E [D (D − 1)] ν := kρ = , (2.1.2) k E [D] k≥0 that will play an important role in what follows. The random rooted tree T (D, ρ, `) is a branching process with ` generations, where the root oﬀspring has distribution D and the vertices in each next generation have oﬀspring that are independent and identically distributed (i.i.d.) with distribution ρ. We write that an event A holds almost surely (a.s.) if P (A) = 1. The ball

Bi (r) of radius r around vertex i of a graph GN is deﬁned as the graph induced by the vertices at graph distance at most r from vertex i. For two rooted trees

5 Ising model on random graphs

∼ T1 and T2, we write that T1 = T2, when there exists a bijective map from the vertices of T1 to those of T2 that preserves the adjacency relations.

Deﬁnition 2.1.1 (Local convergence to random trees). Let PN denote the law induced on the ball Bi (t) in GN centered at a uniformly chosen vertex i ∈ [N] = {1, ..., N}. We say that the graph sequence (GN )N≥1 is locally tree- like with asymptotic degree distributed as D when, for any rooted tree T with t generations, a.s., ∼ ∼ lim PN (Bi (t) = T ) = P (T (D, ρ, t) = T ) . (2.1.3) N→∞ This property implies, in particular, that the degree of a uniformly chosen vertex in GN is asymptotically distributed as D.

Definition 2.1.2 (Uniform sparsity). We say that the graph sequence (GN )N≥1 is uniformly sparse when a.s., 1 X 1 lim lim sup di {di≥`} = 0, (2.1.4) `→∞ N→∞ N i∈[N] where di is the degree of vertex i in GN and 1A denotes the indicator of the event A. A consequence of local convergence and uniform sparsity is that the number of edges per vertex converges a.s. to E[D]/2 as N goes to infinity. Definition 2.1.3 (Strongly finite mean degree distribution). We say that the degree distribution P = (pk)k≥1 has strongly finite mean when there exist constants τ > 2 and c > 0 such that ∞ X −(τ−1) pi ≤ ck . (2.1.5) i=k In Chapter 6 we focus our attention on graphs where the degree distribution satisfies a power law, defined in the following definition.

Deﬁnition 2.1.4 (Power laws). We say that the distribution P = (pk)k≥1 obeys a power law with exponent τ when there exist constants Cp > cp > 0 such that, for all k = 1, 2, ... −(τ−1) X −(τ−1) cpk ≤ pi ≤ Cpk . (2.1.6) i≥k Power law distributions describe settings where nodes of the graph have a highly variable number of neighbours. The eﬀect of this inhomogeneity will be relevant in particular in the analysis at the critical point, as we will see in the following.

Two specific examples of random graph models belonging to the class defined by some of these properties will be considered in detail in the following subsections: the configuration model and the generalized random graph under certain regularity conditions.

6 2.1 Random graph models

2.1.2 Conﬁguration model

The configuration model is a multigraph, that is, a graph possibly having self- loops and multiple edges between pairs of vertices, with fixed degrees. Fix an integer N and consider a sequence of integers d = (di)i∈[N]. The aim is to construct an undirected multigraph with N vertices, where vertex j has degree dj. We assume that dj ≥ 1 for all j ∈ [N] and we define the total degree X `N := di. (2.1.7) i∈[N]

We assume `N to be even in order to be able to construct the graph. Assuming that initially dj half-edges are attached to each vertex j ∈ [N], one way of obtaining a multigraph with the given degree sequence is to pair the half-edges belonging to the different vertices in a uniform way. Two half-edges together form an edge, thus creating the edges in the graph. To construct the multigraph with degree sequence d, the half-edges are numbered in an arbitrary order from 1 to `N . Then we start by randomly connecting the first half-edge with one of the `N − 1 remaining half-edges. Once paired, two half-edges form a single edge of the multigraph. We continue the procedure of randomly choosing and pairing the half-edges until all half-edges are connected, and call the resulting graph the configuration model with degree sequence d, abbreviated as CMN (d). We will consider, in particular, the following models:

(1) The 2-regular random graph, denoted by CMN (2), which is the conﬁgura- tion model with di = 2 for all i ∈ [N].

(2) The conﬁguration model with di ∈ {1, 2} for all i ∈ [N], denoted by

CMN (1, 2), in which, for a given p ∈ [0, 1], we have N − bpNc vertices with degree 1 and bpNc vertices with degree 2.

Remark 2.1.1. The conﬁguration model CMN (1, 2) can be implemented by assigning to each vertex degree 2 with probability p or degree 1 with probability 1 − p, conditioned to having bpNc vertices of degree 2. Another conﬁguration model would be obtained by considering the independent Bernoulli assignment to each vertex. This yields a random graph that has bpNc vertices of degree 2 only on average.

The degree sequence of the conﬁguration model CMN (d) is often assumed to satisfy a regularity condition, which is expressed as follows. Denoting the degree of a uniformly chosen vertex UN ∈ [N] by DN = dUN , we assume that the following property is satisﬁed:

Condition 2.1.1 (Degree regularity). There exists a random variable D with ﬁnite second moment such that, as N → ∞,

D (a) DN −→ D,

7 Ising model on random graphs

(b) E[DN ] → E[D] < ∞,

2 2 (c) E[DN ] → E[D ] < ∞, where −→D denotes convergence in distribution. We call D the asymptotic degree. Further, we assume that P(D ≥ 1) = 1.

As we will see in Section 2.1.4, the properties stated in Condition 2.1.1 are crucial to prove that the conﬁguration models handled in this thesis belong to the general class of locally tree-like random graphs. Moreover, the asymptotic degree D is central to detect the existence of a phase transition.

2.1.3 Generalized random graph

In the generalized random graph, each vertex i ∈ [N] receives a weight wi. Given the weights, edges are present independently, but the occupation probabilities for diﬀerent edges are not identical, but are rather moderated by the weights of the vertices. For a given sequence of weights w = (wi)i∈[N], the graph is denoted by GRGN (w). We call Iij the Bernoulli indicator that the edge between vertex i and vertex j is present and pij = P (Iij = 1) is equal to

wiwj pij = , (2.1.8) lN + wiwj

PN where lN is the total weight of all vertices given by lN = i=1 wi. The degree sequence of the generalized random graph GRGN (w) is often assumed to satisfy a regularity condition, which is expressed as follows. Denoting WN = wUN the weight of an uniformly chosen vertex UN ∈ [N], we introduce the following property:

Condition 2.1.2 (Weight regularity). There exists a random variable W with ﬁnite second moment such that, as N → ∞,

D (a) WN −→ W ,

(b) E[WN ] → E[W ] < ∞,

2 2 (c) E[WN ] → E[W ] < ∞, where −→D denotes convergence in distribution. We call W the asymptotic weight. Further, we assume that E[W ] > 0.

In the following, we will consider deterministic sequences of weights that satisfy Condition 2.1.2. In many cases, one could also work with weights w = (wi)i∈[N] that are i.i.d. random variables. When the weights are themselves random variables, they introduce a double randomness in the random graphs: ﬁrstly there is the randomness introduced by the weights, and secondly there is the

8 2.1 Random graph models randomness introduced by the edge occupation statuses, which are conditionally independent given the weights. The generalized random graph is an example of an inhomogeneous random graph, a large class of models in which the edge probabilities are not identical, but moderated by certain vertex weights. Other instances of inhomogeneous random graphs are studied in the literature (see [4]). For example we can consider the Chung-Lu random graph [7], where the edge probabilities are given by p(CL) = wiwj ∧ 1 or the Norrow-Reittu model [60], with a Poisson number of ij lN edges in between any two vertices i and j, with parameter equal to wiwj /lN . In [64] it is proved that, under suitable hypotheses, these models are asymptotically equivalent to a generalized random graph, therefore the analysis performed in this thesis is still true in these other cases.

2.1.4 Local tree-likeness of CMN (d) and GRGN (w)

Referring to [64] we remark that CMN (d) and GRGN (w) are locally tree-like graphs uniformly sparse when properties (a) and (b) of the Conditions 2.1.1 and 2.1.2 hold. When also property (c) holds, the graphs have strongly ﬁnite mean. It is straightforward to prove that the degree sequences of CMN (2) and CMN (1, 2) satisfy Condition 2.1.1. For example, for CMN (1, 2), we have

P(DN = 2) = 1 − P(DN = 1) = bpNc/N → p, so that the limiting degree D has distribution P(D = 2) = 1 − P(D = 1) = p.

In the case of the GRGN (w) with asymptotic weight W the asymptotic degree D is a mixed Poisson P oi(W ), i.e. W k P(D = k) = E e−W , (2.1.9) k! as shown in [64].

2.1.5 Phase transition in random graphs

Random graphs exhibit a phase transition in the size of the connected components. The existence of a phase transition in the graphs depends on the asymptotic degree D: it is known [4, 42, 51, 52] that when ν > 1 a giant component exists. As we will observe in the following, the phase transition for the Ising model on random graphs is strictly related to the graphs phase transition (see Theorems 2.5.3 (iii) and 5.1.1 (iii) below).

For GRGN (w) the degree distribution is D = P oi (W ) and we have that ν = E[W 2]/E[W ], because E[D] = E[W ] and E [D (D − 1)] = E[W 2]. Thus, depending on W , a giant component for GRGN (w) may exist.

On the other hand, the graph phase transition does not exist for CMN (2) and CMN (1, 2), since ν ≤ 1 in these cases. In particular, ν = 1 for CMN (2) and ν = 2p/(1 + p) < 1 for CMN (1, 2). Note that CMN (2) is critical for the existence of the giant component.

9 Ising model on random graphs

2.1.6 Why these graphs?

The need to work on these particular examples of graphs, i. e., the configuration model and the generalized random graph, arises from the fact that general tools, that we use in the proofs of random quenched theorems, are not available in the averaged quenched and annealed setting. Under these two measures, as we will see in the following, the fluctuations due to the spatial structure become relevant in the study of the central limit theorem. Therefore, due to the lack of general formalism to treat the generic class of locally tree-like graphs, we analyze these specific random graphs as benchmarks, since they allow for explicit computations. We choose the configuration model with vertex degree 2 and the configuration model with vertex degrees 1 and 2 because in these cases only one-dimensional structures, like cycles and lines, appear. Then the partition function of the Ising model on these graphs can be expressed in term of the well-known partition function of the one-dimensional nearest-neighbor Ising model. The choice of the generalized random graph is related to the fact that in this model the edges are independent. This implies that in the annealed setting the Ising model on the generalized random graph is reduced to an inhomogeneous Curie–Weiss model, as we explain in Section 5.4 below.

2.1.7 Other random graph models

We present in this subsection some other examples of random graphs. We are not going to consider them in this thesis, but they should be interesting for future works. One of the simplest random graph model is the Erdős–Rényi random graph, ER [29, 38], denoted by GN (p). This graph has N vertices and each pair of vertices is independently connected with a probability p. If we choose p = c/N the degree of a random vertex in the graph has distribution Bin(N − 1, c/N) that, for N → ∞, converges to a P oiss(c). Despite the simplicity of this model, it is fascinating because of the presence of a phase transition when p varies. The preferential attachment model,[1, 5], is an instance of a graph growing in time. Vertices having a ﬁxed number of edges are sequentially added to the graph. Their edges are attached to a receiving vertex with a probability proportional to the degree of the receiving vertex at that time. Thus, vertices that already have a high degree are more likely to attract edges of new vertices. While the Erdős–Rényi random graph and the preferential attachment model are locally tree-like random graphs, we introduce now a model that does not belong to this class: the random intersection graph,[62, 44]. In this model, we assign to each vertex v a random subset Sv of a given set S. Then two vertices v and w are connected by an edge only if their assigned sets Sv and Sw intersect. This graphs are natural models for social networks. For example, we can consider

10 2.2 Ising model users of a social network as vertices of the graph and social groups as elements of S. Then two users are joined by an edge if they share at least one group.

2.2 Ising model

We start introducing the Ising model on a subset with N sites of the lattice Zd. To each site we assign a spin variable σi which only takes the values +1 (spin-up) or −1 (spin-down). A spin configuration of the system σ = (σ1, σ2, ..., σN ) takes values on the space of spin configurations ΩN , i.e., the set of all spin sequences N N σ. Therefore ΩN = {1, −1} and we have 2 possible configurations. We introduce the Hamiltonian of the system that describes the total energy of a configuration as

N X X HN (σ) = − Ji,jσiσj − h σi. (2.2.1) i,j i=1 ki−jk=1

The first sum is over all pairs of variables that are connected by exactly one bond on the lattice, i.e., we have only nearest-neighbor interactions. If for all (i, j) the coupling constants Ji,j satisfy Ji,j ≥ 0 then the model is ferromagnetic, otherwise if Ji,j < 0 ∀ i, j the model describes an antiferromagnet. For a ferromagnet, if the spins of two nearest neighbors in the configuration have the same orientation (both either up or down), then this pair contributes an energy −Ji,j. If they have opposite orientations, they contribute an energy Ji,j. So, a configuration in which most nearest neighbors have the same orientation has a lower energy level. In this thesis we assume that Ji,j = 1 for each couple (i, j) of nearest neighbors. The parameter h correspond to the presence of an external magnetic field that tends to align the spin variables in its direction. The probability of a particular configuration σ is given by the Boltzmann–Gibbs measure e−βHN (σ) µN,β,h(σ) = , (2.2.2) ZN (β, h) where β = 1/T is the inverse temperature and ZN (β, h) is the partition function given by X −βHN (σ) ZN (β, h) = e . (2.2.3)

σ∈ΩN From the partition function we can derive most of the information about the system. The negative sign in the exponential of the Boltzmann–Gibbs measure gives a higher probability to states with lower energy. A small value of β (which corresponds to a high temperature) tends to make all conﬁgurations more or less equally likely, while a large value of β (corresponding to a low temperature) tends to accentuate the probabilities of the lowest-energy states.

11 Ising model on random graphs

We can also deﬁne the Ising model on a graph G with N vertices. We denote by V (G) the vertex set of G and by E(G) its edge set. Assigning a spin variable σi to each vertex and considering interactions only between vertices that are connected by one edge, we have that the Hamiltonian is given by X X HN (σ) = − Ji,jσiσj − h σi. (2.2.4) (i,j)∈E(G) i ∈V (G)

Then we can proceed as above deﬁning the partition function and the Boltzmann– Gibbs measure.

2.3 Measures and thermodynamic quantities for the Ising model on random graphs

Now we define the Ising model on finite random graphs. In this setting two sources of randomness, i.e., the Ising random variables and the spatial disorder, appear. Then, to describe the system, we have to take into account this double randomness in different ways. We can first consider a fixed realization of the graph. In this case the probability of a spin configuration is given again by the Boltzmann–Gibbs distribution with the sum of the nearest neighbors taken on the edge set. Next we can evaluate a dynamical evolution of the graphs. So the probability of a spin configuration is now obtained by averaging over all possible random graphs with N vertices and different edge sets. As we will see, we can take this average in two different ways. In order to give the definition of the three measures that arise, we denote by

QN the law of the graphs with N vertices belonging to GN , for any N ∈ N. Deﬁnition 2.3.1 (Measures and expectations). For the spin variables σ = N (σ1, ..., σN ) taking values on the space of spin conﬁgurations ΩN = {−1, 1} we consider the following measures:

(i) Random quenched measure. For a given realization GN ∈ GN the random quenched measure coincides with the random Boltzmann–Gibbs distribution h i exp β P σ σ + B P σ (i,j)∈EN i j i∈[N] i µGN (σ) = , (2.3.1) ZGN (β, B) where " # X X X ZGN (β, B) = exp β σiσj + B σi (2.3.2) σ∈ΩN (i,j)∈EN i∈[N]

is the partition function. Here β ≥ 0 is the inverse temperature and B ∈ R is the uniform external magnetic ﬁeld.

12 2.3 Measures and thermodynamic quantities for the Ising model on random graphs

(ii) Averaged quenched measure. This law is obtained by averaging the random Boltzmann–Gibbs distribution over all possible random graphs  h i exp β P σ σ + B P σ (i,j)∈EN i j i∈[N] i PN (σ) = QN (µGN (σ)) = QN   . ZGN (β, B) (2.3.3)

(iii) Annealed measure. This law is obtained by averaging separately the partition function in the Boltzmann–Gibbs distribution h i Q exp β P σ σ + B P σ N (i,j)∈EN i j i∈[N] i PeN (σ) = . (2.3.4) QN (ZGN (β, B))

Note that, in the previous deﬁnitions, the inverse temperature β does not multiply the external ﬁeld. This turns out to be technically convenient and does not change the results, because we are only looking at systems at equilibrium, and hence this would just be a reparametrization.

In the following, with a slight abuse of notation we use the same symbol to denote both a measure and the corresponding expectation. Namely, for any function k : ΩN −→ R, we let X µGN (k) = k(σ)µGN (σ) , σ∈ΩN denote the expectation of k with respect to µGN , and similarly for the measures QN , PN and PeN . Moreover we remark that all the measures deﬁned above depend sensitively on the two parameters (β, B). However, for the sake of notation, we drop the dependence of the measures on these parameters. We denote the partition function by ZN instead of ZGN and sometimes we use Varµ(X) to denote the variance of a random variable X with law µ.

Considering these three measures we can compute the expectation of any function that depends on the spins and on the graphs. In particular, we next present some expectations, called thermodynamic quantities, that are particularly useful to describe the behavior of system. These quantities are functions of the inverse temperature and of the external ﬁeld.

Deﬁnition 2.3.2 (Thermodynamic quantities). For a given N ∈ N we introduce the following thermodynamic quantities at ﬁnite volume:

(i) The random quenched pressure: 1 ψN (β, B) := log ZN (β, B) . (2.3.5) N

13 Ising model on random graphs

(ii) The averaged quenched pressure: 1 ψ (β, B) := QN (log ZN (β, B)) . (2.3.6) N N

(iii) The annealed pressure: 1 ψeN (β, B) := log (QN (ZN (β, B))) . (2.3.7) N

(iv) The random quenched, averaged quenched and annealed magnetization, respectively: SN MN (β, B) := µ , (2.3.8) GN N SN M N (β, B) := PN , (2.3.9) N SN MfN (β, B) := PeN , (2.3.10) N where the total spin is deﬁned as X SN := σi . (2.3.11) i∈[N]

(v) The random quenched susceptibility:

∂ χN (β, B) := MN (β, B). (2.3.12) ∂B

The averaged quenched and annealed susceptibility, respectively χN (β, B) and χeN (β, B), are deﬁned in the same way as the random quenched susceptibility with MN replaced by M N and MfN .

In the following we focus our attention on the thermodynamic limit of these quantities, i.e. their limits as N → ∞. This limit is particularly interesting because critical phenomena, as well as phase transition, may appear.

Remark 2.3.1 (Susceptibility and total spin variance). We remark that in the random quenched setting, the susceptibility coincides with the variance of the sum of the spins rescaled by the square root of N. Indeed, a simple calculation gives

14 2.4 Phase transition and critical behavior

∂ χ (β, B) = M (β, B) N ∂B N P P 2 β σiσj +B σi 1 P P (i,j)∈EN i∈[N] N σ∈Ω i∈[N] σi e = N ZN P P β σiσj +B σi 2 √1 P P (i,j)∈EN i∈[N] ! σ∈Ω i∈[N] σie − N N ZN 2 2 SN SN = µG √ − µG √ N N N N SN = Varµ √ . (2.3.13) GN N SN A similar results is true in the annealed case: χN (β, B) = Var √ . e PeN N In the averaged quenched setting the situation is different. Indeed, accordingly to Definition 2.3.2, the averaged quenched susceptibility is ∂ SN χ (β, B) = M N (β, B) = QN Varµ √ = QN (χN (β, B)) . N ∂B GN N (2.3.14) On the other hand, using the law of total variance, the variance with respect the averaged quenched measure is SN SN SN VarP √ = QN Varµ √ + VarQ µG √ , (2.3.15) N N GN N N N N i.e., we have the sum of two contributions:√ the first is given by the average over graphs of the conditional variance of SN / N (the conditioning is given by the graph realization);√ the second contribution is the variance of conditional mean of SN / N. We can also rewrite the expression in (2.3.15) using the thermodynamic quantities in Definition 2.3.2: SN √ VarP √ = χ (β, B) + VarQ NMN (β, B) . (2.3.16) N N N N

2.4 Phase transition and critical behavior

A phase transition occurs when a small change in a parameter, such as the temperature, causes a qualitative change in a quantity that describes the state of the system. In our case this quantity is called spontaneous magnetization. Suppose we put a piece of magnetic material, like a piece of iron, in a magnetic ﬁeld at a constant temperature. This ﬁeld forces the spins of the material to align with its direction, inducing an amount of magnetization into the material. When

15 Ising model on random graphs we turn oﬀ the external ﬁeld we see that for high temperature the material returns to be unmagnetized. Instead, for low temperature, an amount of magnetization, called spontaneous magnetization, is kept. The temperature that separates these two states is called critical temperature.

From a formal point of view, when M(β, B) := limN→∞ MN (β, B) exists, where

MN (β, B) is any of the magnetizations defined in (2.3.8), (2.3.9) or (2.3.10), criticality manifests itself in the behavior of the spontaneous magnetization + defined as M(β, 0 ) = limB↓0 M(β, B). In fact, the critical inverse temperature is defined as + βc := inf{β > 0 : M(β, 0 ) > 0}, (2.4.1) and thus, depending on the setting, we can obtain the random quenched, averaged rq aq an quenched and annealed critical point denoted by βc , βc and βc respectively. When, for one of the three, 0 < βc < ∞, we say that the system undergoes a phase transition at β = βc. Such transition in mathematical terms is a singularity that occurs when a discontinuity in one of the derivatives of the pressure appears. Note that βc can only exist in the thermodynamic limit, because the finite volume pressure is an analytic function on β and B. Moreover, due to the Z2 symmetry of the Ising model, the magnetization of a finite graph always satisfies + MN (β, 0 ) = 0. The behavior around the critical temperature can be quantified in terms of critical exponents. In particular, we introduce the critical exponents that describe the behavior of the magnetization and susceptibility at the critical inverse temperature βc and at zero external magnetic field. Definition 2.4.1 (Critical exponents). The critical exponents β, δ, γ, γ0 are defined by

+ β M(β, 0 ) (β − βc) , for β & βc; (2.4.2) 1/δ M(βc, B) B , for B & 0; (2.4.3) + −γ χ(β, 0 ) (βc − β) , for β % βc; (2.4.4) + −γ0 χ(β, 0 ) (β − βc) , for β & βc, (2.4.5) where we write f(x) g(x) if the ratio f(x)/g(x) is bounded away from 0 and inﬁnity for the speciﬁed limit.

Remark 2.4.1. We emphasize that there is a diﬀerence between the symbol β for the inverse temperature and the bold symbol β for the critical exponent in (2.4.2). Both notations are standard in the literature, so we decided to follow both of them and distinguish them by the font style.

2.5 Preliminaries

In this section we provide some theorems that are essential to prove our results. In particular, we present two correlation inequalities and a theorem that states

16 2.5 Preliminaries the existence of the limit of the thermodynamical quantities, under the random quenched measure, achieved by Dembo and Montanari first [16] and further by Dommers, Giardinà and van der Hofstad [20]. Moreover, we recall a theorem by Dommers, Giardinà and van der Hofstad [21] that specifies the value of critical exponents for locally tree-like random graph in the random quenched setting. We will compare this result with the annealed critical exponents studied below in Chapter 6. We begin this section by giving first a quick overview on the state of the art for the Ising model on random graphs.

2.5.1 State of the art

The literature on the Ising model is enormous. Since the original study of Ising [41] a massive number of research papers have been published on the subject. We focus our overview on the class of Ising models on random graphs. Ising models on random graphs and complex networks have been first studied with the tools of theoretical physics [22, 47, 23] and more recently they have been the subject of rigorous mathematical studies. A first mathematical approach to the study of the Ising model on random regular graphs was provided by Gerschcnfeld and Montanari in [35]. Subsequently, De Sanctis and Guerra [15] investigated the Ising model on the Erdős–Rényi random graph. They studied the limit of the free energy density at high temperature and at zero temperature. In [16], Dembo and Montanari analyzed the ferromagnetic Ising model for any temperature on random graphs with a finite-variance degree distribution. Later, Dommers, Giardinà and van der Hofstad [20], extended these results to the case where the degree distribution has infinite variance, but strongly finite mean. They computed the thermodynamic limit of the pressure and of various physical quantities, such as the magnetization, the susceptibility and the internal energy. In [18] Dembo, Montanari and Sun generalize this further to more general locally tree-like graphs. An analysis of the critical behavior is reported in [21] by Dommers, Giardinà and van der Hofstad. They studied the critical behavior of the Ising model on power-law random graphs by computing critical exponents for the magnetization and the susceptibility. Regarding the analysis of asymptotic theorems for the sum of the spin, results on lattices or on the complete graph can be found in [26, 28, 27].

2.5.2 Correlation inequalities

We introduce two classical correlation inequalities: the Griﬃths, Kelly, Sherman inequality (GKS) [45] and the Griﬃths, Hurst, Sherman inequality (GHS) [39]. We underline that these inequalities hold in the random quenched setting.

Theorem 2.5.1 (GKS inequality). Consider two Ising model with measures µ 0 0 0 and µ on graphs GN = (VN , EN ) and GN = (VN , EN ), with inverse temperature

17 Ising model on random graphs

0 0 0 0 β and β and external ﬁelds B and B respectively. If EN ⊆ EN , β ≤ β and 0 0 ≤ B ≤ B , then for any U ⊆ V N, ! ! Y 0 Y 0 ≤ µ σi ≤ µ σi . (2.5.1) i∈U i∈U Theorem 2.5.2 (GHS inequality). Let β ≥ 0, then for all B ≥ 0,

∂2 M (β, B) ≤ 0. (2.5.2) ∂B2 N i.e. the magnetization is a concave function of the positive external ﬁeld. Remark 2.5.1. The GHS inequality also holds in the averaged quenched setting. Indeed, using Deﬁnition 2.3.2 and Theorem 2.5.2, it results ∂2 ∂2 S ∂2 M (β, B) = Q µ N = Q (M (β, B)) ∂B2 N ∂B2 N GN N ∂B2 N N ∂2 = Q M (β, B) ≤ 0. (2.5.3) N ∂B2 N These results are particularly useful to study properties of the magnetization and, as we will see, the GHS inequality is crucial to prove the central limit theorem in the random quenched case. Due to the lack of this instrument in the annealed setting, we can not follow the Ellis’ strategy to prove the central limit theorem for the general class of locally tree-like random graphs.

2.5.3 Thermodynamic limits for the random quenched law

In the next theorem we collect some results, taken from [16, 20], that guarantee the existence of the thermodynamic limit of the quantities previously deﬁned in the random quenched setting. In order to state the existence of the limit magnetization, we deﬁne the set

rq rq U := {(β, B) : β ≥ 0, B 6= 0 or 0 < β < βc , B = 0} (2.5.4)

rq where βc is the quenched critical value which is given in the next theorem. Theorem 2.5.3 (Thermodynamic limits for the random quenched law [16, 20]).

Assume that the random graph sequence (GN )N≥1 is locally tree-like, uniformly sparse, and with asymptotic degree distribution D with strongly ﬁnite mean, then the following conclusions hold: (i) For all 0 ≤ β < ∞ and B ∈ R, the quenched pressure exists almost surely in the thermodynamic limit N → ∞ and is given by

ψ(β, B) := lim ψN (β, B). (2.5.5) N→∞ Moreover, ψ(β, B) is a non-random quantity.

18 2.5 Preliminaries

(ii) For all (β, B) ∈ U rq, the random quenched magnetization per vertex exists almost surely in the limit N → ∞ and is given by

M(β, B) := lim MN (β, B). (2.5.6) N→∞ ∂ The limit value M(β, B) equals M(β, B) = ∂B ψ(β, B) for B 6= 0, rq whereas M(β, B) = 0 in the region 0 < β < βc , B = 0. (iii) The critical inverse temperature is given by

rq βc = atanh (1/ν) , (2.5.7) where ν is deﬁned in (2.1.2). (iv) For all (β, B) ∈ U rq, the thermodynamic limit of susceptibility exists almost surely and is given by ∂2 χ(β, B) := lim χN (β, B) = ψ(β, B). (2.5.8) N→∞ ∂B2

Let us remark that, since ν ≤ 1 for both CMN (2) and CMN (1, 2), from (2.5.7) rq it follows that βc = ∞, which means that there is no quenched phase transition in these models.

In the next proposition we also provide an explicit expression for the random quenched pressure and magnetization in the thermodynamic limit. Proposition 2.5.4 (Explicit expression for the thermodynamic quantities [16,

20]). Assume that the random graph sequence (GN )N≥1 is locally tree-like, uniformly sparse, and with asymptotic degree distribution D with strongly ﬁnite mean, then for all 0 ≤ β < ∞ and B > 0 the thermodynamic limit of the pressure is given by E[D] E[D] ψ(β, B) = log cosh(β) − E [log (1 + tanh(β) tanh(h ) tanh(h ))] 2 2 1 2 L h B Y + E log e (1 + tanh(β) tanh(hi)) i=1 L −B Y i + e (1 + tanh(β) tanh(hi)) , (2.5.9) i=1 where (ii) L has distribution D;

∗ ∗ (ii) (hi)i≥1 are i.i.d. copies of the ﬁxed point h = h (β, B) of the distributional recursion

Kt (t+1) d X (t) h = B + atanh(tanh(β) tanh(hi )), (2.5.10) i=1

19 Ising model on random graphs

(0) where h ≡ B, (Kt)t≥0 are i.i.d. random variables with distribution ρ and (t) (t) (hi )i≥1 are i.i.d. copies of h independent of Kt;

(iii) L and (hi)i≥1 are independent. Moreover, taking the derivatives of (2.5.9) with to respect the external ﬁeld B, we obtain the explicit expression for the thermodynamic limit of the magnetization, i.e.

L h X i M(β, B) = E tanh B + atanh(tanh(β) tanh(hi)) . (2.5.11) i=1

2.5.4 Critical exponents for the random quenched law

We next present some critical exponents in the random quenched setting for the Ising model on random graphs with degree distribution satisfying a power law with exponent τ . The values of the critical exponents for magnetization and susceptibility, for diﬀerent values of τ, are stated in the following theorem:

Theorem 2.5.5 (Critical exponents for the random quenched law [21]). Assume that the random graph sequence (GN )N≥1 is locally tree-like, uniformly sparse, and with asymptotic degree distribution D that obeys E[K3] < ∞ or a power law with exponent τ ∈ (3, 5]. Then the critical exponents exist and satisfy

τ ∈ (3, 5) E[K3] < ∞ β 1/(τ − 3) 1/2 δ τ − 2 3 γ 1 1 γ0 ≥ 1 ≥ 1

For the boundary case τ = 5 there are the following logarithmic corrections for β = 1/2 and δ = 3:

1/2 + β − βc rq Mf(β, 0 ) for β & βc , (2.5.12) log 1/(β − βc) B 1/3 Mf(βrq, B) for B & 0. (2.5.13) c log(1/B)

Note that the critical exponents β, δ and γ take the classical mean-ﬁeld values for E[K3] < ∞, but they are diﬀerent for τ ∈ (3, 5).

20 2.6 Other spin models

2.6 Other spin models

Beyond the Ising model, other spin models are interesting to model processes on networks, see [23] for an overview. We now present three such models: the Curie–Weiss model, the Potts model and the XY model.

2.6.1 The inhomogeneous Curie–Weiss model

The Curie–Weiss model is an Ising model that allows interaction between each couple of spins. That means that the spatial distance does not play a role anymore. We next deﬁne the inhomogeneous Curie–Weiss model. This is a generalization of the classical Curie–Weiss model in which the strength of the ferromagnetic interaction between spins is not spatially uniform. As the standard Curie–Weiss model, it is deﬁned on the complete graph with vertex set [N] := {1, ... , N}.

Deﬁnition 2.6.1 (Inhomogeneous Curie–Weiss model). Let σ = {σi}i∈[N] ∈ {−1, 1}N be spin variables. The inhomogeneous Curie–Weiss model, denoted by

CWN (J), is deﬁned by the Boltzmann–Gibbs measure

eHN (σ) µN (σ) = (2.6.1) ZN where the Hamiltonian is 1 X X H (σ) = J (β)σ σ + B σ (2.6.2) N 2 ij i j i i,j∈[N] i∈[N] and ZN is the normalizing partition function. Here β is the inverse temperature, B is the external magnetic ﬁeld and J = {Jij (β)}i,j∈[N] are the spin couplings.

In the above, the interactions Ji,j (β) might be arbitrary functions of the inverse temperature (in particular no translation invariance is required), provided that the thermodynamic limit is well-defined, i.e., the following limit defining the pressure exists and is finite, 1 ψ(β, B) := lim log ZN (β, B) . (2.6.3) N→∞ N

Since the coupling constants J = {Jij (β)}i,j∈[N] are diﬀerent for diﬀerent edges, we speak of an inhomogeneous Curie–Weiss model. In Chapter 5 it is shown that the inhomogeneous Curie–Weiss model arises in the study of the annealed Ising model with network of interactions given by the generalized random graph, also called the rank-1 inhomogeneous random graph. In that case, each vertex i ∈ [N] receives a weight wi and we will take J = {Jij (β)}i,j∈[N] such that wiwj X Jij (β) = β, where `N = wk. (2.6.4) `N k∈[n]

21 Ising model on random graphs

In the case where wi ≡ 1, our model reduces to the homogeneous Curie–Weiss model. We call the coupling constants in (2.6.4) the rank-1 inhomogeneous Curie–Weiss model.

2.6.2 The Potts model

The Potts model is a generalization of the Ising model. It is introduced on the random graph GN as follows: Deﬁnition 2.6.2 (Potts model). Fix an integer q ≥ 2 and assign to each vertex i ∈ [N] a Potts spin variable σi ∈ [q]. Then the Potts model on GN is deﬁned by the Boltzmann–Gibbs measure h i exp β P δ + B P δ i,j∈EN σi,σj i∈[N] σi,1 µN (σ) = , (2.6.5) ZN (β, B) where δa,b is the Kronecker delta i.e. δa,b = 0, 1 if a 6= b and a = b, respectively. Note that the Ising model is equivalent to a Potts model with q = 2, (with σi ∈ {−1, 1} replaced with σi ∈ {1, 2}).

The Potts model has been intensively studied in statistical mechanics because of its importance in the theory of phase transitions and critical phenomena. Moreover, this model is closely related to the graph coloring problem, where the aim is to know if the graph can be colored with q colors in such a way that no two neighboring vertices have the same color. We next mention some results for the Potts model on random graphs. In [9], Contucci, Dommers, Giardinà, and Starr analyze the antiferromagnetic Potts model on the Erdős–Rényi random graph. They show the existence of the thermodynamic limit of the pressure and they identify a phase transition. Dembo, Montanari and Sun [18] study the existence of the free energy density for a Potts model on uniformly sparse graph sequences converging locally to a random tree. Subsequently, Dembo, Montanari, Sly and Sun in [17] provide an explicit formula for the limiting free energy density for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to regular trees of even degree, covering all temperature regimes.

2.6.3 The XY model

Unlike the Ising and Potts models with discrete spins, the XY model is described by continuous spin variables. Indeed, in the XY model, the classical spins can rotate in the interval [0, 2π]. The deﬁnition for the model on a random graph

GN is the following:

22 2.7 Overview

Deﬁnition 2.6.3 (XY model). Assign to each vertex i ∈ [N] a spin variable

σi ∈ [0, 2π]. Then the XY model on GN is deﬁned by the Boltzmann–Gibbs measure h i exp β P cos(σ − σ ) + B P cos(σ ) i,j∈EN i j i∈[N] i µN (σ) = . (2.6.6) ZN (β, B) There are few studies of the XY model on complex networks. We can ﬁnd an overview of them in [23].

2.7 Overview

In this section we discuss about the main topics of this thesis. We provide a wide analysis on the three measures and their properties. We present our principal results and we give an idea of the strategy of the proofs. Furthermore an examination for the scenario at the critical point and in the low-temperature region is performed.

2.7.1 Discussion on the measures

To understand the difference between the quenched and annealed settings, it is convenient to think of a microscopic dynamics yielding the equilibrium state. For instance, one could imagine that the spins are subject to a Glauber dynamics with a reversible Boltzmann–Gibbs distribution and the graph also has its own dynamical evolution approaching the graph’s stationary distribution. In general, these two dynamics are intertwined and both concur to determine the equilibrium state, i.e., the asymptotic value of an ergodic dynamical time average. The quenched and annealed state arise as follows: (a) In the quenched state, the changes of the graph happen on a time-scale that is infinitely longer than the time-scale over which the changes of the spin variables occur. Thus in the quenched state the graph viewed by the evolving spins is frozen. One distinguishes between the random quenched measure, i.e., the random Boltzmann–Gibbs distribution of a given realization of the graph, and the averaged quenched measure, i.e., the average of the Boltzmann–Gibbs distribution over the graph ensemble. Several thermodynamic observables (e.g., the free energy per particle, the internal energy per particle, etc.) are self-averaging, and therefore the random quenched values and their averaged quenched expectations do coincide in the thermodynamic limit. In the study of the fluctuations of the properly rescaled magnetization one finds a Gaussian limiting law. Interestingly, the asymptotic variances of the random quenched and averaged quenched state might be different due to local Gaussian fluctuations of graph properties. (b) In the annealed state, the random environment evolves much faster than the spin variables. As a consequence, the environment seen by the spins

23 Ising model on random graphs

includes all possible arrangements of the random graph. The annealed measure (defined in (2.3.4)) is given by the stationary reversible measure of a Glauber spin dynamics in which the transition from a configuration σ to another configuration σ0 occurs with probability

0 E[e−βH(σ )] ∧ 1, (2.7.1) E[e−βH(σ)]

where H is the Hamiltonian and E[·] represents the average over the graph ensemble. The above dynamics corresponds to an extremely fast random graph dynamics in which we do not even observe the graph at any time, but merely see it averaged over the random graph distribution. This is equivalent to an effective Glauber dynamics with (annealed) Hamiltonian equal to 1 Han(σ) = − log(E[e−βH(σ)]). (2.7.2) β Thus, by construction, the annealed measure is necessarily non-random and its normalization includes the average over the graphs ensemble. In this thesis, we will study annealed central limit theorems for the ferromagnetic Ising model on random graphs, in order to deduce what the effect of annealing on the macroscopic properties of the Ising model is. The definition of the annealed measure in the context of Ising models on random graphs is thus different than in other classes of problems with disorder. Usually, this can be interpreted in terms of the dynamics of the environment and the process on it being equally fast. In the context of random walks in random environment [8], annealing is rather similar to what we here have called the averaged quenched measure. This is due to the lack of a partition function for random walks in random environment. In disordered systems (such as spin glasses [50, 12]), annealed disorder is usually considered to be easier to deal with mathematically, since the average on the disorder and the thermal average are treated on the same footing. This is true whenever the edges of the graph are independent, due to the form of the Hamiltonian that allows a factorization of expectations w.r.t. the bond variables. If instead the edge distribution in the graph does not have a product structure, then the annealed case can actually be more difficult than the quenched case. Indeed, whereas the random-quenched case is dominated by the typical realization of the graph (often having the local structure of a random tree), in the annealed case (as in the averaged quenched case) rare graph samples having a very low energy actually give a contribution that can not be ignored. This is due to the fact that the Ising model gives rise to exponential functionals on the random graph, and expectations of exponential functionals tend to be dominated by rare events in which the exponential functional is larger than it would be under the quenched law. Deriving such statement rigorously requires a deep understanding of the large deviation properties of random graphs, a highly

24 2.7 Overview

interesting but also challenging topic. In this thesis, we consider graph ensembles of both types, i.e., random graphs with independent edges (the generalized random graph) or dependent edges (the conﬁguration model).

2.7.2 Overview of results

The primary purpose of this thesis is to study asymptotic results for the total spin SN under the three different measures introduced in Section 2.3. First of all, we prove the existence of the law of large numbers. Namely, for all settings, we prove that the total spin normalized by the volume N converges to the magnetization of the model. Depending on the measure involved, the convergence can be in probability or almost surely. The scenario for the central limit theorem is more complex. For each measure we work in the region of the parameters (β, B) such that there exists a unique Gibbs measure. For the general class of the locally tree-like random graphs, working under√ the random quenched measure, we obtain that the total spin SN rescaled by N converges to a Gaussian random variable with variance equal to the susceptibility of the system. In order to prove this, we mimic the strategy used by Ellis for the Ising model on Zd in [26], getting the same result. On the 2-regular configuration model, for all three measures the central limit theorems show the convergence to the same Gaussian random variable. Indeed, because the degree is not allowed to fluctuate, there is no contribution from the graph and, as a result, the limiting variance coincides with the random quenched susceptibility.

The situation is different for the CMN (1, 2). Here the fluctuations of the degree of the graph become relevant providing a different central limit theorem for each of the three measures. In the random quenched setting the variance will still coincide with the random quenched susceptibility, while it will be larger in the averaged quenched setting and different in the annealed one. The Ising model on the generalized random graph under the annealed measure is reduced to an inhomogeneous Curie–Weiss model. Then we can again follow Ellis’ strategy for the central limit theorem, obtaining as limiting variance of the Gaussian the annealed susceptibility of the model. In this setting we discover that the model presents an annealed critical temperature which is larger than the quenched one. So we are not able to compare the two different variances of the central limit theorems, i.e. the susceptibilities under the random quenched and annealed measures. Thanks to this link between the Ising model on the generalized random graph and the Curie–Weiss model, we can go further and see what happens at the critical point. Adapting the strategy in [21] to our case, we prove that the annealed critical exponents match those of quenched setting and, following [26], we show a non-classical limit theorem: rescaling in an appropriate way the total spin we obtain the convergences to a non-Gaussian random variable.

25 Ising model on random graphs

The role of the inhomogeneity of the graphs, expressed in terms of the power law distribution of the degrees, is crucial for the study at the critical point. Indeed, the analysis of the critical exponents and the non-classical limit theorem depends signiﬁcantly on the value of the exponent τ of the power law. On the contrary, this inhomogeneity does not have any inﬂuence when we investigate the scenario in the uniqueness regime.

2.7.3 CLT proof strategy

The proofs of the central limit theorems for the three different settings are different (in fact they require the control of the fluctuations of SN with respect to different ensembles), but are based on the same main idea [26]. This idea consists in using the moment generating function of the random variables S −E(S ) N √ N , where E is the average in the chosen measure, and in showing that N with respect to the same measure these moment generating functions converge to those of Gaussian random variables. Whenever possible, this step requires the computation of the variances of the limiting Gaussian variables. This could be achieved by considering the scaled cumulant generating functions of SN , given by 1 c (t) = log µ [exp (tS )] (2.7.3) N N GN N in the random quenched setting, 1 c (t) = log P [exp (tS )] (2.7.4) N N N N in the averaged quenched setting and 1 cN (t) = log PeN [exp (tSN )] (2.7.5) e N in the annealed setting. By taking the second derivative of (2.7.3), (2.7.4) SN and (2.7.5) and evaluating it in zero one obtains, respectively, Varµ √ , GN N SN SN VarP √ and Var √ . The crucial argument in the proof of the theo- N N PeN N rems is to show the existence of these variances in the limit N → ∞. This in 00 turn is a consequence of the existence of the limit of the sequences (cN (t))N≥1, 00 00 (cN (t))N≥1 and (ecN (t))N≥1 for t = tN = o(1). While the existence of the limit c(t) := limN→∞ cN (t) can be established for the Ising model on locally tree-like random graphs as a simple consequence of the existence of the random quenched pressure, the existence of the limit c(t) := limN→∞ cN (t) is more challenging. In particular, it requires detailed knowledge not only of the typical local structure of the graph around a random vertex, but also of the fluctuations around that structure. Moreover, the general argument of [26] relies on concavity of the first derivatives of the cumulant generating functions. This can be achieved 0 for cN (t), i.e., in the random quenched setting, thanks to the GHS inequality, which holds for the ferromagnetic Boltzmann–Gibbs measures µGN . On the

26 2.7 Overview other hand, under the averaged quenched measure, the ﬁrst derivative of the cumulant generating functions can not be expressed in term of the averaged quenched magnetization to exploit the GHS inequality, and we are not able 0 to prove the concavity of cN (t). Thus in the averaged quenched setting we focus on two speciﬁc models, i.e. the CMN (2) and CMN (1, 2) random graphs, which allow for explicit computations of the relevant quantities. In fact in these cases, since the typical structure in the graphs are cycles (for CMN (2)) and lines and cycles (for CMN (1, 2)), the averaged quenched pressure of the Ising model on these graphs can be expressed in terms of the Ising model pressure ψd=1(β, B) of the one-dimensional nearest-neighbor Ising model. It turns out however that whereas the averaged quenched pressure of the regular random d=1 graph CMN (2) exactly equals ψ (β, B), in the case of CMN (1, 2) the pressure is more involved. Indeed, besides ψd=1(β, B), a new term appears that depends (N) on a set of random variables (p` )`≥1 whose value depends on the realization (N) of the random graph. More precisely, Np` is the number of lines of length ` in the graph (the cycles give a vanishing contribution in the thermodynamic limit).

Then, in order to prove the CLT for CMN (1, 2), it is of pivotal importance to (N) control the fluctuations of the random variables p` in the thermodynamic limit. This result is obtained in [14], where it is proven that the joint limit law of the number of connected components in a graph with vertices of degrees 1 and 2 is Gaussian. Relying on this result, we can complete our proof of the averaged quenched CLT for CMN (1, 2). In the annealed setting we can again rewrite the cumulant generating function in terms of the pressure, since cÑ (t) = ψeN (β, B + 1) − ψeN (β, B). We will show by an explicit computation that the annealed pressure of GRGN (w) coincides with that of an inhomogeneous Curie–Weiss model. From this fact, the thermodynamic limit of the annealed pressure, magnetization and susceptibility can be obtained. This again relies on the GHS inequality that is valid also for this inhomogeneous ferromagnetic system. Thus, for the generalized random graph the annealed CLT can be proven in a similar way as for the random quenched measure. On the other hand, the proofs of the CLT for the configuration models do not follow from the abstract argument based on the GHS inequality, since GHS is not available in the general annealed context. Because of that, we have 00 to explicitly control the limit (cÑ (tN ))N≥1 throughout the computation of the annealed pressure. It is relatively simple to accomplish this task in the case of the regular CMN (2) graph. The fluctuating degree of CMN (1, 2) makes the 00 computation of the pressure and of the limit (cÑ (tN ))N≥1 much more compli- cated. While the cycles give a vanishing contribution to the thermodynamic limit, the distribution of the length of the lines has to be carefully analyzed and its Gaussian fluctuations appear in the CLT for the total spin.

27 Ising model on random graphs

2.7.4 Diﬀerences between random quenched and averaged quenched setting

Here we explain the distinction between the random quenched and the averaged quenched CLTs. As we said before, in the central limit theorem for the total spin in the random quenched setting, the variance of the Gaussian random variable SN coincides with the random quenched susceptibility χ. The variance VarP √ , √ N N expressing the fluctuations of SN / N with respect to the averaged quenched measure, has instead two contributions. The first is given√ by the average over random graphs of the conditional variance of SN / N with respect to the Boltzmann–Gibbs measure (the conditioning is given by the graph realization);√ the second contribution is given by the variance of conditional mean of SN / N (see (2.3.15)). If a CLT with respect to the averaged quenched measure holds, then the thermodynamic limit of the first term in the right-hand side of (2.3.15) equals the magnetic susceptibility χ, which is a self-averaging quantity. It is clear from (2.3.15) that one expects a different variance in the CLT in the averaged quenched case whenever the thermodynamic limit of the second term on the right-hand side is different from zero. The analysis that we perform on the configuration models under consideration leads us to conjecture that when the vertex degrees are not all equal, one has that lim Var µ √SN is a strictly positive number. This means N→∞ QN GN N that when there are non-vanishing fluctuations of the rescaled total spin with respect to the graph measure, those will be determined by the fluctuations of the degrees distribution of the graph. On the contrary, we predict that when the degrees are fixed, such as in the r-regular random graph, the two limiting variances are equals, thus yielding no distinction between random and averaged quenched central limit theorems. We prove this in the case where r = 2.

2.7.5 Properties of annealing

Working with the annealed measure some interesting phenomena emerge. First of all, in the presence of a ferromagnetic phase transition, annealing can an rq change the critical temperature, meaning that βc < βc . We proved this for the rank-1 inhomogeneous graph. For the configuration models with vertex degrees an rq at most two that we have analyzed, it holds βc = βc = ∞. We conjecture that in the general case when there is a positive proportion of vertices of degree at least 3 and ν > 1 (so that there exists a giant component), an annealed positive critical temperature exists. We believe that this annealed critical temperature is strictly larger than the quenched critical temperature whenever the vertex degrees fluctuate and a positive proportion of the vertices have at least degree three. Furthermore, the annealed state satisfies a central limit theorem for the rescaled magnetization, as the quenched state does. Unfortunately, we can only

28 2.7 Overview prove this for certain random graph sequences, but we believe this to be true in general. The variance of the annealed CLT and the variance of the quenched CLT are different whenever the degrees are allowed to fluctuate. We showed this in the case of the generalized random graph, where they can not be ordered because the quenched and annealed critical temperatures are different and the quenched and annealed susceptibilities diverge at the critical point. For CMN (1, 2) having zero critical temperature and fluctuating degrees the variances are also different, however we have not been able to compare them.

From the analysis of the CMN (2), we see that both the annealed critical temperature and the annealed variance are the same of their quenched counterparts. We conjecture this behavior to occur for all random regular graphs.

2.7.6 Universality classes for power-law random graphs

Universality is a key concept in the theory of phase transitions, with application to a large variety of physical systems. Informally, universality means that in the thermodynamic limit different systems show common properties close to criticality. The theory based on the renormalization group suggests that systems fall into universality classes, defined by the values of their critical exponents describing the nature of the singularities of measurable thermodynamic quantities at the critical point. In our study we investigate universality for spin system on power-law random graphs, i.e., graph sequences where the fraction of nodes that have k neighbors is proportional to k−τ for some τ > 1, displaying phase transitions. In [21] the quenched critical exponents have been rigorously analyzed for a large class of random graph models. More precisely in [21] it is proved that the critical exponent, describing the behavior of magnetization and susceptibility near the critical point, take the same values as the mean-field Curie–Weiss model whenever the degree distribution has a finite fourth moment. For power law random graphs it is proved that for τ > 5 the model is in the mean-field universality class, whereas the critical exponent are different from the mean-field values for τ ∈ (3, 5). In Chapter 6 we provide the analysis of the critical behavior but in the annealed setting. Our results are fully compatible with the universality conjecture. The annealed critical temperature is different (actually higher) than the quenched critical temperature, but the set of annealed critical exponents are the same as the quenched critical exponents.

2.7.7 Violation of CLT at βc

In the case of the random quenched CLT, the variance is given by the sus- rq ceptibility of the model. If B = 0 and β = βc the variance diverges because the susceptibility becomes inﬁnite at the critical temperature. So, for these parameters, the CLT breaks down and a diﬀerent scaling of the total spin SN is

29 Ising model on random graphs needed to obtain a non-trivial limiting distribution. For example, in [26], it is 3/4 shown that for the Curie–Weiss model, at B = 0, β = βc, the quantity SN /N converges in distribution to the random variable X with density proportional to exp −x4/12. We conjecture a similar behavior for the Ising model on locally tree-like random graphs with finite fourth moment of the degree distribution. Indeed, as said before, those graphs have been shown to be in the same universality class as the Curie–Weiss model, i.e., their critical exponents agree with the mean-field critical exponents of the Curie–Weiss model [21]. We investigate this problem in detail for the generalized random graphs under the annealed measure in Chapter 6. We prove that the scaling with N 3/4 is also correct when E[W 4] < ∞, but different for τ ∈ (3, 5). Furthermore we show that when E[W 4] = ∞, different asymptotic distributions arise in the scaling limit. We characterize them for a weight deterministic sequence in which the weights follows a precise power-law. Such a sequence is rather generic in the sense that it produces an asymptotic weight that is also power-law distributed. The analysis shows that the fluctuations of the total spin decrease as the exponent τ becomes smaller and the distribution seen in the scaling limit has tails proportional to τ−1 e−Cx .

2.7.8 Low-temperature region

When the graph sequence (GN )N≥1 is such that there exists a ﬁnite critical inverse temperature βc, then the law of large numbers and the central limit theorem do not apply to the low temperature region corresponding to B = 0, d β > βc. For the Ising model on Z [54, 48] or on the complete graph [26], in the low temperature region the law of large numbers for the empirical sum of the spin breaks down because, in the thermodynamic limit, the Boltzmann–Gibbs measure 1 + − becomes a mixture of two pure states µ = 2 (µ + µ ). As a consequence, the empirical magnetization is distributed like the sum of two Dirac deltas at the ∗ ∗ symmetric values ±M (β), with M (β) = limB&0 M(β, B), whereas the CLT with respect to the Boltzmann–Gibbs measure breaks down. However, by using general properties of ferromagnetic systems (e.g. GKS inequalities) it is possible to prove a CLT with respect to the measure µ+, respectively µ− [54, 48]. For the Ising model on random graphs it is believed that a similar picture applies. For instance, for the Ising model on regular random graphs it has been proved in [53] that the low-temperature measure is a convex combination of + and − states and therefore, by appealing to the general results of [54] we conclude that a CLT holds in the pure phase.

2.8 Organization of the thesis

This thesis is organized as follows. In each chapter we present the theorems and the proofs concerning a particular measure.

30 2.8 Organization of the thesis

In Chapter 3, in the general context of locally tree-like random graphs, we present our results with respect to the random quenched measure. We establish the rate at which the law of large numbers for the total spin is reached, and we prove a central limit theorem for the (centered) total spin rescaled by the square root of the volume. As in the case of the Ising model on the lattice or on the complete graph [26], the variance appearing in the CLT result is given by the spin susceptibility. In Chapter 4 we obtain asymptotic results with respect to the averaged quenched measure. While the law of large numbers can be easily formulated for the entire class of locally tree-like random graphs, the scaling to a normal random variable turns out to be much more challenging. Thus, we restrict ourselves to the configuration models with degrees at most 2 and consider the two simplest cases. In particular, we will consider the configuration model with all vertices having degree two, i.e. the 2-regular random graph, and the case where a fraction of the vertices has degrees one and the remaining fraction has degree two. In both cases we obtain the central limit theorem by an explicit computation. In the first case, we find that the CLT with respect to the averaged quenched measure is the same as the CLT with respect to the random quenched measure. In particular the variance of the Gaussian law is given by the susceptibility of the Ising model in one dimension. In the second case, we prove that the asymptotic variance of the rescaled total spin is larger than the susceptibility. This difference originates from the fluctuations of the connected component sizes of the configuration model with degrees 1 and 2, that also follows a Gaussian law in the limit of very large graphs. In Chapter 5 we deal with the annealed measure in the uniqueness regime outside the critical region. First we analyze the generalized random graph model for which we compute the pressure and magnetization in the thermodynamic limit, identify the critical temperature and then prove the law of large number and the central limit theorem. All of these results rely on the fact that the Ising model on the generalized random graph in the annealed setting turns into an inhomogeneous Curie–Weiss model. Then we continue by studying the thermodynamic quantities and the central limit theorems for the 2-regular configuration model and for the model with vertices of degrees 1 and 2. In the former case we show that the variance of the limiting normal variable is the susceptibility of the one-dimensional Ising model. In the latter case, which is much more difficult, the varying degrees of the vertices affect the pressure and the limiting distribution. Finally in Chapter 6 we analyze what happens at the critical point. As mentioned before, at the critical point the CLT breaks down and a different scaling of the total spin is needed to obtain a non-trivial limiting distribution. We investigate this problem for the Ising model on generalized random graph in the annealed setting. We show that the critical exponents for this model match those of an Ising model on locally tree-like random graphs in the random quenched setting. The second aim is to prove a non-classical limit theorem at

31 Ising model on random graphs the critical point. It results that we need different scalings for the total spin according if the degree distribution satisfies a prescribed power law or it has a finite fourth moment In Chapter 7 we introduce a speculative analysis on the distribution of the fixed point of the stochastic recursion equation for the cavity field of the Ising model on a locally tree-like random graph. We provide some analytical results in the zero temperature limit equipped with numerical simulation.

32 RANDOMQUENCHEDMEASURE 3

In this chapter we present our results in the random quenched setting on the general class of the locally tree-like random graphs. We study the asymptotic behavior of the total spin SN under diﬀerent scalings, i.e. we prove the law of large numbers and the central limit theorem. A full control for the quenched pressure and the main thermodynamical observables has been achieved in [16, 20], see Theorem 2.5.3. We leverage on the knowledge about the quenched pressure in the proofs of both random quenched asymptotic theorems for the total spin. Furthermore, a key role is played by the GHS inequality in the proof of the central limit theorem. Thanks to these ingredients, it turns out that our proofs can be performed along the lines of Ellis’ proofs in the Zd case [26, Section V.7].

3.1 LLN and CLT for locally tree-like random graphs

Our first result in the random quenched setting is the strong law of large numbers for SN /N. In [20] it is proved that this sum normalized by N converges almost surely to a number that is the magnetization of the model as given in (2.5.6). As a preliminary step we will prove here a similar result with a different approach based on large deviation theory, which leads to exponentially fast convergence in probability. Exponential convergence is defined as follows:

Deﬁnition 3.1.1 (Exponential convergence). We say that a sequence of random variables XN with laws µN converges in probability exponentially fast to a exp constant x0 w.r.t. µN , and write XN −→ x0, if for any ε > 0 there exists a number L = L(ε) > 0 such that

−NL µN (|XN − x0| ≥ ε) ≤ e for all suﬃciently large N.

Then we can present the statement for the strong law of large numbers:

Theorem 3.1.1 (Random quenched SLLN). Let (GN )N≥1 be a sequence of random graphs that are locally tree-like, uniformly sparse, and with asymptotic degree distribution with strongly ﬁnite mean. Then, for all (β, B) ∈ U qu,

S exp N −→ M w.r.t. µ , as N → ∞, N GN where M = M(β, B) is deﬁned in (2.5.6). √ To see ﬂuctuations of the total spin, one needs to rescale SN − NMN by N. To prove the CLT, we restrict to the uniqueness regime U qu of parameters (β, B)

33 Random quenched measure such that there exists a unique Gibbs measure [53]. As in the case of the Ising model on the lattice or on the complete graph [26], the variance appearing in the CLT result is given by the spin susceptibility:

Theorem 3.1.2 (Random quenched CLT). Let (GN )N≥1 be a sequence of random graphs that are locally tree-like, uniformly sparse, and with asymptotic degree distribution with strongly ﬁnite mean. Then, for all (β, B) ∈ U qu,

SN − NMN D √ −→ N (0, χ) , w.r.t. µG , as N → ∞, N N where χ = χ(β, B) is deﬁned in (2.5.8) and N (0, χ) denotes a centered Gaussian random variable with variance χ.

The remainder of the chapter is devoted to the proofs of these two theorems.

3.2 Proofs

In order to prove the strong law of large lumbers in the random quenched setting, we present a preliminary theorem that guarantees exponential convergence under general hypotheses.

3.2.1 Exponential convergence

Theorem 3.2.1 (Exponential convergence and cumulant generating functions).

Let W = (Wn)n≥1 be a sequence of random vectors which are deﬁned on prob- D ability spaces {(Ωn, Fn, Pn)}n≥1 and which take values in R . We deﬁne the cumulant generating functions as

1 D cn(t) = log En [exp(ht, Wni)] , n = 1, 2, ... , t ∈ R , (3.2.1) an where (an)n≥1 is a sequence of positive real numbers tending to inﬁnity, En denotes expectation with respect to Pn, and h−, −i is the Euclidean inner product on RD. We assume that the following hypotheses hold:

D (a) Each function cn(t) is ﬁnite for all t ∈ R ;

D (b) c(t) = limn→∞ cn(t) exists for all t ∈ R and is ﬁnite. Then the following statements are equivalent: exp (1) Wn/an −→ z0;

(2) c(t) is diﬀerentiable at t = 0 and ∇c(0) = z0.

34 3.2 Proofs

See [26, Theorem II.6.3] for a proof based on a large deviation argument.

Now we are ready to prove the strong law of large numbers using the cumulant generating function. As we will see, the existence of the limiting cumulant generating function of the total spin with respect to the random quenched measure will be a direct consequence of the existence of the thermodynamic limit of the quenched pressure.

3.2.2 Random Quenched SLLN: Proof of Theorem 3.1.1

According to Theorem 3.2.1, exponential convergence can be obtained by proving the existence of the limit of the random quenched cumulant generating function cN (t) defined in (2.7.3), and proving the differentiability of the limiting function in t = 0. We have 1 c (t) = log µ [exp (tS )] N N GN N P P exp β σiσj + (B + t) σi 1 X (i,j)∈eN i∈[N] = log N ZN (β, B) σ∈ΩN 1 Z (β, B + t) = log N , (3.2.2) N ZN (β, B) and recalling the definition (2.3.5), the function cN (t) can be rewritten as a difference of random quenched pressures as

cN (t) = ψN (β, B + t) − ψN (β, B). (3.2.3)

The existence of the limit

c(t) := lim cN (t) = ψ(β, B + t) − ψ(β, B) a.s. (3.2.4) N→∞ is then obtained from the existence of the pressure in the thermodynamic limit, as stated in Theorem 2.5.3. Moreover, it results that c(t) is finite. From the differentiability of the infinite-volume random pressure with respect to B (see Theorem 2.5.3), we also obtain ∂ ∂ c0(t) = [ψ(β, B + t) − ψ(β, B)] = [ψ(β, B + t)] , ∂t ∂B and hence ∂ c0(0) = [ψ(β, B)] = M(β, B). ∂B Thus, by Theorem 3.2.1, we obtain the exponential convergence.

Note that, as a consequence of Borel–Cantelli Lemma, the exponential convergence in probability implies the convergence almost surely (see [26, Theorem II.6.4]).

35 Random quenched measure

Below, we show the central limit theorem proving the convergence of the moment generating function. In order to do this, we again use the cumulant generating function and its derivatives together with the GHS inequality.

3.2.3 Random Quenched CLT: Proof of Theorem 3.1.2

We give the proof for B ≥ 0 only, the case B < 0 is handled similarly. The strategy of the proof is to show that w.r.t. the random quenched measure the moment generating function of the random variable

SN − NMN (β, B) VN = √ (3.2.5) N converges to the moment generating function of a Gaussian random variable with variance χ(β, B) given in (2.5.8), i.e., 1 2 lim µG (exp (tVN )) = exp χ(β, B)t for all t ∈ [0, α) , (3.2.6) N→∞ N 2 and some α > 0. This can be done by expressing µGN (exp (tVN )) in terms of the second derivative of the cumulant generating function cN (t). A simple computation shows that 0 1 µGN (SN exp (tSN )) SN cN (t) = = µGN (β,B+t) , (3.2.7) N µGN (exp (tSN )) N where, in order to stress the dependence on the magnetic ﬁeld, we have used the symbol µGN (β,B+t) (·) to denote the µGN -average in the presence of the ﬁeld B + t. Thus,

2 2 µ S exp (tSN ) µ (exp (tSN )) − µ (SN exp (tSN )) 00 1 GN N GN GN cN (t) = 2 N µ (exp (tSN )) GN SN = Varµ (β,B+t) . (3.2.8) GN N

In particular, the derivatives in t = 0 of cN (t) equal

0 cN (0) = MN (β, B), and 1 c00 (0) = µ S2 − µ2 (S ) = χ (β, B). N N GN N GN N N By Theorem 2.5.3,

00 cN (0) = χN (β, B) → χ(β, B) as N → ∞. (3.2.9)

36 3.2 Proofs

√ Let us take t > 0 and set tN = t/ N. By using the fact that cN (0) = 0 and applying Taylor’s theorem with Lagrange remainder, we obtain tSN − tNMN (β, B) log µG (exp (tVN )) = log µG exp √ N N N √ = log [µGN (exp (tN SN ))] − t NMN (β, B) 0 = N cN (tN ) − tN cN (0) t2 t2 = N N c00 (t∗ ) = c00 (t∗ ), (3.2.10) 2 N N 2 N N √ ∗ for some tN ∈ [0, t/ N]. 00 ∗ In order to control the limiting behavior of cN (tN ), we exploit the following property of cN (t): Proposition 3.2.2 (Convergence of double derivative cumulant generating qu function). For β ≥ 0, B > 0 and for 0 < β < βc , B = 0, there exists some 00 α > 0 such that limN→∞ cN (tN ) = χ(β, B) for all tN ∈ [0, α) with tN → 0 and almost all sequences of graphs (GN )N≥1. Proposition 3.2.2 immediately implies that (3.2.6) holds, which in turn proves Theorem 3.1.2.

The remainder of this section is devoted to the proof of Proposition 3.2.2. It 0 relies on the concavity of the functions cN (t), as proven in the following lemma: 0 Lemma 3.2.1. For B ≥ 0, cN (t) is concave on [−B, ∞). 0 Proof. For B ≥ 0, the concavity of cN (t) on [0, ∞) can be obtained by observing that this function is the magnetization per particle w.r.t. µGN in the presence of the magnetic ﬁeld B + t, see (3.2.7), and then by applying the GHS inequality. Indeed, for t ≥ −B, the GHS inequality (see Theorem 2.5.2) implies that

∂2 1 X ∂2 c0 (t) = µ (σ ) ≤ 0, (3.2.11) ∂t2 N N ∂B2 GN (β,B+t) i i∈[N]

0 so that cN (t) is concave on [−B, ∞) for B ≥ 0. The proof of Proposition 3.2.2 further requires the following lemma that is proved in [26, Lemma V.7.5]:

Lemma 3.2.2 (Convergence of derivatives of convex functions). Let (fn)n≥1 be a sequence of convex functions on an open interval A of R such that f(t) = limn→∞ fn(t) exists for every t ∈ A. Let (tn)n≥1 be a sequence in A that 0 0 0 converges to a point t0 ∈ A. If fn(tn) and f (t0) exist, then limn→∞ fn(tn) 0 exists and equals f (t0).

37 Random quenched measure

Proof of Proposition 3.2.2. In the following proof, we use subscripts to denote the dependence on β and B of the functions cN (t) and c(t) and we write, e.g., cN (t) = cN,β,B (t).

We ﬁrst consider B > 0. By Hölder’s inequality, t 7→ cN,β,B (t) is a convex function on R and, as noted in (3.2.3) and (3.2.4), cN,β,B (t) −→ cβ,B (t) = ∂ ψ(β, B + t) − ψ(β, B) as N → ∞. For B > 0, the derivative ∂B ψ(β, B0) exists for all B0 in some neighborhood of B. Hence, there exists α > 0 such that 0 ∂ c (t) = ∂B ψ(β, B + t) exists for all −α < t < α. Then, Lemma 3.2.2 implies that

0 0 cN,β,B (t) −→ cβ,B (t) for all − α < t < α , as N → ∞. (3.2.12)

0 According to Lemma 3.2.1, each function −cN,β,B (t) is convex on the interval [−B, ∞), which contains the origin in its interior since B > 0. By (2.5.8) we ∂2 know that ∂B2 ψ(β, B) = χ(β, B), hence ∂2 c00 (0) = ψ(β, B) = χ(β, B). (3.2.13) β,B ∂B2 This completes the proof of the proposition for B > 0, because Lemma 3.2.2 implies that, for any 0 ≤ tN < α with tN → 0,

00 00 −cN,β,B (tN ) −→ −cβ,B (0) = −χ(β, B) as N → ∞. (3.2.14)

The proof of Proposition 3.2.2 for B > 0 used the fact that the function 0 t 7→ −cN,β,B (t) is convex on an interval that contains the origin in its interior. 0 But for B = 0, −cN,β,B (t) is convex on [0, ∞) and by symmetry is concave on qu (−∞, 0]. Hence the above proof must be modified. We fix 0 < β < βc and notice that χ(β, 0) is finite. qu ∂ Since for 0 < β < βc , ∂B ψ(β, B) exists for all B real, cβ,0(t) = ψ(β, t) − ψ(β, 0) is differentiable for all t real. We define new functions ( −c0 (t) for t ≥ 0, h (t) = N,β,0 (3.2.15) N 00 −cN,β,0(0) · t for t < 0, ( −c0 (t) for t ≥ 0, h(t) = β,0 (3.2.16) −χ(β, 0) · t for t < 0.

0 Then hN is continuous at 0 since cN,β,0(0) = M(β, 0) = 0, and convex on R. By 00 (3.2.9), cN,β,B (0) → χ(β, B), and so hN (t) → h(t) for all t real. Since χ(β, 0) is 0 00 ﬁnite, h (0) = −cβ,0(0) = −χ(β, 0). Lemma 3.2.2 implies that for any tN ≥ 0 with tN → 0

0 00 0 hN (tN ) = −cN,β,0 (tN ) → h (0) = −χ(β, 0) as N → ∞. (3.2.17)

This proves the proposition for B = 0.

38 AVERAGEDQUENCHEDMEASURE 4

This chapter is devoted to the study of asymptotic theorems in the averaged quenched setting. We start by showing the existence of thermodynamic limits and a law of large numbers for the class of locally tree-like random graphs. Unlike the quenched case, here exponentially fast convergence does not hold and we present a weak law of large numbers. To state a central limit theorem, we work on two particular random graph models: the 2-regular random graph and the configuration model with degrees 1 and 2. The reason, as already explained in Section 2.7.3, is that we cannot follow the previous strategy to prove the CLT for locally tree-like random graphs, because the proof for the concavity of the first derivative of the cumulant generating function cannot be obtained with the same argument used in the random quenched setting. So we choose two simple models where we can handle the problem using explicit calculations. Indeed the computation for the Ising model on the two selected configuration models is based on the solution of the one-dimensional Ising model.

4.1 Thermodynamic limits and LLN for locally tree-like random graphs

The next theorem states the existence of the limit of the thermodynamic quantities in the averaged quenched setting, showing also that the averaged quenched pressure, magnetization and susceptibility coincide with their random quenched counterparts. As a consequence, the averaged quenched critical inverse tempera- aq aq rq ture βc coincides with the random quenched one, i.e., βc = βc . We denote qu qu this unique critical value by βc , and denote the uniqueness set by U . Theorem 4.1.1 (Thermodynamic limits for the averaged quenched law). As- sume that the random graph sequence (GN )N≥1 is locally tree-like, uniformly sparse, and with asymptotic degree distribution D with strongly ﬁnite mean. Then the following conclusions hold: (i) The thermodynamic limit of the averaged quenched pressure exists almost surely and is given by

ψ(β, B) := lim ψN (β, B) = ψ(β, B). (4.1.1) N→∞

(ii) For all (β, B) ∈ U qu, the thermodynamic limit of the averaged quenched magnetization exists almost surely and is given by

M(β, B) := lim M N (β, B) = M(β, B). (4.1.2) N→∞

39 Averaged quenched measure

(iii) For all (β, B) ∈ U qu, the thermodynamic limit of the averaged quenched susceptibility exists almost surely and is given by

χ(β, B) := lim χN (β, B) = χ(β, B). (4.1.3) N→∞ As a consequence, using the law of total variance in (2.3.16), SN lim VarP √ ≥ χ(β, B). (4.1.4) N→∞ N N

We next state the weak law of large numbers for locally tree-like random graphs. A counterexample showing that the convergence at exponential rate is lost is presented in Section 4.4.2.

Theorem 4.1.2 (Averaged quenched WLLN). Assume that the law QN of the random graph is such that almost all sequences (GN )N≥1 are locally tree-like, uniformly sparse, and with asymptotic degree distribution D with strongly ﬁnite mean. Then, for all (β, B) ∈ U qu, S N −→P M(β, B) w.r.t. P , as N → ∞, N N where −→P denotes convergence in probability, i.e., for all ε > 0, SN lim PN − M(β, B) > ε = 0. (4.1.5) N→∞ N

4.2 CLT for CMN (2)

We present now the central limit theorem for CMN (2). We observe that the theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for this system, similarly to the one-dimensional Ising model.

Theorem 4.2.1 (Averaged quenched CLT for CMN (2)). Let (GN )N≥1 be sequences of CMN (2) graphs. Then, for any β ≥ 0 and B ∈ R,

SN − PN (SN ) D √ −→ N (0, χ) , w.r.t. PN , as N → ∞, N where χ = χ(β, B) is the thermodynamic limit of the susceptibility (2.5.8) specialized to the Ising model on CMN (2). Moreover, χ(β, B) is also equal to the susceptibility of the one-dimensional Ising model (see Section 4.5.1), i.e., cosh(B)e−4β χ(β, B) = χd=1(β, B) = . (4.2.1) (sinh(B) + e−4β )3/2 The identiﬁcation of the variance χ(β, B) as χd=1(β, B) also holds for the random quenched setting in Theorem 3.1.2, so that we see that, for CMN (2), the averaged quenched and random quenched variances in the CLT are equal.

We next investigate a case where this is not true, i.e., for the CMN (1, 2).

40 4.3 CLT for CMN (1, 2)

4.3 CLT for CMN (1, 2)

From the CMN (1, 2) example we see that while averaging with respect to the qu graph measure QN does not change βc nor ψ(β, B), (cf. Theorem 4.1.1), it does change the variance of the limiting Gaussian distribution of the total spin since, when passing from random to averaged quenched measure, the fluctuations of the graph are taken into account. Thus, Theorem 4.3.1 shows that the total spin rescaled by the square root of the volume also has fluctuations with respect to 2 the random graph measure. Those fluctuations, quantified by σG in Theorem 4.3.1, are in turn forced by the unequal degrees in CMN (1, 2).

We remark that also for the CMN (1, 2) there is no quenched phase transition, qu because βc = ∞. Indeed, as for the CMN (2) case, the Ising model on the

CMN (1, 2) is reduced to the one-dimensional Ising model.

Theorem 4.3.1 (Averaged quenched CLT for CMN (1, 2)). Let (GN )N≥1 be sequences of CMN (1, 2) graphs. Then, for any β ≥ 0 and B ∈ R,

SN − PN (SN ) D 2 √ −→ N 0, σ , w.r.t. PN , as N → ∞, N aq

2 2 where σaq = χ + σG, with χ = χ(β, B) the thermodynamic limit of the susceptibility of the Ising model on CMN (1, 2) (whose explicit expression is given in 2 2 (4.6.19) below) and σG = σG(β, B) a positive number that is deﬁned in (4.6.39) below.

4.4 Proofs for locally tree-like random graphs

The proofs for locally tree-like random graphs are quite simple and they follow from the results in the random quenched setting.

4.4.1 Averaged quenched thermodynamic limits: Proof of Theorem 4.1.1

Since ψN (β, B) = QN (ψN (β, B)), by Theorem 2.5.3(i) and the Bounded Con- vergence Theorem we obtain ψ(β, B) = ψ(β, B). Item (ii) can be proved in the same way. To prove the statement (iii) we use the following lemma:

Lemma 4.4.1 (Interchanging limits and derivatives). Let (fn)n≥1 be a sequence of functions that are twice diﬀerentiable in x. Assume that

(a) limn→∞ fn(x) = f(x) for some function y 7→ f(y) that is diﬀerentiable in x;

d (b) dx fn(x) is monotone in [x − h, x + h] for all n ≥ 1 and some h > 0. Then, d d lim fn(x) = f(x). (4.4.1) n→∞ dx dx

41 Averaged quenched measure

See [20] for a proof of this lemma.

Now we apply Lemma 4.4.1 with n = N and fn equal to B 7→ M N (β, B). We ∂ combine Theorem 4.1.1(ii) and the fact that B 7→ ∂B M N (β, B) is non-increasing by the GHS inequality (see Remark 2.5.1). Then we have

∂ ∂ lim χN (β, B) = lim M N (β, B) = M(β, B) = χ(β, B), N→∞ N→∞ ∂B ∂B where the last equality is obtained from Thorem 2.5.3.

4.4.2 Averaged quenched WLLN: Proof of Theorem 4.1.2

The proof of this theorem follows from the random quenched SLLN. Indeed, by the deﬁnition of the averaged quenched measure, SN SN PN − M(β, B) > ε = QN µG − M(β, B) > ε , (4.4.2) N N N that, combined with Theorem 3.1.1, i.e. lim µ SN − M(β, B) > ε = N→∞ GN N 0, and the Bounded Convergence Theorem leads to the result.

In the random quenched setting, we prove that the ﬁnite-volume total spin SN normalized by N converges in probability exponentially fast to the magnetization M, which is a non-random quantity. For the averaged quenched setting, instead, it is easy to give examples for which the convergence at exponential rate is lost.

Counterexample 4.4.1. Let us consider the sequence (GN )N≥1 given by ( CM (r) with probability 1 − 1 , G = N N (4.4.3) N 1 KN with probability N , where CMN (r) is the r-regular random graph (r ∈ N), i.e., a conﬁguration model with degree sequence di = r for all i ∈ [N], and KN is the complete graph. The inﬁnite-volume magnetization M of the Ising model (2.3.1) on the sequence

(GN )N≥1 coincides with the limiting magnetization of CMN (r), hence M 6= 1 for any β < ∞. On the other hand, the inﬁnite-volume magnetization on the sequence of complete graphs (KN )N≥1 is 1 and, thus, there exists ε > 0 such that S P N − M > ε G = K = 1 + o(1), N N N N as N → ∞. Therefore,

42 4.5 Proofs for CMN (2)

S S P N − M > ε = P G = K · P N − M > ε G = K N N N N N N N N N + PN GN = CMN (r) S · P N − M > ε G = CM (r) N N N N 1 ≥ 1 + o(1) , N which prevents the sequence from converging exponentially fast.

4.5 Proofs for CMN (2)

In this section, we prove the CLT with respect to the averaged quenched measure for the 2-regular random graph CMN (2). We start by computing the partition function for the one-dimensional Ising model. This will be used in the proofs of CLT’s because the structures formed in CMN (2) and CMN (1, 2), are lines (indicated with the letter l) and cycles (or tori, indicated with the letter t).

4.5.1 Partition functions for the one-dimensional Ising model

The partition function of the one-dimensional Ising model is given by

N N ! (b) X X X X ZN (β, B) = ... exp β σiσi+1(b) + B σi , (4.5.1) σ1=±1 σN =±1 i=1 i=1 where for periodic boundary conditions (b = t) we put σN+1(t) = σ1, whereas we put σN+1(l) = 0 for free boundary condition (b = l). We often omit (β, B) (t) (l) from the notation and simply write ZN and ZN . Let D be the 2 × 2 matrix 2 deﬁned, for (σi, σi+1) ∈ {−1, +1} , by B D = exp βσ σ + (σ + σ ) . σi,σi+1 i i+1 2 i i+1

With this deﬁnition we may rewrite (4.5.1) for periodic boundary conditions in the form

(t) X X ZN = ··· Dσ1,σ2 Dσ2,σ3 ··· DσN ,σN+1 σ1=±1 σN =±1 N N N = Trace(D ) = λ+ + λ− , (4.5.2) where λ+ and λ− are the two eigenvalues of D, given by q β 2 −4β λ± = λ±(β, B) = e cosh(B) ± sinh (B) + e . (4.5.3)

43 Averaged quenched measure

Obviously, λ+(β, B) > λ−(β, B). In the case of free boundary conditions, we observe that

(l) X X B B 2 σ1 2 σN ZN = ··· e Dσ1,σ2 Dσ2,σ3 ··· DσN−1,σN e σ1=±1 σN =±1 X X B σ N−1 B σ T N−1 = e 2 1 D e 2 N = v D v, (4.5.4) σ1,σN σ1=±1 σN =±1 where the vector v is deﬁned by vT = (eB/2, e−B/2). This can be written as

T 2 (l) N N (v · v±) ZN = A+λ+ + A−λ− , with A± = , (4.5.5) λ± where v± denote the two orthonormal eigenvector of the matrix D, and therefore

−2β ±B β+B 2 ∓B −β β+B e e + (λ+ − e ) e ± 2e (λ+ − e ) A± = A±(β, B) = −2β β+B 2 . (4.5.6) [e + (λ+ − e ) ]λ±

Let us remark that, since β > 0, λ±(β, B) > 0 and A± (β, B) > 0. From (4.5.2) and (4.5.5), it follows that the pressure of the one-dimensional Ising model is, independently of the boundary conditions, given by

d=1 ψ (β, B) = log λ+(β, B) . (4.5.7)

4.5.2 Quenched pressure

t Any 2-regular random graph is formed by cycles only. Thus, denoting by KN the random number of cycles in the graph, we can enumerate them in an arbitrary t order from 1 to KN and call LN (i) the length (i.e. the number of vertices) of t the i-th cycle. The random variable KN is given by

N t X KN = Ij, (4.5.8) j=1 where Ij are independent Bernoulli variables, i.e.,

1 I = Bern . (4.5.9) j 2N − 2j + 1

Indeed, at every step j in the construction of the 2-regular random graph, namely at each pairing of two half-edges, we have one and only one possibility to close a cycle. This possibility corresponds to drawing exactly one half-edge out of the remaining 2N − 2j + 1 unpaired half-edges. The indicator of this event is the N Bernoulli variable Ij. Obviously, the variables (Ij )j=1 are independent, but not t identical, and the number of the cycles KN is distributed as their sum.

44 4.5 Proofs for CMN (2)

Since the random graph splits into (disjoint) cycles, its partition function factorizes into the product of the partition functions of each cycle. Therefore,

t KN Y (t) ZN (β, B) = Z (β, B), (4.5.10) LN (i) i=1 where, by (4.5.2) the partition function of the i-th cycle is

Z(t) (β, B) = λLN (i)(β, B) + λLN (i)(β, B). (4.5.11) LN (i) + −

Because β > 0, we have 0 < λ−(β, B) < λ+(β, B), so that, for every i,

λLN (i)(β, B) ≤ Z(t) (β, B) ≤ 2λLN (i)(β, B). (4.5.12) + LN (i) + As a result, we can bound the the pressure by

t t t KN KN KN Y L (i) Y (t) Y L (i) λ N (β, B) ≤ Z (β, B) ≤ 2λ N (β, B), (4.5.13) + LN (i) + i=1 i=1 i=1

Kt P N and, since i=1 LN (i) = N, we ﬁnally obtain

N Kt N λ+ (β, B) ≤ ZN (β, B) ≤ 2 N λ+ (β, B). (4.5.14) Now we are ready to compute the quenched pressure in the thermodynamic limit, deﬁned as in (2.5.5). By the previous inequality,

Kt log λ (β, B) ≤ ψ (β, B) ≤ log 2 · N + log λ (β, B), (4.5.15) + N N + where ψN (β, B) is the random quenched pressure deﬁned in (2.3.5). Now, by (4.5.8) and (4.5.9),

N 1 1 X 1 log N Q Kt = ∼ → 0 as N → ∞. (4.5.16) N N N N 2N − 2i + 1 N i=1

t KN Therefore, by applying Markov’s inequality to the non-negative variable N , Kt N −→P 0, w.r.t. Q . (4.5.17) N N Hence, taking the limits in (4.5.15), we obtain the inﬁnite-volume random quenched pressure as

d=1 ψ(β, B) = ψ (β, B) ≡ log λ+(β, B). (4.5.18)

Moreover, from Corollary 4.1.1 we also obtain the averaged quenched pressure

ψ(β, B) = ψ(β, B) = ψd=1(β, B). (4.5.19)

45 Averaged quenched measure

4.5.3 Cumulant generating functions

As already said in Section 2.7.3, in order to prove the averaged quenched CLT for

CMN (2), we need to calculate the limit for N → ∞ of the cumulant generating function in the averaged quenched random setting. We write this function in the following form: " # 1 1 Z(t) (β, B + t) c (t) = log P [exp(tS )] = log Q N , (4.5.20) N N N N N N (t) ZN (β, B) where, in the right-hand side of the previous equation, we have expressed the random quenched average of exp(tSN ) as a ratio of partition functions, i.e., (t) (t) µGN [exp(tSN )] = ZN (β, B + t) /ZN (β, B). Again using (4.5.14), we bound the ratio of the partition functions at diﬀerent ﬁelds as N N −Kt λ+ (β, B + t) ZN (β, B + t) Kt λ+ (β, B + t) 2 N ≤ ≤ 2 N . λ+ (β, B) ZN (β, B) λ+ (β, B) Therefore, N ! 1 −Kt λ+ (β, B + t) log QN 2 N ≤ cN (t) N λ+ (β, B) N ! 1 Kt λ+ (β, B + t) ≤ log QN 2 N , N λ+ (β, B) and, recalling that λ+(β, B) and λ+(β, B + t) are not random, we obtain

1 −Kt log Q 2 N ≤ c (t) − log λ (β, B + t) + log λ (β, B) N N N + + 1 Kt ≤ log Q 2 N . N N Now we can compute the limit of the averages in the previous equation recalling t that KN is a sum of independent Bernoulli variables Ii, c.f. (4.5.8). Indeed, N 1 Kt 1 Y I log Q 2 N = log Q 2 i N N N N i=1 N 1 Y 2 1 = log + 1 − N 2N − 2i + 1 2N − 2i + 1 i=1 N 1 X 1 N→∞ = log 1 + −→ 0, (4.5.21) N 2N − 2i + 1 i=1 where the limit can be obtained using the inequality log(1 + x) ≤ x for x ≥ 0, and the same asymptotic estimate already used in (4.5.16). In a similar fashion we can also prove that

1 −Kt lim log QN 2 N = 0. (4.5.22) N→∞ N

46 4.5 Proofs for CMN (2)

By (4.5.21) and (4.5.22) we ﬁnally obtain the limit of the cumulant generating function in the averaged quenched setting as

c(t) = lim cN (t) = ψ(β, B + t) − ψ(β, B) N→∞ = log λ+(β, B + t) − log λ+(β, B) . (4.5.23)

On the other hand, since by (3.2.4), we also have that c(t) = ψ(β, B + t) − ψ(β, B), we conclude that in the thermodynamic limit the averaged quenched and the random quenched cumulant generating functions of the total spin are the same and are given by:

c(t) = c(t) = log λ+(β, B + t) − log λ+(β, B). (4.5.24)

4.5.4 Averaged quenched CLT: Proof of Theorem 4.2.1

According to our general strategy, in order to prove Theorem 4.2.1, we need to show that, for all t ∈ R, SN − PN (SN ) 1 2 2 lim PN exp t √ = exp σaq(β, B)t . (4.5.25) N→∞ N 2 As in the proof of Theorem 3.1.2, the limit can be computed by expressing the expectation on the left-hand side in terms of the proper generating function.

Here, this is the averaged quenched cumulant generating function, i.e., cN (t).

Rewrite in terms of cumulant generating functions. Using the fact 0 SN that cN (0) = 0 and cN (0) = PN ( N ), see (4.5.20), by Taylor’s theorem with Lagrange remainder, tSN − tPN (SN ) t t log PN exp √ = log PN exp √ SN − √ PN (SN ) N N N √ t 0 = NcN √ − t Nc (0) N N t2 = c00 (t ), (4.5.26) 2 N N for some t ∈ [0, √t ]. In order to compute the limit of the sequence c00 (t ), N N N N we consider " # 1 Z(t)(β, B + t) λ (β, B + t) c (t) − c(t) = log Q N − log + N N N (t) λ (β, B) ZN (β, B) +  (t)  ZN (β,B+t) Kt N " N LN (i) # 1  (λ+(β,B+t))  1 Y 1 + (rB+t) = log QN   = log QN , N  Z(t)(β,B)  N LN (i) N i=1 1 + (rB ) N (λ+(β,B))

47 Averaged quenched measure where we have used (4.5.2), (4.5.10), (4.5.20), (4.5.23), and, omitting the dependence on β, we have deﬁned

λ−(β, B) rB = r(β, B) = . (4.5.27) λ+(β, B) Then, recalling (4.5.23), we can rewrite

Kt " N LN (i) # 1 Y 1 + (rB+t) cN (t) = log λ+(β, B + t) − log λ+(β, B) + log QN . N LN (i) i=1 1 + (rB ) (4.5.28) The second derivative of (4.5.28) is

2 00 ∂ 1 IIIN (t) cN (t) = 2 log λ+(β, B + t) + IN (t) + IIN (t) + , ∂t NDN (t) DN (t) (4.5.29) where we have introduced

t KN LN (i)−2 0 2 00 X LN (i)(rB+t) [(LN (i) − 1)(rB+t) + rB+trB+t] IN (t) = QN 1 + (r )LN (i) i=1 B t KN LN (j) Y 1 + (rB+t) · , 1 + (r )LN (j) j=1 B j6=i

t t KN KN LN (i)+LN (j)−2 0 2 LN (l) X LN (i)LN (j)(rB+t) (rB+t) Y 1 + (rB+t) IIN (t) = QN , (1 + (r )LN (i))(1 + (r )LN (j)) 1 + (r )LN (l) i,j=1 B B l=1 B j6=i l6=i,j

t t KN KN 2 LN (i)−1 0 LN (j) X LN (i)(rB+t) rB+t Y 1 + (rB+t) IIIN (t) = QN , 1 + (r )LN (i) 1 + (r )LN (j) i=1 B j=1 B j6=i

t KN LN (i) Y 1 + (rB+t) DN (t) = QN . 1 + (r )LN (i) i=1 B

Analysis of the second derivative of the cumulant generating function. The ﬁnal step in the proof of the theorem requires to compute the limit of the second term in the right-hand-side of (4.5.29). This step is taken in the following lemma:

Lemma 4.5.1.√Let t > 0 and let (tN )N≥1 be a sequence of real numbers such that tN ∈ [0, t/ N]. Then, 1 IIIN (tN ) lim IN (tN ) + IIN (tN ) + = 0. (4.5.30) N→∞ NDN (tN ) DN (tN )

48 4.5 Proofs for CMN (2)

Proof. We ﬁrst consider the term IN (tN ). From (4.5.27), it is clear that 0 < LN (i)−2 r(β, B) < 1. As a consequence, the terms LN (i)(LN (i) − 1)(rB+tN ) and LN (i)−1 LN (i)(rB+tN ) , appearing in the numerator of IN (t) are uniformly bounded 0 00 in N. Moreover, also the sequences (r )N and (r )N , being convergent, B+tN B+tN are bounded. Therefore, there exists a constant C > 0 such that

LN (i)−2 0 2 LN (i)−1 00 LN (i)(LN (i) − 1)(rB+t ) (r ) + LN (i)(rB+t ) r ≤ C. N B+tN N B+tN Therefore, also

t K LN (i)−2 0 2 LN (i)−1 00 N LN (i)(LN (i) − 1)(rB+t ) (r ) + LN (i)(rB+t ) r X N B+tN N B+tN 1 + (r )LN (i) i=1 B t ≤ C · KN , (4.5.31) and thus Kt ! N LN (i) t Y 1 + (rB+tN ) |IN (tN )| ≤ C · QN KN . (4.5.32) 1 + (r )LN (i) i=1 B We now proceed by ﬁrst estimating each term appearing in the product. Let us

ﬁx an arbitrary length l ∈ N to be chosen later on. Recalling that 0 < rB < 1 we obtain ( LN (j) l 1 + (rB+t ) 1 + (r ) if L (j) > l, N ≤ B+tN N (4.5.33) 1 + (r )LN (j) B 1 + δN (l) if LN (j) ≤ l, 1 + (r )k where δ (l) = max B+tN − 1 N−→→∞ 0. Then, N : k k≤l 1 + (rB )

L (j) N l 1 + (rB+t ) l (r ) ∨δ (l) N ( B+tN N ) L (j) ≤ 1 + (rB+tN ) ∨ δN (l) ≤ e , (4.5.34) 1 + (rB ) N where a ∨ b = max{a, b}. Finally, using (4.5.34) and the Cauchy-Schwarz inequality Kt ! N LN (i) t Y 1 + (rB+tN ) |IN (tN )| ≤ C · QN KN 1 + (r )LN (i) i=1 B Kt ! P N (r )l∨δ (l) t i=1 ( B+tN N ) ≤ C · QN KN e

Kt (r )l∨δ (l) t N ( B+tN N ) ≤ C · QN KN e 1 2 1 2 h 2i / h 2Kt (r )l∨δ (l) i / t N ( B+tN N ) ≤ C · QN KN QN e . (4.5.35)

We next consider DN (tN ). As before, ( LN (j) l −1 1 + (rB+t ) (1 + (r ) ) if L (j) > l , N ≥ B N (4.5.36) 1 + (r )LN (j) −1 B (1 + δ¯N (l)) if LN (j) ≤ l ,

49 Averaged quenched measure

k where now δ¯ (l) = min 1+(rB ) − 1 N−→→∞ 0. Then, N : k≤l 1+(r )k B+tN

LN (j) 1 + (rB+t ) 1 l ¯ N ≥ ≥ e−((rB ) ∨δN (l)), (4.5.37) L (j) l 1 + (rB ) N 1 + ((rB ) ∨ δ¯N (l))

1 −x because 1+x ≥ e for x > −1. From the previous bound, we obtain

Kt N Kt ! l P N l Y −((rB ) ∨δ¯N (l)) − ((rB ) ∨δ¯N (l)) DN (tN ) ≥ QN e = QN e i=1 i=1 t l −K ((rB ) ∨δ¯N (l)) = QN e N . (4.5.38)

Collecting (4.5.35) and (4.5.38) we obtain

1 1 h 2i 2 h 2Kt (r )l∨δ (l) i 2 t N ( B+tN N ) |I (t )| C · QN KN QN e lim N N ≤ lim . t l ¯ N→∞ NDN (tN ) N→∞ −K ((rB ) ∨δN (l)) N · QN e N (4.5.39) The averages in the right-hand side can be bounded using the distribution of t KN in (4.5.8) and (4.5.9). Indeed, using that, for ﬁxed l, δN (l) → 0 and that l l ¯ (rB+tN ) can be made small by taking l large, we can make (rB ) ∨ δN (l) ≤ ε for N large enough. Thus,

N ε 2Kt (r )l∨δ (l) Y (e − 1) Q e N ( B+tN N ) ≤ 1 + N 2(N − i) + 1 i=1 1 ε √ Γ N + 2 e π 1 (eε−1) = ∼ N 2 , (4.5.40) 1 1 ε Γ(N + 2 )Γ( 2 e ) where the product is expressed in terms of the Gamma function, and the asymptotic relation Γ(N+a) = N a−b(1 + o(1)), valid when a, b are uniformly Γ(N+b) bounded, has been used. In a similar fashion the average in the denominator of (4.5.39) can be bounded by t l 1 −ε −K ((rB ) ∨δ¯N (l)) (e −1) QN e N ≥ N 2 , (4.5.41)

1/2 h t 2i while QN KN = O(log N). Thus, we conclude that the right-hand 3 (eε−1)−1 side of (4.5.39) is bounded by O(log N)N 4 = o(1) by taking ε > 0 to be suﬃciently small. This proves the bound on IN (tN ) in (4.5.30). With similar calculations, that we omit for the sake of brevity, we can also show that the terms involving IIN (tN ) and IIIN (tN ) vanish, thus concluding the proof of Lemma 4.5.1.

50 4.6 Proofs for CMN (1, 2)

Completion of the proof of Theorem 4.2.1. Equations (4.5.26), (4.5.29) and Lemma 4.5.1 complete the proof of the averaged quenched CLT, since

2 SN − PN (SN ) t 00 lim log PN exp t √ = lim cN (tN ) N→∞ N N→∞ 2 2 2 t ∂ = · 2 log λ+(β, B + t) 2 ∂t t=0 t2 cosh(B)e−4β = · . (4.5.42) 2 (sinh(B) + e−4β )3/2

Since the thermodynamic limits of the two cumulant generating functions c(t) and c(t) are the same, cf. (4.5.24), also the variances of the limiting normal distribution in the random quenched and the averaged quenched are equal.

4.6 Proofs for CMN (1, 2)

In this section, we consider the conﬁguration model CMN (1, 2), introduced in Section 2.1.2. In this graph, the connected components are either tori (t) connecting vertices of degree 2, or lines (l) formed joining vertices with degree 2 and ending with two vertices with degree 1. In order to state some properties of the number of lines and tori, we need to introduce some notation. Recall that the number of vertices of degree 1 and 2 are given by

n1 := # {i ∈ [N] : di = 1} = N − bpNc, (4.6.1)

n2 := # {i ∈ [N] : di = 2} = bpNc, (4.6.2) and the total degree of the graph equals X `N = di = 2n2 + n1 = N + bpNc. (4.6.3) i∈[N]

The number of edges is given by `N /2, so we assume n1 to be even. Let us also (l) denote by KN the number of connected components in the graph and by KN (t) and KN the number lines and tori. Obviously,

(l) (t) KN = KN + KN . (4.6.4)

Because every line uses up two vertices of degree 1, the number of lines is given (l) by n1/2, i.e., KN = (N − bpNc)/2, almost surely. Regarding the number of (t) t cycles, we have that KN /N has the same distribution of KN¯ /N, where N is the (random) number of vertices with degree 2 that do not belong to any line t and KN¯ is the number of tori on this set of vertices. Then, since this subset (t) P forms a CMN¯ (2) graph, we can apply (4.5.17), obtaining that KN /N −→ 0, so that also K (1 − p) N −→P . (4.6.5) N 2

51 Averaged quenched measure

The partition function can be computed as in the case of CMN (2), and is given by (l) (t) KN KN Y (l) Y (t) ZN (β, B) = Z (β, B) · Z (β, B), (4.6.6) L(l)(i) L(t)(i) i=1 N i=1 N where by (4.5.5) and (4.5.2)

(l) (l) (l) LN (i) LN (i) Z (l) (β, B) = A+λ+ + A−λ− , (4.6.7) LN (i) (t) (t) (t) LN (i) LN (i) Z (t) (β, B) = λ+ + λ− , (4.6.8) LN (i)

(l) (t) and LN (i) and LN (j) are the lengths (i.e., the number of vertices) of the ith line and jth torus (in any arbitrary labeling).

4.6.1 Quenched pressure

Starting from the expression of the partition function (4.6.6) and with some algebraic manipulations, we obtain the random quenched pressure at volume N as

N X (N) l ψN (β, B) = log λ+(β, B) + pl log A+(β, B) + A−(β, B) (rB ) l=2 K(t) N 1 X L(t)(i) + log 1 + (r ) N . (4.6.9) N B i=1 Here the ﬁrst sum is taken over the possible line lengths and we deﬁne

(l) KN (N) 1 X p := 1 (l) (4.6.10) l N {L (i)=l} i=1 N to be the normalized number of lines of length l in the CMN (1, 2) graph. The (N) random variables (pl )l≥2 play a fundamental role in the proofs, therefore we start by investigating their behavior in the thermodynamic limit:

Lemma 4.6.1 (Convergence of lines of given lengths). For every l ≥ 2, as N → ∞, 2p l−2 1 − p 1 − p p(N) −→P p∗ = . (4.6.11) l l p + 1 p + 1 2 Proof. The result can be obtained by applying the Second Moment Method, (N) ∗ which states that the convergence of the averages QN pl → pl and the

52 4.6 Proofs for CMN (1, 2)

(N) vanishing of the variances VarQN pl →0 together imply the required conver- (N) gence in probability. We start by computing the average of pl . From (4.6.10), we obtain Q (K (l)) Q (p(N)) = N N Q (L (1) = l) N l N N N (1 − p) = Q (L (1) = l) (1 + o(1)), (4.6.12) 2 N N

(l) (l) because KN /N → (1 − p)/2 a.s., and the distribution of lengths LN (i) of the ith line is independent of the label i. The average on the right hand side of the previous equation can be computed as l−3 ! Y 2n2 − 2i n1 − 1 Q (L (1) = l) = . (4.6.13) N N ` − 1 − 2i ` − 1 − 2(l − 2) i=0 N N

n2 n1 Recalling (4.6.1) we have that N → p and N → 1 − p as N → ∞, therefore, l−2 2p 1−p by deﬁnition (4.6.3) of `N , we obtain that QN (LN (1) = l) → p+1 p+1 , (N) concluding the computation of the limit of QN (pl ). Also the variance of (4.6.10) (l) can be computed explicitly. Indeed, exploiting the fact that KN is a.s. constant, we obtain

(N) VarQN (pl ) 1 = K(l)Q (L (1) = l)(1 − Q (L (1) = l)) N 2 N N N N N 1 + K(l)(K(l) − 1) Q (L (1) = L (2) = l) − Q (L (1) = l)2 N 2 N N N N N N N 1 = K(l)Q (L (1) = l)(1 − Q (L (1) = l)) N 2 N N N N N 1 + K(l)(K(l) − 1)Q (L (1) = l) N 2 N N N N · QN (LN (2) = l| LN (1) = l) − QN (LN (1) = l) , (4.6.14) where the conditional probability in the previous equation is given by

QN ( LN (2) = l| LN (1) = l) l−3 Y 2n2 − 2(l − 2) − 2i n1 − 3 = · . (4.6.15) ` − 2(l − 2) − 3 − 2i ` − 2(l − 2) − 3 − 2(l − 2) i=0 N N Again recalling (4.6.1) and (4.6.3), we see that the conditional probability l−2 2p 1−p QN ( LN (2) = l| LN (1) = l) converges to p+1 p+1 as N → ∞, so that (N) N→∞ VarQN (pl ) −→ 0.

With the help of the previous lemma we will be able to control the contribution of the lines to ψN (β, B) in the thermodynamic limit (second term in the right-hand

53 Averaged quenched measure side of (4.6.9)). From the next lemma, we deduce that in (4.6.9) the contribution of the tori is negligible when N → ∞: Lemma 4.6.2 (Bounding the contribution due to tori). Conditionally on (l) (l) (l) (t) LN (1), ... , LN (KN ), the number of tori KN in the CMN (1, 2) graph is stocha- t (t) stically smaller than the number of tori KN in CMN (2) (and we write KN t KN ). Therefore, K(t) N 1 X L(t)(i) P log 1 + (r ) N −→ 0. (4.6.16) N B i=1

Proof. Since 0 < rB = λ−/λ+ < 1,

K(t) N (t) (t) 1 X L (i) KN log 2 0 ≤ log 1 + (r ) N ≤ . (4.6.17) N B N i=1 Kt Recalling that K(t) Kt and N −→P 0 by (4.5.17), we complete the proof of N N N the lemma. By applying Lemma 4.6.1 and Lemma 4.6.2, we ﬁnally identify the thermodynamic limit of the random quenched pressure as 1 log Z (β, B) −→P ψ(β, B) (4.6.18) N N X ∗ l ≡ log λ+(β, B) + pl log A+(β, B) + A−(β, B) (rB ) . l≥2 Moreover, recalling that the thermodynamic limit of the averaged quenched pressure is equal to the random quenched pressure, we also have ψ(β, B) = ψ(β, B). The random quenched susceptibility is then obtained by calculating the second derivative of the quenched pressure, i.e., X ∂2 ∂2 χ(β, B) = p∗ log A (β, B) + A (β, B) (r )l + log λ (β, B). l ∂B2 + − B ∂B2 + l≥2 (4.6.19) This will allow us to compare the variances in the random and averaged quenched settings.

4.6.2 Averaged quenched CLT: Proof of Theorem: 4.3.1

Once more, to prove the CLT in the averaged quenched setting we must show that, for all t ∈ R, 2 SN − PN (SN ) t 2 lim PN exp t √ = exp σaq , (4.6.20) N→∞ N 2 2 for some σaq.

54 4.6 Proofs for CMN (1, 2)

Analysis of the moment generating function. We start by rewriting the moment generating function as a product of two terms that will be analyzed separately, i.e.,

" t # ZN β, B + √ SN − PN (SN ) N t PN exp t √ = QN exp −√ PN (SN ) . N ZN (β, B) N (4.6.21) From (4.6.9), we separate the contribution of lines and tori to the partition function, as

(N) ZN (β, B) = exp NFβ,B p + NEN (β, B) , (4.6.22) where

(N) X (N) l Fβ,B p = log λ+(β, B) + pl log A+(β, B) + A−(β, B) (rB ) , l≥2 (4.6.23)

K(t) N 1 X L(t)(i) E (β, B) = log 1 + (r ) N , (4.6.24) N N B i=1 and p(N) = p(N) is deﬁned in (4.6.10). The ﬁrst factor in (4.6.21) is l l≥2 "Z β, B + √t # N N QN ZN (β, B) " (N) (N) t # N F t (p )−F (p ) +N E β,B+ √ −E (β,B) β,B+ √ β,B N N N N = QN e

" N F (p(N))−F (p(N)) t # β,B+ √t β,B N EN β,B+ √ −EN (β,B) N N (N) = QN e QN e p " N F (p(N))−F (p(N)) # β,B+ √t β,B N = QN e (1 + o(1)), (4.6.25)

t N EN β,B+ √ −EN (β,B) because the contribution of e N as N → ∞ is negligible. Indeed, by Taylor expansion, t N EN β, B + √ − EN (β, B) (4.6.26) N (t) (t) (t) L (i)−1 K N d N LN (i) (rB ) dt rB+ √t (t) t X N t tCK √ t=0 √ √ N = (t) + o ≤ , N LN (i) N N i=1 1 + (rB )

55 Averaged quenched measure since the summands in the previous equation are uniformly bounded in N. Therefore, by (4.5.21),

(t) t tCK N EN β,B+ √ −EN (β,B) √ N N→∞ N (N) N (N) QN e p ≤ QN e p −→ 1, (4.6.27) and then (4.6.25) follows. The next step is to estimate the expectation appearing in (4.6.25). To this end, we introduce the notation

(N) (N) (N) ∆FB+ √t (p ) := Fβ,B+ √t (p ) − Fβ,B (p ), (4.6.28) N N l fl(B) := log A+ (B) + A− (B)(rB ) , (4.6.29) where, here and in the rest of the section, the dependence on β is dropped for simplicity. X ∗ t (N) Adding and subtracting pl fl B + √ to ∆FB+ √t (p ), yields N N l≥2

(N) t X (N) ∗ t ∆FB+ √t (p ) = log λ+ B+ √ − log λ+(B) + pl − pl fl B+ √ N N N l≥2

X ∗ t X (N) + p fl B+ √ − p fl(B). (4.6.30) l N l l≥2 l≥2

Expanding around t = 0, we obtain

2 2 (N) ∂ t ∂ ∆FB+t(p ) = t (log λ+ (B + t)) + (log λ+ (B + t)) ∂t t=0 2 ∂t2 t=0 X (N) ∗ ∂ + t p − p (fl (B + t)) l l ∂t t=0 l≥2 2 2 t X (N) ∗ ∂ + p − p (fl (B + t)) 2 l l ∂t2 l≥2 t=0

X ∗ ∂ + t p (fl (B + t)) l ∂t l≥2 t=0 2 2 t X ∗ ∂ 2 + p (fl (B + t)) + o(t ). (4.6.31) 2 l ∂t2 l≥2 t=0 √ Using the above for t replaced with t/ N, recalling the expression for the random quenched susceptibility given in (4.6.19), taking the exponential of (N) N∆FB+ √t (p ) and the average with respect to QN , we get N

56 4.6 Proofs for CMN (1, 2)

N(F (p(N))−F (p(N))) h β,B+ √t β,B i N QN e (4.6.32) √ √ " ∂ P (N) ∗ ∂ t2 t N log λ+(B+t) +t N (p −p ) (f (B+t)) χ ∂t t=0 l l l ∂t l t=0 = e 2 · QN e

√ t2 P (N) ∗ ∂2 P ∗ ∂ # (p −p ) (fl(B+t)) + t N p (fl(B+t)) + o(1) · e 2 l l l ∂t2 t=0 l l ∂t t=0 .

Next we consider the factor exp − √t P (S ) appearing in (4.6.21). Since N N N

h ∂ ZN (B + t) i PN (SN ) = QN t=0 , ∂t ZN (B) we can write

∂ PN (SN ) = N (log λ+ (B + t)) ∂t t=0 h X (N) ∂ ∂ i + QN N p (fl(B + t)) + N EN (B + t) . l ∂t t=0 ∂t t=0 l≥2 Z β,B+ √t h N N i As we have already done for QN , we can show, by bounding the ZN (β,B) derivative of EN (β, B), that the term corresponding to tori gives a vanishing contribution to the limit of √t P (S ). Thus, we can write N N N √ t ∂ − √ PN (SN ) −t N (log λ+(B+t)) e N = e ∂t t=0 √ h i P (N) ∂ −t NQN p (fl(B+t)) · e l≥2 l ∂t t=0 (1 + o(1)). (4.6.33)

Combining (4.6.32) and (4.6.33), the moment generating function can be rewritten as

h SN − PN (SN ) i PN exp t √ N √ (N) (N) t N P p −Q (p ) ∂ (f (B+t)) t2χ/2 h l≥2 l N l ∂t l t=0 i = e · QN e (1 + o(1)). (4.6.34)

S −P (S ) As the previous equation shows, the moment generating function of N √N N N is expressed in terms of that of

√ N X (N) (N) ∂ XN := N p − QN (p ) (fl (B + t)) , (4.6.35) l l ∂t t=0 l=2

57 Averaged quenched measure

l l Since fl(B) = log(A+(B) + A−(B)(rB ) ) = log(A+(B)) + log(1 + aB (rB ) ) with aB = A−(B)/A+(B), and log(A+(B)) is independent of l, so that

N X (N) (N) pl − QN (pl ) log(A+(B)) ≡ 0, (4.6.36) l=2 we can take ∂ l γl(B) := log(1 + aB+t(rB+t) ) . (4.6.37) ∂t t=0

In particular, γl(B) is exponentially small for large l. Thus, in order to complete the proof of the theorem we need to show that the sequence (XN )N≥1 converges to a normal limit. This result is provided by the next lemma:

(N) Lemma 4.6.3 (CLT for linear combinations of (pl )l≥2). As N → ∞,

√ N X (N) (N) D 2 XN ≡ N pl − QN pl γl(B) −→ N 0, σG , (4.6.38) l=2 with

∞ " l−2# 1 − p X α2l−4 (l − 2 − α)2 1 α σ2 := γ2(B) − 1 + + G 2 l (1 + α)2l−2 α(1 + α) 1 + α 1 + α l=2 ∞ 1 − p X αl+j−4 (l − 2 − α)(j − 2 − α) + γ (B)γ (B) − 1 + , 2 l j (1 + α)l+j−2 α(1 + α) l,j=2 (4.6.39) where α = 2p/(1 − p) and p is deﬁned in (4.6.1). Also the moment generating 2 function of XN converges to that of N 0, σG .

Completion of the proof of Theorem 4.3.1. Lemma 4.6.3 implies that for any t ∈ R

√ t N P p(N)−Q p(N) γ (B) h l l N l l i t2σ2 /2 lim QN e = e G . (4.6.40) N→∞

This completes the proof of the averaged quenched CLT for CMN (1, 2), since h SN − PN (SN ) i 2 2 lim PN exp t √ = exp t σaq/2 , (4.6.41) N→∞ N

2 2 with σaq = χ + σG. The remaining part of this section is devoted to the proof of Lemma 4.6.3.

58 4.6 Proofs for CMN (1, 2)

A result by de Panafieu and Broutin [14]. The proof of Lemma 4.6.3 is based on the following theorem by de Panaﬁeu and Broutin [14] that states a joint CLT for the number of connected components in graphs with degrees 1 and 2: Theorem 4.6.1 (CLT for connected components in random graphs [14, Theorem 1]). Let the random vector Cτ denote the number of connected components of size τ in a random graph with n1 vertices of degree 1 and n2 vertices of degree 2 and no other degrees allowed. For some T ∈ N and all 2 ≤ τ ≤ T , let 1 ατ−2 n ∗ √ 1 Cτ = Cτ − τ−1 (4.6.42) n1/2 (1 + α) 2

∗(N) ∗ ∗ where α = 2n2/n1 is a ﬁxed real positive value, and C = (C2 , ... , CT ). Then, as N = n1 + n2 → ∞, and with φ (N) (s2, ... , sT ) denoting the joint C∗ moment generating function of C∗(N),

T lim φ s , ... , s esH¯ s¯ /2, (4.6.43) ∗(N) ( 2 T ) = n1→∞ C T where s¯ = (s2, ... , sT ), s¯ is its transpose, and H = H(α) is the (T − 1) × (T − 1)-sized covariance matrix given by

r+t−4 r−2 α (r − 2 − α)(t − 2 − α) 1l{r=t} α H (α) = − 1 + + , r,t (1 + α)r+t−2 α(1 + α) 1 + α 1 + α (4.6.44) and the convergence is uniform for s¯ in a neighborhood of 0¯ = (0, ... , 0). As a result, C∗(N) −→D N (0¯, H(α)), the multivariate normal distribution with covariance matrix H(α), α = 2p/(1 − p). Theorem 4.6.1 cannot be applied directly to our case since it deals with the number of all the connected components of the random graphs, while we are only interested in the lines of CMN (1, 2). Therefore, some work is needed in order to adapt Theorem 4.6.1 to our setting. To deduce this result, let us introduce some notation. Denote by Λl the number of lines of length l in CMN (1, 2) (omitting ∗ the dependence of Λl √on N to lighten the notation) and by Λl the centered variable normalized by n1/2 as in (4.6.42). Then, we have the following lemma:

Lemma 4.6.4 (CLT for number of lines of ﬁxed lengths in CMN (1, 2)). Let Λl be the number of lines of length l in a CMN (1, 2) graph with n1 vertices of degree one and n2 of degree two, as in (4.6.1). For some T ∈ N and all 2 ≤ l ≤ T , let

∗ 1 Λl = √ (Λl − QN (Λl)) . (4.6.45) n1/2

Then, as N → ∞, with α = 2n2/n1 ﬁxed, the moment generating function of ∗(N) ∗ ∗ the vector Λ ≡ (Λ2, ... , ΛT ) satisﬁes the following limit

(N) s·Λ∗ sH¯ s¯T /2 lim QN e = e , (4.6.46) N→∞

59 Averaged quenched measure

T where s¯ = (s2, ... , sT ), s¯ is its transpose, and H = H(α) is the (T − 1) × (T − (N) D 1)-sized covariance matrix given in (4.6.44). As a result, Λ∗ −→ N (0¯, H(α)).

Proof. Consider the number Θl of tori of length l in CMN (1, 2) and center (N) p n1 ∗ ∗ ∗ ∗ and normalize it by 2 to obtain Θl and let Θ = (Θ2, ... , ΘT ). Then, (N) (N) (N) (N) C∗ = Λ∗ + Θ∗ , where C∗ is deﬁned in (4.6.42). In order to compute (N) the limit of the generating function of Λ∗

(N) (N) (N) s·Λ∗ s·C∗ −s·Θ∗ QN e = QN e e (4.6.47) we follow the same strategy that will be used in the proof of Lemma 4.3. In particular we use Hölder’s inequality to obtain

(N) p s·C∗ /p QN e (N) (N) 1/p (N) 1/q s·Λ∗ s·C∗ p −s·Θ∗ q ≤ QN e ≤ QN e QN e (N) p/q s·Θ∗ q/p QN e (4.6.48) where p = 1 + ε and 1/p + 1/q = 1, so that q = (1 + ε)/ε and ε > 0. Now, for ﬁxed s∗ > 0, we want to prove that

(N) s·Θ∗ lim QN e = 1 (4.6.49) N→∞ √ ∗ ∗ for all s¯ with maxi |si| = s . Since Θl = (Θl − QN (Θl))/ n1/2, we can write

(N) PT PT s·Θ∗ l=2 slQN (Θl) l=2 slΘl QN e = exp − √ QN exp √ . (4.6.50) n1/2 n1/2 P Since the number of tori KN (t) can be written as KN (t) = l Θl and when ∗ maxi |si| = s , we have

PT slQN (Θl) l=2 QN (KN (t)) √ < s∗ √ , (4.6.51) n1/2 n1/2

(1−p) from which, recalling that QN (KN (t)) ∼ log N (see (4.5.16)), and n1 ∼ 2 N, we conclude that the ﬁrst factor in the right-hand side of (4.6.50) converges to 1 ∗ as N → ∞. In order to deal with the second factor, and again with maxi |si| = s ,

PT ∗ KN (t) l=2 slΘl ∗ KN (t) QN exp − s √ ≤ QN exp √ ≤ QN exp s √ . n1/2 n1/2 n1/2

By convexity of x 7→ ex and Jensen’s inequality, we have that E[eX ] ≥ eE[X], so the lower bound is at least 1 − o(1) by (4.6.51).

60 4.6 Proofs for CMN (1, 2)

For the upper bound, we note that KN (t) is stochastically dominated by KN , the number of tori in a graph of N vertices of degree 2. Thus, recalling that KN is the sum of independent Bernoulli variables, see (4.5.8) and (4.5.9), √ N s/ n1/2 KN (t) Y e − 1 Q exp s√ ≤ 1 + . (4.6.52) N n 2 2N − 2i + 1 1/ i=1 We apply this to s = s∗ > 0 and consider √ N s/ n1/2 KN (t) X e − 1 0 ≤ log Q exp s√ = log 1 + N n 2 2N − 2i + 1 1/ i=1 √ N−1 X 1 ≤ es/ n1/2 − 1 , (4.6.53) 2j + 1 j=0

PN−1 1 where we use that log(1 + x) ≤ x for x > 0. Since j=0 2j+1 ∼ log N and √ q exp s/ n /2 − 1 ∼ 2 √s , we conclude that the right-hand side of the 1 1−p N last display converges to 0, as required.

Proof of Lemma 4.6.3. We split XN into two parts as √ k (1) X (N) (N) XN (k) = N pl − QN (pl ) γl(B), (4.6.54) l=2 (2) (1) XN (k) = XN − XN (k), (4.6.55) where we ﬁx k ∈ {2, ... , N − 1}. Then, we use Hölder’s inequality to bound

(1) (2) (1) (2) XN X (k) X (k) pX (k)1/p qX (k)1/q QN e = QN e N e N ≤ QN e N QN e N , (4.6.56) where p = 1 + ε and 1/p + 1/q = 1, so that q = (1 + ε)/ε. Further, with the same choices of p, q, we can also bound

(1) (2) (2) X (k)/p XN /p −X (k)/p XN 1/p X (k)(q/p)1/q QN e N = QN e e N ≤ QN e QN e N , so that

(1) (2) XN X (k)/pp X (k)(q/p)p/q QN e ≥ QN e N /QN e N . (4.6.57) We aim to prove that, for every b ∈ R and k ∈ N,

(1) bX (k) b2σ2 (k)/2 lim QN e N = e G , (4.6.58) N→∞ 2 2 2 2 where limk→∞ σG(k) = σG (we give the explicit expressions of σG(k) and σG at the end of the present section) and, for every b ∈ R,

bX(2)(k) lim sup lim sup QN e N = 1. (4.6.59) k→∞ N→∞

61 Averaged quenched measure

Substituting (4.6.58)–(4.6.59) into (4.6.56)–(4.6.57) and letting ε & 0 with p = 1 + ε, q = (1 + ε)/ε concludes the proof of Lemma 4.6.3. We continue with the proofs of (4.6.58) and (4.6.59).

Proof of (4.6.58). This is a direct consequence of Lemma 4.6.4.

Proof of (4.6.59). Here we need to show that we can eﬀectively truncate the sum over k. We start by eﬀectively going to the single variable case. We denote X γ>k = |γl|, ql = |γl|/γ>k. (4.6.60) l>k

Then, we note that,√ with Y the random variable for which P(Y = l) = ql and denoting bN = b N, we can rewrite

(2) n (N) (N) o bXN (k) QN e = QN exp bN γ>kEY sign(γY ) pY − QN (pY )

h n (N) (N) oi ≤ QN EY exp bN γ>ksign(γY ) pY − QN (pY ) , where the inequality follows by Jensen’s inequality and EY denotes the expectation w.r.t. Y (keeping all other randomness ﬁxed). Thus,

(N) (N) (2) b γ sign(γ ) p −Q (p ) bX (k) X N >k l l N l QN e N ≤ qlQN e . (4.6.61) l>k

(N) (N) Thus, it suﬃces to bound the moment generating function of pl − QN (pl ) for a single l. (N) We continue by studying this moment generating function. Denote Nl = Npl , so that

(N) (N) √ bN γ>ksign(γl) pl −QN (pl ) bγ>ksign(γl)(Nl−QN (Nl))/ N QN e = QN e .

For θ ∈ R, we let (l) θ(Nl−QN (Nl)) Mn1,n2 (θ) = QN e . (4.6.62)

We prove by induction on n1 ≥ 2 that there exists A, ε > 0 such that, uniformly in l, for all |θ| ≤ ε and all n2 ≥ 0,

2 (l) An1θ Mn1,n2 (θ) ≤ e . (4.6.63)

(l) For n1 = 0, Mn1,n2 (θ) ≡ 1. This initiates the induction hypothesis. To advance the induction hypothesis, we note that, with

pn1,n2 (k) = QN (L(1) = k), (4.6.64)

62 4.6 Proofs for CMN (1, 2)

(l) the moment generating function Mn1,n2 (θ) satisﬁes the recursion

(l) X (l) θ −θp (l) M (θ) = M (θ)p (k) 1l + 1l e e n1,n2 . n1,n2 n1−2,n2−k+2 n1,n2 {k6=l} {k=l} k≥2 (4.6.65)

We use the induction hypothesis, which is allowed since the summand in (4.6.65) is non-negative, to arrive at

2 X (l) A(n1−2)θ θ −θpn1,n2 (l) Mn1,n2 (θ) ≤ e pn1,n2 (k) 1l{k6=l} + 1l{k=l}e e k≥2 2 A(n1−2)θ θ −θpn1,n2 (l) = e 1 + (e − 1)pn1,n2 (l) e . (4.6.66)

Choose ε > 0 suﬃciently small, so that eθ ≤ 1 + θ + θ2 for all |θ| ≤ ε. Then,

2 (l) A(n1−2)θ 2 2 2 Mn1,n2 (θ) ≤ e 1 + (θ + θ )pn1,n2 (l) [1 − θpn1,n2 (l) + θ pn1,n2 (l) ] 2 A(n1−2)θ 2 2 ≤ e [1 + 4θ pn1,n2 (l) ]. (4.6.67) Choosing A > 0 suﬃciently large, we can advance the induction hypothesis uniformly in l√. We conclude that (4.6.63) holds. Applying (4.6.63) with θ = bγ>ksign(γl)/ N leads to

(2) 2 2 bX (k) X A(bγ>k) n1/N A(bγ>k) QN e N ≤ qle ≤ e , (4.6.68) l>k since (ql)l>k is a probability measure and n1 ≤ N. This completes the proof of (4.6.59), since γ>k is exponentially small in k.

2 The limiting variance σG. We conclude this section reporting the explicit 2 (1) computation of σG(k), the limiting variance of XN (k) appearing in (4.6.58) and 2 of σG, the limiting variance of XN , see (4.6.39). (1) (N) We express XN (k) as a linear combination of Λl = Npl recalling that the ∗ ∗ p n1 variables Λl in Lemma 4.6.4, are deﬁned as Λl = (Λl − QN (Λl)) / 2 . Sub- stituting in (4.6.54), we obtain

r k (1) 1 − p X X (k) = (1 + o(1)) γ (B)Λ∗, (4.6.69) N 2 l l l=2 and notice that the variance

k (1) 1 − p X (N)∗ Var (X (k)) = (1 + o(1)) γ2(B)Var (Λ ) QN N 2 l QN l l=2 k ! X (N)∗ (N)∗ + γl(B)γj (B)CovQN Λl , Λl (4.6.70) l,j=2 l6=j

63 Averaged quenched measure

has a limit as N → ∞. Indeed, using the fact that the functions γl(B) are independent of N, by Theorem 4.6.1,

k k ! 2 (1) 1 − p X 2 X σG(k) := lim VarQ (XN (k)) = γl (B)Hl,l + γl(B)γj (B)Hl,j , N→∞ N 2 l=2 l,j=2 l6=j (4.6.71) where Hl,j are the elements of the covariance matrix explicitly given in (4.6.44) 2p α l+j with α = 1−p . Because of the exponentially small factor ( 1+α ) in Hl,j, and the exponential smallness of γl(B) for large l, see (4.6.37), the sums in (4.6.71) 2 converge absolutely as k → ∞. Thus, we conclude that the limiting variance σG exists and is given by (4.6.39).

64 ANNEALEDMEASURE:UNIQUENESSREGIME 5

The main aim of this chapter is to prove a central limit theorem√ with respect to the annealed measure for the magnetization rescaled by N of the Ising model on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of the thermody- an namic limits and of a finite annealed inverse critical temperature 0 ≤ βc < ∞. Then we prove the law of large numbers and the central limit theorem, in the uniqueness regime. In the case of our configuration models, the central limit theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for these systems as they are closely related to the one-dimensional Ising model. For both these configuration models we focus our analysis first on the study of the thermodynamic limits of pressure and magnetization and then we prove the asymptotic theorems. The proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie–Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.

5.1 Thermodynamic limits, SLLN and CLT for GRGN (w)

The proofs of the SLLN and CLT for GRGN (w) require to investigate the uniqueness regime for GRGN (w). For this, we ﬁrst investigate the existence of the thermodynamic quantities in the inﬁnite volume limit with respect to the annealed law. These results will be obtained in the next theorem. They show, in particular, that annealing changes the critical inverse temperature. Indeed, the an annealed critical inverse temperature βc is strictly smaller than the quenched qu critical inverse temperature βc , when the latter exists. In the statement of the theorem below, we will use the notation U an for the annealed uniqueness regime, i.e.,

an an U := {(β, B) : β ≥ 0, B 6= 0 or 0 < β < βc , B = 0} . (5.1.1)

Theorem 5.1.1 (Thermodynamic limits for the annealed GRGN (w)). Let (GN )N≥1 be a sequence of GRGN (w) satisfying Condition 2.1.2. Then the following conclusions hold:

65 Annealed measure: uniqueness regime

(i) For all 0 ≤ β < ∞ and for all B ∈ R, the annealed pressure exists in the thermodynamic limit N → ∞ and is given by

ψe(β, B) := lim ψeN (β, B), (5.1.2) N→∞ its value is given in (5.4.14). (ii) For all (β, B) ∈ U an, the magnetization per vertex exists in the limit N → ∞, i.e.,

Mf(β, B) := lim MfN (β, B). (5.1.3) N→∞ ∂ For B 6= 0 the limit value Mf(β, B) equals Mf(β, B) = ∂B ψe(β, B) and is given by " s !# sinh (β) Mf(β, B) = E tanh W z∗ + B , (5.1.4) E [W ]

where z∗ = z∗(β, B) is the solution of the ﬁxed-point equation " s ! s # sinh (β) sinh (β) z = E tanh W z + B W (5.1.5) E [W ] E [W ]

and W is the limiting random variable deﬁned in Condition 2.1.2. (iii) The spontaneous magnetization is given by ( 0 if β ∈ U an Mf(β) := lim Mf(β, B) = B→0+ 6= 0 if β ∈/ U an and the annealed critical inverse temperature is an βc = asinh (1/ν) , (5.1.6) E[W 2] where ν, deﬁned in (2.1.2), is given by ν = E[W ] and W is the limiting random variable introduced in Condition 2.1.2. In particular, if ν > 1, then an qu βc < βc . (iv) For all (β, B) ∈ U an, the thermodynamic limit of the susceptibility exists and is given by ∂2 χ(β, B) := lim χN (β, B) = ψe(β, B). (5.1.7) e N→∞ e ∂B2

Remark 5.1.1. Theorem 5.1.1, holds for the inhomogeneous Curie–Weiss model

CWN (J) deﬁned in 2.6.1 with β replaced with asinh(β). For the rank-1 inhomogeneous Curie–Weiss model in the special case where wi ≡ 1, Theorem 5.1.1 reproves the classical result for the Curie–Weiss model. When the weights are inhomogeneous, the critical value is instead given by βc = 1/ν.

66 5.2 Thermodynamic limits, SLLN and CLT for CMN (2)

Having investigated the phase diagram of the annealed Ising model on the

GRGN (w), we next state the SLLN and CLT for the total spin in the following two theorems:

Theorem 5.1.2 (Annealed SLLN). Let (GN )N≥1 be a sequence of GRGN (w) graphs satisfying Condition 2.1.2 then, for all (β, B) ∈ U an, for any ε > 0 there exists a number L = L(ε) > 0 such that the total spin is exponentially concentrated in the form SN −NL PeN − Mf ≥ ε ≤ e for all suﬃciently large N, N where Mf = Mf(β, B) is the annealed magnetization deﬁned in (5.1.3).

Theorem 5.1.3 (Annealed CLT). Let (GN )N≥1 be a sequence of GRGN (w) graphs satisfying Condition 2.1.2. Then, for all (β, B) ∈ U an, the total spin satisﬁes a CLT of the form

SN − PeN (SN ) D √ −→ N (0, χ), w.r.t. PeN , as N → ∞, N e where χe = χe(β, B) is the thermodynamic limit of the annealed susceptibility deﬁned in Deﬁnition 2.3.2 and N (0, σ2) denotes a centered normal random variable with variance σ2.

5.2 Thermodynamic limits, SLLN and CLT for CMN (2)

As ﬁrst step, we present our results concerning the thermodynamic limits for the CMN (2). From the analysis of the limit of the magnetization, we remark that there is no annealed phase transition for CMN (2). Then we can state a SLLN and a CLT for total spin for each choice of the parameters β and B. It turns out that the central limit theorem under the annealed measure again has a variance that coincides with the random quenched susceptibility, as in the random quenched and averaged quenched settings.

Theorem 5.2.1 (Thermodynamic limits for the annealed law for CMN (2)). Let (GN )N≥1 be a sequence of CMN (2) graphs. Then, for all β > 0, B ∈ R, the following hold:

(i) The annealed pressure exists in the thermodynamic limit N → ∞ and is given by

ψe(β, B) := lim ψeN (β, B) = log λ+(β, B), (5.2.1) N→∞ where q β 2 −4β λ+(β, B) = e cosh(B) + sinh (B) + e . (5.2.2)

67 Annealed measure: uniqueness regime

(ii) The magnetization per vertex exists in the limit N → ∞ and is given by

sinh(B) Mf(β, B) := lim MfN (β, B) = q . (5.2.3) N→∞ sinh2(B) + e−4β

Remark 5.2.1. Since limB→0+ Mf(β, B) = 0 for all β > 0, by deﬁnition (2.4.1) we conclude that there is no annealed phase transition for CMN (2). This is not surprising, since CMN (2) consists of a collection of disjoint cycles, and the Ising model does not have a phase transition in dimension one.

Next we state the SLLN and the CLT for the total spin in CMN (2):

Theorem 5.2.2 (Annealed SLLN for CMN (2)). Let (GN )N≥1 be a sequence of CMN (2) graphs. Then, for all β ≥ 0, B ∈ R, for any ε > 0 there exists a number L = L(ε) > 0 such that the total spin is exponentially concentrated in the form SN −NL PeN − Mf ≥ ε ≤ e for all suﬃciently large N, N where Mf = Mf(β, B) is the annealed magnetization deﬁned in (5.2.3).

Theorem 5.2.3 (Annealed CLT for CMN (2)). Let (GN )N≥1 be a sequence of CMN (2) graphs. Then, for all β ≥ 0, B ∈ R, the total spin satisﬁes a CLT of the form

SN − PeN (SN ) D √ −→ N (0, χ) , w.r.t. PeN , as N → ∞, N where χ = χ(β, B) is the thermodynamic limit of the random quenched susceptibility of the Ising model on CMN (2). Moreover, χ(β, B) is also equal to the susceptibility of the one-dimensional Ising model, i.e.,

cosh(B)e−4β χ(β, B) = χd=1(β, B) = . (5.2.4) (sinh(B) + e−4β )3/2

5.3 Thermodynamic limits, SLLN and CLT for CMN (1, 2)

Our main result for CMN (1, 2) again concerns its thermodynamic limits, a SLLN and a CLT for its total spin. In particular, we present two different computations for the limit of the pressure, the second one based on a result by Broutin and de Panafieu [14]. Regarding the central limit theorem, we show that the varying degrees of the vertices of CMN (1, 2) affect the limiting distribution, thus quantifying the role of inhomogeneity in the Ising model. In fact, the limiting variance is the sum of that of the one-dimensional Ising model and of an extra term emerging from the fluctuations of the connected components of the graph.

68 5.3 Thermodynamic limits, SLLN and CLT for CMN (1, 2)

Theorem 5.3.1 (Thermodynamic limits for the annealed law for CMN (1, 2)). Let (GN )N≥1 be a sequence of CMN (1, 2) graphs for a given p ∈ (0, 1). Then, for all β > 0, B ∈ R, the following hold:

(i) The annealed pressure exists in the thermodynamic limit N → ∞ and is given by

ψe(β, B) : = lim ψeN (β, B) N→∞ 1 − p = log λ (β, B) + log A (β, B) + H(s∗, t∗). (5.3.1) + 2 +

1−p where A+(β, B) is deﬁned in (4.5.6), the function H : [0, 2 ] × [0, p] → R is deﬁned in (5.6.38) below, and (s∗, t∗) is the unique maximizer of 1−p (s, t) 7→ H(s, t) on [0, 2 ] × [0, p]. (ii) The magnetization per vertex exists in the limit N → ∞, i.e.,

∂ Mf(β, B) := lim MfN (β, B) = ψe(β, B), (5.3.2) N→∞ ∂B and is given in (5.6.83) below.

Remark 5.3.1. Since limB→0+ Mf(β, B) = 0 for all β > 0 (the explicit expression of the magnetization is given in (5.6.83)), we conclude that there is no annealed phase transition for CMN (1, 2). Again this is not surprising, since

CMN (1, 2) consists of a collection of one-dimensional lines and cycles, and the one-dimensional Ising model does not have a phase transition.

In the next theorem we present a different way to obtain the limit of the annealed pressure. In [14] Broutin and de Panafieu proved a CLT for the number of lines of given lengths in CMN (1, 2). Leveraging on this result, we proved the averaged quenched CLT for the total spin of the Ising model on CMN (1, 2) in Chapter 4. We apply the result of [14] to compute also the annealed pressure (2.3.7) as well, but we obtain a result different from (5.3.1). While we are able to see numerically that the two formulas agree, we have no analytic proof that they coincide.

Theorem 5.3.2 (Limit for the annealed pressure for CMN (1, 2): way 2). Let (GN )N≥1 be a sequence of CMN (1, 2) graphs for a given p ∈ (0, 1). Then, for all β > 0, B ∈ R, the annealed pressure exists in the thermodynamic limit N → ∞ and is given by

(1 − p) ψe(β, B) = log λ+ + α log(α) − (1 + α) log(1 + α) + log(A+) 2 (1 − p) h 2p i + log (1 + arB(ζ)) − log(1 − ζ) − log(ζ) , (5.3.3) 2 1 − p

69 Annealed measure: uniqueness regime where α = 2p/(1 − p), 1 − y B(y) = r (5.3.4) 1 − ry and ζ is the unique solution in (0, 1) of

y 1 + arB(y)2 2p = . (5.3.5) 1 − y 1 + arB(y) 1 − p

We next state the SLLN and the CLT for the total spin in CMN (1, 2):

Theorem 5.3.3 (Annealed SLLN for CMN (1, 2)). Let (GN )N≥1 be a sequence of CMN (1, 2) graphs. Then, for all β ≥ 0, B ∈ R, for any ε > 0 there exists a number L = L(ε) > 0 such that the total spin is exponentially concentrated in the form SN −NL PeN − Mf ≥ ε ≤ e for all suﬃciently large N, N where Mf = Mf(β, B) is the annealed magnetization deﬁned in (5.3.2).

Theorem 5.3.4 (Annealed CLT for CMN (1, 2)). Let (GN )N≥1 be a sequence of CMN (1, 2) graphs. Then, for all β > 0, B ∈ R,

SN − PeN (SN ) D 2 √ −→ N 0, σ , w.r.t. PeN , as N → ∞, N 2

2 where σ2 is deﬁned in (5.6.89) below. 2 Remark 5.3.2. By observing the expression of σ2 in (5.6.89), we note that it corresponds to the double derivative with respect to the external ﬁeld of the annealed 2 pressure in (5.3.1). Therefore, the variance σ2 of the central limit theorem for CMN (1, 2) should be equal to the thermodynamic limit of the susceptibility of the model, as for the GRGN (w) and CMN (2) cases. Unfortunately, we are not able 2 2 to prove that χ β, B ∂ ψ β, B , converges to χ β, B ∂ ψ β, B , eN ( ) = ∂B2 eN ( ) e( ) = ∂B2 e( ) ∂ because we cannot prove the monotonicity of ∂B MfN (β, B) in order to apply Lemma 4.4.1 that allows the interchange between limit and derivative.

5.4 Proofs for GRGN (w)

As we announced before, the study of the Ising model on the generalized random graph under the annealed measure is reduced to the analysis of an inhomogeneous Curie–Weiss model on the complete graph. The inhomogeneous Curie–Weiss model is a generalization of the classical Curie–Weiss model in which the strength of the ferromagnetic interaction between spins is not spatially uniform (see Deﬁnition 2.6.1). This mapping arises from the computation of the annealed partition function, exploiting the independence of the edges in the

70 5.4 Proofs for GRGN (w) generalized random graph. After the proof of the thermodynamic limits, we demonstrate the strong law of large numbers and the central limit theorem using the general strategy adopted for the locally tree-like random graphs in the random quenched setting. Indeed, looking at our model as a Curie–Weiss model we are again allowed to use the GHS inequality.

5.4.1 Annealed thermodynamic limits: Proof of Theorem 5.1.1

To prove the existence of the limit of the thermodynamical quantities, we start by studying the annealed partition function. The proof is divided into several steps.

Annealed partition function. We start by analyzing the average of the partition function for GRGN (w). By remembering that in this random graph the edges are independent and denoting by Iij the Bernoulli indicator that the edge between vertex i and vertex j is present, we compute

X h X X i QN (ZN (β, B)) = QN exp β Iijσiσj + B σi

σ∈ΩN i

σ∈ΩN B P σ X i∈[N] i Y βIij σiσj = e QN e

σ∈ΩN i

−β −βij β βij e pij + (1 − pij ) = Cije , and e pij + (1 − pij ) = Cije .

Now, by adding and dividing the two equations of the system above, we get

Cij cosh (βij ) = pij cosh(β) + (1 − pij ) ,

β 1 e pij + (1 − pij ) βij = log −β . 2 e pij + (1 − pij )

71 Annealed measure: uniqueness regime

Then, using the symmetry βij = βji we arrive at

B P σ P β σ σ Y X i∈[N] i i

σ∈ΩN

Q Q −βii/2 where G(β) = i

CWN (J), deﬁnitined in 2.6.1, where the coupling constant Jij (β) between vertices i and j is equal to βij. Our proof below shows that the coupling βij is close to the form in (2.6.4) with β replaced by sinh(β), so that the study of the annealed generalized random graph reduces to the rank-1 inhomogeneous Curie–Weiss model. In the next step, we analyze this partition function in detail.

Towards an inhomogeneous Curie–Weiss model. We continue by showing that βij is close to factorizing into a contribution due to i and to j. For this, by a Taylor expansion of x 7→ log(1 + x), 1 1 β = log 1 + p (eβ − 1) − log 1 + p (e−β − 1) ij 2 ij 2 ij 1 1 = p (eβ − 1) − p (e−β − 1) + O(p2 ) = sinh(β)p + O(p2 ). (5.4.4) 2 ij 2 ij ij ij ij Then,

QN (ZN (β, B)) P 1 sinh(β) P p σ σ +O P p2 σ σ X B σi 2 i,j∈[N] ij i j i,j∈[N] ij i j = G2(β) e i∈[N] e .

σ∈ΩN where G2(β) = G(β)G1(β). To control the error in the exponent, we use pij ≤ wiwj /`N and the assumptions in Condition 2.1.2, to obtain

P 2 2 w w 2 w X 2 X i j i∈[N] i pijσiσj ≤ = = o(N). (5.4.5) `N `N i,j∈[N] i,j∈[N] Then,

w w B P σ 1 sinh(β) P i j σ σ o(N) X i∈[N] i 2 i,j∈[N] ` i j QN (ZN (β, B)) = G2(β)e e e N

σ∈ΩN 2 1 sinh(β) P P wiσi o(N) X B σi 2 `N i∈[N] = G2(β)e e i∈[N] e .

σ∈ΩN

72 5.4 Proofs for GRGN (w)

When wi ≡ w for all i, so that GRGN (w) is the Erdős–Rényi random graph, we retrieve the Curie–Weiss model at inverse temperature β0 = sinh(β)w. In our inhomogeneous setting, we obtain an inhomogeneous Curie–Weiss model that we will analyze next.

Analysis of the inhomogeneous Curie–Weiss model. We use the Hub- 2 bard–Stratonovich identity, i.e., we write et /2 = E[etZ ], with Z standard Gaussian. Then, we ﬁnd

q sinh(β) P B P σ h wiσi Z i o(N) X i∈[N] i `N i∈[N] QN (ZN (β, B)) = G2(β)e e E e

σ∈ΩN N s h Y sinh (β) i = G (β)eo(N)2N E cosh w Z + B 2 ` i i=1 N N s h n X sinh(β) oi = G (β)eo(N)2N E exp log cosh w Z + B . 2 ` i i=1 N

We rewrite the sum in the exponential, using the fact that WN = wVN , where we recall that VN is a uniform vertex in [N], to obtain

QN (ZN (β, B)) s h n h sinh(β) ioi o(N) N = G2(β)e 2 E exp NE log cosh WN Z + B Z NE[WN ] NF √Z o(N) N N N = G2(β)e 2 E e , (5.4.6) where s h sinh (β) i FN (z) = E log cosh WN z + B . (5.4.7) E [WN ]

Here we emphasize the fact that in (5.4.7), the expectation is w.r.t. WN only.

We continue by analyzing FN (z). We claim that, uniformly for |z| ≤ a and any a < ∞,

sup |FN (z) − F (z)| = o(1), (5.4.8) |z|≤a where s h sinh (β) i F (z) = E log cosh W z + B . (5.4.9) E[W ]

To see (5.4.8), we note that FN (z) → F (z) for every z ﬁxed by Condition 2.1.2(a)-(b), and the fact that log cosh(x) ≤ |x|. Further, s 0 sinh(β) h sinh (β) i |FN (z)| ≤ E tanh WN z + B WN ≤ sinh(β), E[WN ] E[WN ]

73 Annealed measure: uniqueness regime

0 since tanh(x) ≤ 1 for all x, so that |FN (z)| is uniformly bounded in N and z. Therefore, (FN )N≥1 forms a uniformly equicontinuous family of functions, so that (5.4.8) follows from Arzelà–Ascoli. Since FN (z) ≤ sinh(β)|z|, it further follows that, for a > 4 sinh(β), Z √ NFN √ h i N √ N sinh(β)|Z| √ E e 1l{|Z|>a N} ≤ E e 1l{|Z|>a N} h √ i N sinh(β)Z √ = 2E e 1l{Z>a N} Z ∞ √ 2 N sinh(β)z −z2/2 = √ √ e e dz 2π a N ∞ √ 2 Z ≤ ea sinh(β)N−a N/2 e N(sinh(β)−a)xdx 0 2 ≤ e−a N/4, (5.4.10) Z h NFN √ i N √ which, for a suﬃciently large, is negligible compared to E e 1l{|Z|≤a N} . We conclude that NF √Z o(N) N h N i QN (ZN (β, B)) = G2(β)e 2 E e . (5.4.11)

A large deviation analysis. The expectation in (5.4.11) is an expectation of an exponential functional,√ to which we apply large deviation machinery. The Gaussian variable Z/ N satisﬁes a large deviation principle with rate √ 2 d 1 function I(z) = z /2 and speed N, because Z/ N = N (Z1 + ··· + ZN ), where (Zi)i∈[N] are i.i.d. standard Gaussian variables. Using Varadhan’s Lemma and the fact that z 7→ F (z) is continuous, we calculate the thermodynamic limit of the pressure as 1 lim log QN (ZN (β, B)) N→∞ N 1 = log 2 + lim log G2(β) + sup [F (z) − I(z)] N→∞ N z s h h sinh(β) i z2 i = log 2 + α (β) + sup E log cosh W z + B − . (5.4.12) z E [W ] 2

1 where α (β) = limN→∞ N log G2 (β). The equation that deﬁnes the supremum is s s h sinh (β) sinh (β) i z∗ = z∗(β, B) = E tanh W z∗ + B W , (5.4.13) E[W ] E[W ] and the annealed pressure is obtained by substituting the supremum point z∗ in the right hand side of (5.4.12) as

74 5.4 Proofs for GRGN (w)

s h sinh (β) i ψe(β, B) = log 2 + α(β) + E log cosh W z∗(β, B) + B E[W ] z∗(β, B)2 − . (5.4.14) 2 This completes the proof of Theorem 5.1.1(i).

an The critical inverse temperature. To identify βc as stated in Theorem 5.1.1(ii), we evaluate (5.4.13) when B & 0 to obtain s s sinh (β) sinh (β) z∗ = H(z∗) where H(z) = E tanh W z W . E [W ] E [W ] (5.4.15) We investigate the solutions of z∗ = H(z∗) in (5.4.15). We note that z 7→ H(z) is an increasing and concave function in [0, ∞). When H0(0) > 1, we have three solutions of (5.4.15), i.e., ±z∗ and 0, where z∗ = z∗(β, 0+) > 0. When H0(0) ≤ 1, instead, z∗ = 0 is the only solution. This leads us to compute

E W 2 H0(0) = sinh (β) = sinh (β) ν. (5.4.16) E [W ]

an an Thus, the annealed critical temperature βc satisﬁes sinh (βc ) = 1/ν. Since qu qu an tanh (βc ) = 1/ν, and tanh(x) < sinh(x) ∀ x > 0, we obtain βc > βc , unless an qu when ν = ∞, in which case βc = βc = 0.

Thermodynamic limit of the magnetization. To prove the existence of the magnetization in the thermodynamic limit stated in Theorem 5.1.1(ii), we follow the strategy used by Dommers, Giardinà and van der Hofstad in [21]. We apply Lemma 4.4.1 with n = N and fn equal to B 7→ ψeN (β, B). We verify the conditions in Lemma 4.4.1 and start by noting that

∂ MfN (β, B) = PeN SN /N = ψeN (β, B), (5.4.17) ∂B and limN→∞ ψeN (β, B) = ψe(β, B) by Theorem 5.1.1(i) with B 7→ MfN (β, B) non-decreasing:

∂ 1 h 2 2i MfN (β, B) = PeN S − PeN (SN ) ≥ 0. (5.4.18) ∂B N N Thus, we can indeed conclude that ∂ ∂ Mf(β, B) = lim MfN (β, B) = lim ψeN (β, B) = ψe(β, B). (5.4.19) N→∞ N→∞ ∂B ∂B

75 Annealed measure: uniqueness regime

The limit magnetization Mf(β, B) can be explicitly computed by taking the derivative of ψe(β, B), (5.4.14) and using the ﬁxed point equation (5.4.13), to obtain s sinh (β) Mf(β, B) = E tanh W z∗ + B . (5.4.20) E [W ]

Thermodynamic limit of the susceptibility. Finally, the thermodynamic limit of the susceptibility in Theorem 5.1.1(iv) is proved using Lemma 4.4.1 ∂ by combining Theorem 5.1.1(ii) and the fact that B 7→ ∂B MfN (β, B) is non- increasing by the GHS inequality. Indeed, by the explicit computation in (5.4.3), we see that the annealed partition function can be viewed as the partition function of an inhomogeneous Curie-Weiss model, where the ﬁeld is homogeneous and the coupling constants depend on the edges. Since such an inhomogeneous Ising model also satisﬁes the GHS inequality, the same follows for the annealed partition function for GRGN (w). Therefore,

∂2 1 X ∂2 MfN (β, B) = PeN (σi) ≤ 0. (5.4.21) ∂B2 N ∂B2 i∈[N]

5.4.2 Annealed SLLN: Proof of Theorem 5.1.2

With Theorem 5.1.1 in hand, we now have all the hypotheses to prove Theorem 5.1.2 following the strategy used for the random quenched setting in Section 3.2.2 of Chapter 3 verbatim. Indeed, for the proof of the annealed SLLN, referring to Section 3.2.2, we obtain the existence of the thermodynamic limit of the annealed cumulant generating function 1 cN (t) = log PeN [exp (tSN )] = ψeN (β, B + t) − ψeN (β, B) (5.4.22) e N by Theorem 5.1.1(i). Then, from [26, Theorem II.6.3] and Theorem 5.1.1(ii) we conclude the proof.

5.4.3 Annealed CLT: Proof of Theorem 5.1.3

To prove the annealed CLT we mimic again the strategy used for the random quenched measure (see Section 3.2.3 for the proof of the random quenched CLT). We use the existence in the thermodynamic limit of the pressure, the magnetization and the susceptibility given by Theorem 5.1.1 together with the GHS inequality that is still true in the annealed setting thanks to the mapping to the inhomogeneous Curie–Weiss model.

76 5.5 Proofs for CMN (2)

5.5 Proofs for CMN (2)

In this section we prove the CLT with respect to the annealed measure for the 2-regular random graph. We start by computing the annealed pressure using the partition functions for the one-dimensional Ising model with periodic boundary conditions.

5.5.1 Annealed thermodynamic limits: Proof of Theorem 5.2.1

From Section 4.5 in Chapter 4, we remember that any 2-regular random graph is formed by cycles only. Thus, as in Section 4.5, denoting the random number t of cycles in the graph by KN , we can enumerate them in an arbitrary order from t 1 to KN and call LN (i) the length (i.e., the number of vertices) of the ith cycle. t The random variable KN has distribution given by

N t X KN = Ij, (5.5.1) j=1

1 where Ij are independent Bernoulli variables given by Ij = Bern 2N−2j+1 . See Section 4.5 for a proof of this fact. Since the random graph splits into (disjoint) cycles, its quenched partition function factorizes into the product of the partition functions of each cycle. Therefore, t KN Y (t) ZN (β, B) = Z (β, B). (5.5.2) LN (i) i=1 We remember that the partition function of the one-dimensional Ising model (t) with periodic boundary conditions ZN is given by

(t) N N ZN (β, B) = λ+ (β, B) + λ− (β, B), (5.5.3) where q β 2 −4β λ±(β, B) = e cosh(B) ± sinh (B) + e . (5.5.4)

Then, following the same calculation of Section 4.5 we obtain

N Kt N λ+ (β, B) ≤ ZN (β, B) ≤ 2 N λ+ (β, B). (5.5.5)

The thermodynamic limit of the annealed pressure ψeN (β, B), deﬁned in (2.3.7), can be computed along the same lines of the averaged quenched one in Section −1 4.5. Indeed, by applying the monotone operator N log(QN (·)) to (5.5.5) and using the fact that λ+(β, B) is non random, we obtain

1 Kt log λ+(β, B) ≤ ψeN (β, B) ≤ log QN 2 N + log λ+(β, B). (5.5.6) N

77 Annealed measure: uniqueness regime

Now using the fact

N 1 Kt 1 Y I log Q 2 N = log Q 2 i N N N N i=1 N 1 Y 2 1 = log + 1 − N 2N − 2i + 1 2N − 2i + 1 i=1 N 1 X 1 N→∞ = log 1 + −→ 0, (5.5.7) N 2N − 2i + 1 i=1 we conclude that the annealed pressure of CMN (2) coincides with the pressure of the one-dimensional Ising model, i.e.,

d=1 ψe(β, B) = ψ (β, B) ≡ log λ+(β, B). (5.5.8)

Therefore, it also agrees with the averaged and random quenched pressures, i.e.,

ψe(β, B) = ψ(β, B) = ψ(β, B). (5.5.9)

It also straightforwardly follows that the annealed cumulant generating function of CMN (2) coincides with the random and averaged quenched ones, i.e.,

ec(t) = c(t) = c(t) = log λ+(β, B + t) − log λ+(β, B). (5.5.10) The existence of the magnetization in the thermodynamic limit (Theorem 5.2.1(ii)) can be proved, as in the previous section, using Lemma 4.4.1 and the existence of the thermodynamic limit of the pressure (5.5.8), so we obtain

∂ sinh(B) Mf(β, B) = ψe(β, B) = q , (5.5.11) ∂B sinh2(B) + e−4β as required.

5.5.2 Annealed SLLN: Proof of Theorem 5.2.2

The proof, as in Theorem 5.1.2 for the generalized random graph, is based again on the same argument of the random quenched setting (see Section 3.2.2), following immediately from the existence of the annealed pressure in the thermodynamic limit and its diﬀerentiability with respect to B.

5.5.3 Annealed CLT: Proof of Theorem 5.2.3

To prove the CLT in the annealed setting, we adopt the strategy for the averaged quenched CLT on CMN (2) in Section 4.5.4.

78 5.5 Proofs for CMN (2)

Rewrite in terms of cumulant generating functions. Using the annealed cumulant generating function and using a Taylor expansion, we write 2 tSN − tPeN (SN ) t 00 log PeN exp √ = c (tN ), (5.5.12) N 2 eN √ 00 where tN ∈ [0, t/ N]. Then the aim is to prove that limN→∞ ecN (tN ) exists as a ﬁnite limit. (t) N N By expressing ecN (t) in terms of ZN = λ+ + λ− and using (5.5.10), we can compute the diﬀerence as

(t) Z (β,B+t) Kt N Q N LN (i) QN N QN 1 + (rB+t) 1 (λ+(β,B+t)) 1 i=1 cN (t) − c(t) = log = log t , e e N Z(t)(β,B) N K N Q Q N 1 + (r )LN (i) QN N N i=1 B (λ+(β,B)) where, as in Chapter 4, we have deﬁned

λ−(β, B) rB = r(β, B) = . (5.5.13) λ+(β, B) Then Kt Q N LN (i) 1 QN i=1 1 + (rB+t) c (t) = log λ (β, B + t) − log λ (β, B) + log . eN + + Kt N Q N L (i) QN i=1 (1 + (rB ) N ) (5.5.14) Our aim is to show that the double derivative arises from the ﬁrst term only, the second derivative of the last term vanishes.

Computation of the second derivative of the cumulant generating function. The second derivative of (5.5.14) is

2 " # 00 ∂ 1 IIIgN (t) ecN (t) = 2 log λ+(β, B + t) + IeN (t) + IIfN (t) + , ∂t NDeN (t) DeN (t) (5.5.15) where

Kt N X LN (i)−2 0 2 00 IeN (t) = QN LN (i)(rB+t) [(LN (i) − 1)(rB+t) + rB+trB+t] i=1 Kt N Y LN (j) · 1 + (rB+t) , j=1 j6=i

Kt Kt N N X LN (i)+LN (j)−2 0 2 Y LN (l) IIfN (t) = QN LN (i)LN (j)(rB+t) (rB+t) 1 + (rB+t) , i,j=1 l=1 j6=i l6=i,j

79 Annealed measure: uniqueness regime

Kt Kt N N 2 X LN (i)−1 0 Y LN (j) IIIgN (t) = QN LN (i)(rB+t) rB+t 1 + (rB+t) , i=1 j=1 j6=i

Kt N Y LN (i) DeN (t) = QN 1 + (rB+t) . i=1

Uniform bound on the averaged normalized partition function. To analyze the contributions above we show that the averaged normalized partition function of CMN (2) is uniformly bounded: Lemma 5.5.1 (The partition function on tori). For every γ < 1 and α ∈ (0, ∞), there exists a constant A = A(α, γ) such that, uniformly in N,

t " KN # Y LN (i) QN 1 + αγ ≤ A. (5.5.16) i=1

Kt Q N LN (i) Proof. Denote ZN = QN i=1 1 + αγ . For the proof we use induction in N. The induction hypothesis is that there exists an A > 1 such that 1 ZN ≤ A 1 − √ . (5.5.17) 2 3 N + 1 Fix M ≥ 1 large. We note that we can fix A so large that the inequality is trivially satisfied for N ≤ M. To advance the induction hypothesis we first derive a recursion relation for ZN . We have

t N K X YN LN (i) ZN = QN (LN (1) = l) QN 1 + αγ LN (1) = l l=1 i=1 N X l = QN (LN (1) = l) 1 + αγ ZN−l. (5.5.18) l=1

Kt Q N LN (i) Indeed, the average of i=1 1 + αγ conditioned on LN (1) = l, reduces to the average on a CMN (2) graph with N − l vertices of a similar product. This average gives rise to the factor ZN−l in (5.5.18), while the term corresponding to the ﬁrst cycle is factorized, being 1 + αγl. Substituting the induction hypothesis into (5.5.18) leads to

N X l 1 ZN ≤ A QN (LN (1) = l) 1 + αγ 1 − √ (5.5.19) 2 3 N − l + 1 l=1 N N X l X 1 ≤ A QN (LN (1) = l) 1 + αγ − A QN (LN (1) = l) √ . 2 3 N − l + 1 l=1 l=1

80 5.5 Proofs for CMN (2)

It is not hard to see that

N X l QN (LN (1) = l)γ ≤ c/(N + 1), (5.5.20) l=1 while there exists a constant θ > 1 such that

N X 1 θ QN (LN (1) = l) √ ≥ √ . (5.5.21) 3 N − l + 1 3 N + 1 l=1

D Indeed, by [65, Exercise 4.1], or an explicit computation, LN (1)/N −→ T , where

T has density fT (x) given by 1 f (x) = √ . (5.5.22) T 2 1 − x Therefore, rewriting the sum in (5.5.21) we have

N " # X 1 1 1 QN (LN (1) = l) √ = √ QN , 3 N − l + 1 3 N + 1 p3 l=1 1 − LN (1)/(N + 1) and by Fatou’s Lemma and weak convergence, we obtain " # 1 1 lim inf Q ≥ E √ > 1. (5.5.23) N p3 3 N→∞ 1 − LN (1)/(N + 1) 1 − T

Since we can assume that N ≥ M, which is suﬃciently large, we thus obtain (5.5.21). Thus, c θ 1 ZN ≤ A 1 + − √ ≤ A 1 − √ , (5.5.24) N + 1 2 3 N + 1 2 3 N + 1 when N is suﬃciently large. This advances the induction hypothesis and completes the proof of the lemma.

Analysis of the second derivative of the cumulant generating function. Armed with Lemma 5.5.1, it is now easy to show that all the contributions in the second term of the r.h.s. of (5.5.15) indeed vanish for a sequence t = o(1). To see this, let t > 0 and (t ) a sequence of real N √ N N≥1 numbers such that tN ∈ [0, t/ N]. We consider ﬁrst the term IeN (tN ). As in Lemma 4.5.1 in Section 4.5.4, there exists a constant C > 0 such that

t KN X L (i)−2 0 2 L (i)−1 00 L (i)(L (i) − 1)(r ) N (r ) + L (i)(r ) N r N N B+tN B+tN N B+tN B+tN i=1 t ≤ C · KN , (5.5.25)

81 Annealed measure: uniqueness regime

since rB+tN < 1. Then, using the Cauchy-Schwarz inequality and recalling that 0 < r < 1, we obtain

Kt N t Y LN (i) |IeN (tN )| ≤ C · QN KN 1 + (rB+tN ) i=1 t KN 21/2 2 1/2 t Y LN (i) ≤ C · QN KN · QN 1 + (rB+tN ) i=1 t KN 2 1/2 1/2 t Y LN (i) ≤ CQN KN · QN 1 + 3(rB+tN ) . (5.5.26) i=1

Using Lemma 5.5.1 with α = 3 and γ = rB+tN we conclude that

t KN Y 1/2 1 LN (i) 2 QN 1 + 3(rB+tN ) ≤ A . (5.5.27) i=1

Finally, since DeN (tN ) ≥ 1,

1 |IeN (tN )| C · A 2 · log N N→∞ ≤ −→ 0. (5.5.28) NDeN (tN ) N

Similar computations allow us to estimate IIfN (tN ) and IIIgN (tN ) to obtain

IIfN (tN ) IIIgN (tN ) lim = 0, lim = 0 . (5.5.29) N→∞ N→∞ 2 NDeN (tN ) N DeN (tN )

Completion of the proof of Theorem 5.2.3. Having proved that " # 1 IIIgN (tN ) lim IeN (tN ) + IIfN (tN ) + = 0 , (5.5.30) N→∞ NDeN (tN ) DeN (tN ) the combination of (5.5.12) and (5.5.15) yields the proof of the annealed CLT, i.e., " !# S − P (S ) t2 ∂2 N √eN N lim log PeN exp t = 2 log λ+(β, B + t) N→∞ N 2 ∂t t=0 t2 cosh(B)e−4β = . 2 (sinh(B) + e−4β )3/2

Therefore, we conclude that the annealed CLT has the same variance as in averaged quenched case, i.e., the variance in both cases is the susceptibility of the one-dimensional Ising model.

82 5.6 Proofs for CMN (1, 2)

5.6 Proofs for CMN (1, 2)

In this section, we consider the Configuration Model CMN (1, 2), already studied in the averaged quenched setting. We first recall some of its features. The connected components of this graph are either cycles or tori (which we indicate by a superscript (t)) connecting vertices of degree 2, or lines (indicated by a superscript (l)) having vertices of degree 2 between two vertices of degree 1. In order to state some properties of the number of lines and tori, we remember some notation introduced in Section 4.6. By taking p ∈ (0, 1), let us define the number of vertices of degree 1 and 2 by

n1 := # {i ∈ [N] : di = 1} = N − bpNc, (5.6.1)

n2 := # {i ∈ [N] : di = 2} = bpNc, (5.6.2) and the total degree of the graph by X `N = di = 2n2 + n1 = N + bpNc. (5.6.3) i∈[N]

Then, the number of edges is given by `N /2. Let us also denote by KN the (l) (t) number of connected components in the graph and by KN and KN the number lines and tori. Obviously,

(l) (t) KN = KN + KN . (5.6.4)

Because every line uses up two vertices of degree 1, the number of lines is given (l) by n1/2, i.e., KN = (N − bpNc)/2 a.s. Regarding the number of cycles, we have (t) t that KN has the same distribution of KN¯ , where N is the (random) number of t vertices with degree 2 that do not belong to any line and KN¯ is the number of tori on this set of vertices. Then, since this subset forms a CMN¯ (2) graph, we (t) P can apply (4.5.17) from Section 4.5, obtaining that KN /N −→ 0, so that also

P KN /N −→ (1 − p)/2. (5.6.5)

Denoting the length (i.e. the number of vertices) in the ith line and jth torus (l) (t) (for an arbitrary labeling) by LN (i) and LN (j), the partition function can be computed as

(l) (t) KN KN Y (l) Y (t) ZN (β, B) = Z (β, B) · Z (β, B), (5.6.6) L(l)(i) L(t)(i) i=1 N i=1 N where, by (4.5.2),

(t) (t) (t) LN (i) LN (i) Z (t) (β, B) = λ+ (β, B) + λ− (β, B), (5.6.7) LN (i)

83 Annealed measure: uniqueness regime while the partition function on each line is obtained using the partition function of the one-dimensional Ising model with free boundary conditions, (see (4.5.5)), as (l) (l) (l) LN (i) LN (i) Z (l) (β, B) = A+λ+ + A−λ− , (5.6.8) LN (i) where −2β ±B β+B 2 ∓B −β β+B e e + (λ+ − e ) e ± 2e (λ+ − e ) A±(β, B) = −2β β+B 2 . (5.6.9) [e + (λ+ − e ) ]λ± This is the starting point of our analysis of the annealed Ising measure on

CMN (1, 2).

5.6.1 Annealed partition function

In order to prove the thermodynamic limits, the SLLN and CLT in the annealed setting, an analysis of the annealed partition function is necessary. Omitting the dependence on β, we start by writing

(N) NFB (p )+NEN (B) QN [ZN (B)] = QN [e ] (5.6.10) where, by Chapter 4,

(N) X (N) l FB (p ) = log λ+(B) + pl log A+(B) + A−(B) (r(B)) , (5.6.11) l≥2 (t) KN 1 X L(t)(i) E (B) = log 1 + r(B) N , (5.6.12) N N i=1

(N) (N) with r(B) = rB deﬁned in (5.5.13) and p = p the empirical distribu- l l≥2 tion of the line lengths given by

(l) KN (N) 1 X p := 1 (l) . (5.6.13) l N {L (i)=l} i=1 N We have ∞ (N) Nl NFB (p ) N Y l e = (λ+(B)) A+(B) + A−(B) (r(B)) l=2 ∞ ∞ Nl N Y Nl Y l = (λ+(B)) (A+(B)) 1 + a(B) (r(B)) . (5.6.14) l=2 l=2 where A (B) a(B) = − (5.6.15) A+(B)

84 5.6 Proofs for CMN (1, 2)

(N) and Nl = Npl is the number of lines of length l. We rewrite the second factor in (5.6.14) as ∞ Y Nl n1/2 (A+(B)) = (A+(B)) , (5.6.16) l=2 P since l≥2 Nl = n1/2. Therefore, we arrive at

∞ (N) Nl NFB (p ) N n1/2 Y l e = λ+ (B)A+ (B) 1 + a(B)r (B) l=2 ∞ N n1/2 Y Nl = λ+ (B)A+ (B) cl(B) , (5.6.17) l=2 where we deﬁne l cl(B) := 1 + a(B)r (B) . (5.6.18) Next, deﬁne X MN = N − lNl (5.6.19) l≥2

(2) for the number of vertices that are not part of a line. Then, denoting by ZN (B) the partition function of CMN (2),

(N) Q [Z (B)] = Q [eNFB (p )Z¯ (2) (B)] (5.6.20) N N N MN ∞ (2) Y = λN (B)An1/2(B)Q Z¯ (B) c (B)Nl , + + N MN l l=2 where we write ¯ (2) −N (2) ZN (B) = λ+ ZN (B). (5.6.21)

Asymptotic behavior of the annealed partition function. The key result for the proofs of our theorems is the following proposition that establishes the exponential growth of the annealed partition function with polynomial corrections: Proposition 5.6.1. The following holds true:

(a) For B 6= 0, there exist I = I(B) and J = J(B) such that, as N → ∞,

∞ (2) Y Q Z¯ (B) c (B)Nl = J(B)eI(B)N (1 + o(1)). (5.6.22) N MN l l=2

The function B 7→ J(B) is continuous, while B 7→ I(B) is inﬁnitely diﬀerentiable.

85 Annealed measure: uniqueness regime

(b) Given t ∈ R there exist I¯ = I¯(t) and J¯ such that, as N → ∞,

∞ Nl √ (2) t Y t I¯(t/ N)N QN Z¯ √ cl √ = J¯e (1 + o(1)). (5.6.23) MN N N l=2

The function t 7→ I¯(t) is inﬁnitely diﬀerentiable.

Strategy to prove asymptotic behavior. To prove Proposition 5.6.1 we ﬁrst use the law of total probability to write

∞ (2) Y Q Z¯ (B) c (B)Nl N MN l l=2 n2 ∞ X (2) Y Nl = Em[Z¯m (B)]QN cl(B) | MN = m QN (MN = m), (5.6.24) m=0 l=2 where we denotes Em the expectation with respect to an independent CMm(2). Our aim is to prove that the asymptotic behavior of (5.6.24) is essentially dominated by the term with m = 0, which gives the exponential growth J(B)eNI(B) stated in Proposition 5.6.1. To achieve a full control we analyze in the following the three contributions whose product gives rise to the summand of (5.6.24):

(2) i) Em[Z¯m (B)]: this is subdominant in the limit N → ∞ since, by Lemma (2) 5.5.1, supm Em[Z¯m (B)] is bounded. Therefore it will appear only in the prefactor J(B).

ii) QN (MN = m): we study the distribution of the number of vertices in tori

MN in Lemma 5.6.1; in particular we prove the existence of a limiting distribution function in the limit N → ∞.

Q∞ Nl iii) QN l=2 cl(B) | MN = m : this is rewritten explicitly in Lemma 5.6.2 and its asymptotics is computed in Lemmas 5.6.3 and 5.6.4.

The number of vertices in tori. We start by analyzing the random variable MN . Lemma 5.6.1 (The number of vertices in tori). When N → ∞, there exists a random variable M such that

D MN −→ M, (5.6.25)

M and for every b ∈ R such that |b| < (1 + p)/(2p) we have that QN [b ] < ∞.

86 5.6 Proofs for CMN (1, 2)

Further, 1 n 2 n2−m QN (MN = m) = 2 (n2 − m)!(n1 − 1)!!(2m − 1)!! (n1 + 2n2 − 1)!! m n /2 + n − m − 1 · 1 2 (5.6.26) n2 − m (n − 1)!!n ! 2mn /2 + n − m − 1 = 2n2 1 2 2−2m 1 2 . (n1 + 2n2 − 1)!! m n2 − m Proof. Number the vertices of degree 2 in an arbitrary way. We write

∞ n2 X (t) (t) X MN = lNl , where Nl = Ji(l), (5.6.27) l=1 i=1 and Ji(l) is the indicator that vertex i is in a cycle of length l of which vertex i has the smallest label. We compute that

(t) n2 1 l QN [N ] = QN (vertex 1 is in cycle of length l) → (2p/(1 + p)) ≡ λ . l l 2l l

(t) It is not hard to see, along the lines of [64, Proposition 7.12], that (Nl )l≥1 converges in distribution to a collection of independent Poisson random variables (t) 1 l (Pl)l≥1 with parameters (λl)l≥1. Further, since QN [Nl ] ≤ 2l (n2/`N ) , which decays exponentially, the contribution from large l equals zero whp, i.e., QN (∃l > (t) T : Nl > 0) is small uniformly in N for T large. This shows that

D X MN −→ lPl ≡ M. (5.6.28) l≥1 Note that ∞ ∞ l P (bl−1)λ M Y lPl Y (b −1)λl l≥1 l QN [b ] = QN [b ] = e = e , (5.6.29) l=1 l=1 which is ﬁnite only when |b| < (1 + p)/(2p). To prove (5.6.26), we note that 1 QN (MN = m) = N(n1, n2, m), (5.6.30) (n1 + 2n2 − 1)!! where N(n1, n2, m) is the number of ways in which the half-edges can be paired such that there are precisely m degree 2 vertices in cycles. We claim that n n 2 + n − m − 1 2 n2−m 1/ 2 N(n1, n2, m)= 2 (n2 − m)!(n1 − 1)!!(2m − 1)!! . m n2 − m (5.6.31) For this, note that

n2 (1) there are ( m ) ways to choose the m vertices of degree 2 that are in cycles;

87 Annealed measure: uniqueness regime

(2) there are (2m − 1)!! ways to pair the half-edges that are incident to vertices in cycles;

(3) there are (n1 − 1)!! ways to pair the vertices of degree 1 (and this corresponds to the pairing of degree 1 vertices in lines);

(4) there are (n2 − m)! ways to order the vertices that are in lines; (5) there are 2 ways to attach the half-edges of a degree 2 vertex inside a line, n −m and there are in total n2 − m degree 2 vertices in lines, giving 2 2 ways to attach their half-edges; and

(6) ﬁnally, there are (n1/2+n2−m−1) ways to create n /2 lines with n − m n2−m 1 2 vertices of degree 2.

Multiplying these numbers out gives (5.6.31). This completes the proof of Lemma 5.6.1.

Combinatorial expression of the partition function. To perform the Q∞ Nl asymptotic analysis of the partition function QN l=2 cl(B) | MN = m , we rewrite it as a double sum in Lemma 5.6.2 and then we investigate the asymptotics of the summand in Lemma 5.6.3 by Stirling’s formula. Finally, in Lemma 5.6.4, we use the Laplace method to estimate the asymptotics of the double sum.

Lemma 5.6.2 (Generating function of number of lines in CMN (1, 2)). For l every a, r, for cl = 1 + ar for every l ≥ 2,

∞ n1/2 n2−m Y Nl X X (N) QN cl(B) | MN = m = B`,k (n2 − m), (5.6.32) l=2 `=0 k=0 where

(`+k−1)(n1/2−`+n2−m−k−1) (N) n1/2 2 ` k k n2−m−k B (n2 − m) = (ar ) r . (5.6.33) `,k ` (n1/2+n2−m−1) n2−m

Proof. When MN = m, we have that n2 − m vertices of degree 2 have to be divided over n1/2 lines. Number the lines as 1, ... , n1/2 in an arbitrary way. Denote the number of degree 2 vertices in line j by Yj and rewrite

∞ Y Nl QN cl(B) | MN = m l=2

n1/2 X Y ij +2 = QN Y1 = i1, ... , Yn1/2 = in1/2 (1 + ar ) (5.6.34) (i ,...,i ) j=1 1 n1/2

88 5.6 Proofs for CMN (1, 2)

where (i1, ... , in1/2) is such that i1 + ··· + in1/2 = n2 − m. Let [n1/2] = Qn1/2 ij +2 {1, ... , n1/2}, and expand out j=1 (1 + ar ) to obtain

X X 2 |Γ| Y ij QN Y1 = i1, ... , Yn1/2 = in1/2 (ar ) r . (5.6.35) (i ,...,i ) ⊆[n 2] j∈Γ 1 n1/2 Γ 1/ where the sum over Γ is over all subsets of [n1/2]. We denote

n1/2 X Nn1,n2−m = # (i1, ... , in1/2) : ij ≥ 0 ∀ j and ij = n2 − m j=1 n /2 + n − m − 1 = 1 2 , (5.6.36) n2 − m so that (5.6.35) is equal to

n −m X 1 X X2 (ar2)|Γ| rk1l n 2+n −m−1 {P i =k} ( 1/ 2 ) j∈Γ j (i ,...,i ) n2−m ⊆[n 2] k=0 1 n1/2 Γ 1/ n1/2 n2−m X 1 X n1/2 2 ` X k = (ar ) r 1l{(i +...+i =k)} (n1/2+n2−m−1) ` 1 ` (i ,...,i ) n2−m `=0 k=0 1 n1/2 1 = · (n1/2+n2−m−1) n2−m n1/2 n2−m X n1/2 X ` + k − 1 n1/2 − ` + n2 − m − k − 1 · (ar2)` rk ` k n2 − m − k `=0 k=0 n1/2 n2−m `+k−1 n1/2−`+n2−m−k−1 X X n1/2 ( k )( n −m−k ) = (ar2)`rk 2 . ` (n1/2+n2−m−1) `=0 k=0 n2−m

Asymptotics by Stirling’s formula. We continue the analysis by investi- (N) gating the asymptotics of B`,k (n2) in (5.6.33) when `, k and n2 are of the same (N) (N) asymptotic order. To alleviate the notation we write Ba,b (n2) := Bbac,bbc(n2) when a, b are not necessarily integers.

(N) Lemma 5.6.3 (Asymptotics of B`,k (n2)). Let Dp = [0, (1 − p)/2] × [0, p]. For external ﬁelds B 6= 0, there exists a function H(s, t) continuous in Dp and ◦ ◦ smooth in Dp (the interior of Dp) and a function C(s, t) smooth in Dp, such that C(s, t) B(N) (n ) = exp {NH(s, t)} (1 + o(1)), as N → ∞, (5.6.37) sN,tN 2 N

Moreover, H(s, t) is strictly concave on its domain Dp and its (unique) maximum ∗ ∗ ◦ ◦ point (s , t ) lies in the interior Dp. In Dp, the functions are deﬁned as follows:

89 Annealed measure: uniqueness regime

1 − p 1 − p 1 − p H(s, t) = (1 − p) log − 2s log(s) − 2 − s log − s 2 2 2 + s log(ar2) + t log(r) + (s + t) log(s + t) − t log(t) 1 + p 1 + p + − s − t log − s − t − (p − t) log(p − t) 2 2 1 + p 1 + p − log + p log(p), (5.6.38) 2 2 and q 1−p ( 1+p − s − t)(s + t)p 1 2 2 C(s, t) = q . (5.6.39) 2π s 1−p − s 1+p 2 ( 2 )(p − t)t

Finally, uniformly in (s, t) ∈ Dp,

(N) 1/2 BsN,tN (n2) ≤ CN exp {NH(s, t)} . (5.6.40) √ Proof. Using Stirling’s approximation in the form n! = e−nnn 2πn (1 + o(1)) for n large, taking a, b ∈ N, we can rewrite the binomial coeﬃcients as √ b n b (1 + o(1)) = e[b log b−a log a−(b−a) log (b−a)]n · √ √ √ . (5.6.41) a n a b − a 2πn

Plugging the previous formula into (5.6.33), then (5.6.37) follows. By inspection, ◦ H(s, t) and C(s, t) are smooth functions in Dp for B 6= 0. The function H(s, t) can be further extended by continuity to the boundary ∂Dp of Dp, while C(s, t) ◦ cannot be deﬁned in Dp \ Dp since it is unbounded there. In order to prove concavity of H(s, t), we check that its Hessian matrix Q(s, t) is negative deﬁnite ◦ on each point of Dp. For this, we compute   1 + 1 − 2 − 2 1 + 1 1−p −s+p−t s+t 1−p −s s 1−p −s+p−t s+t Q(s, t) =  2 2 2  .  1 1 1 1 1 1  1−p + s+t 1−p + s+t − p−t − t 2 −s+p−t 2 −s+p−t The eigenvalues µ+ and µ− of Q(s, t) are " 1 2 2 2 2 1 1 µ± = + − − − − 2 1+p s + t 1−p s p − t t 2 − s − t 2 − s v  u !2 !2 u 2 2 1 1 1 1 ± t + − − + 4 +  . 1−p s p − t t 1+p s + t  2 − s 2 − s − t

We can easily see that µ− < 0, and in order to show that also µ+ is negative, we observe that the determinant of the Hessian matrix is positive. Therefore, H(s, t)

90 5.6 Proofs for CMN (1, 2)

◦ is strictly concave in Dp and, by continuity, concave in Dp. Concavity implies that (s, t) 7→ H(s, t) has a unique global maximum in Dp, the uniqueness follows by strict concavity and the fact that the maximizer is not on the boundary. ∗ ∗ ◦ In order to ﬁnd (s , t ) := argmax H(s, t), and to prove that it lies in Dp, we (s,t)∈Dp calculate ∂H(s, t) 1 − p 1 + p = 2 log − s − log − s − t + log(s + t) − 2 log(s) ∂s 2 2 + log(ar2), ∂H(s, t) 1 + p = log(s + t) − log(t) − log − s − t + log(p − t) + log(r), ∂t 2 so that (s∗, t∗) is a solution of the system

2  ( 1−p −s)  2 (s+t) = 1 ,  s2 1+p −s−t ar2 ( 2 ) (5.6.42) (p−t) (s+t) 1  t 1+p = r . ( 2 −s−t) Since B 6= 0, and then both ar2 and r are ﬁnite and larger than zero, it easy to see from (5.6.42) that the maximum point cannot be attained on√ the boundary. −n n The√ proof of (5.6.40) follows similarly, now using that e n 2πn ≤ n! ≤ −n n 1 e n 2πn(1 + 12n ) for every n ≥ 1. The power of N is needed to make the estimate uniform, e.g., by bounding s((1 − p)/2 − s) ≥ c/N uniformly for s ≥ 1/N.

Asymptotics by Laplace method. In the next lemma we compute the asymptotic behavior of (5.6.32) by using the discrete analogue of the Laplace method.

Q∞ Nl Lemma 5.6.4 (Asymptotics of QN [ l=2 cl(B) | MN = m]). For every m ≥ 0 ﬁxed, ∞ 1 Y Nl ∗ ∗ ∗ ∗ − ∗ m QN [ cl(B) | MN = m] = 2π C(s , t )(det Q(s , t )) 2 (b ) l=2 · exp{NH(s∗, t∗)}(1 + o(1)), (5.6.43) where (s, t) 7→ H(s, t) and (s, t) 7→ C(s, t) are deﬁned in Lemma 5.6.3, (s∗, t∗) is the maximum point of H(s, t), Q(s, t) is the Hessian matrix of H and ! 1 + p p − t∗ b∗ = . (5.6.44) 2p 1+p ∗ ∗ 2 − s − t Proof. We start by proving (5.6.43) for m = 0. Due to Lemma 5.6.3, we may estimate the asymptotic behavior of the double sum

n1/2 n2 X X (N) K(N) := B`,k (n2), (5.6.45) `=0 k=0

91 Annealed measure: uniqueness regime

−1 by making use of the function fN (s, t) = N C(s, t) exp{NH(s, t)} that ap- peared in (5.6.37). The correspondence between the two sets of variables (`, k) and (s, t) is given by the simple transformation s = `/N and t = k/N. We denote this transformation by TN . ∗ ∗ ∗ ∗ Let us deﬁne `N := bs Nc, kN := bt Nc and introduce 0 < δ < min{p, (1 − p)/2}. The precise value of δ will be chosen later on. We partition the domain (N) of the summation appearing in the sum of B`,k (n2)

ΛN = {(`, k) : ` = 1, ... , n1/2, k = 1, . . . n2}, (5.6.46) into two subsets

∗ ∗ Uδ,N = {(`, k) ∈ ΛN : |` − `N | ≤ δN + 1, |k − kN | ≤ δN + 1}, c Uδ,N = ΛN \Uδ,N . ∗ ∗ The set Uδ,N is to be considered as a neighborhood of (`N , kN ), the “maximum” (N) point of B`,k (n2). We observe that TN (Uδ,N ) is contained in the neighborhood ∗ ∗ of (s , t ) in Dp, i.e.,

∗ 1 ∗ 1 Wδ+ 1 = {(s, t) ∈ Dp : |s − s | ≤ δ + , |t − t | ≤ δ + }, (5.6.47) N N N c c while TN (Uδ,N ) is contained in its complement W 1 := Dp\Wδ+ 1 . We rewrite δ+ N N (5.6.45) as K(N) = K1(δ, N) + K2(δ, N) where

X (N) K1(δ, N) := B`,k (n2), (5.6.48) (`,k)∈Uδ,N X (N) K2(δ, N) := B`,k (n2). (5.6.49) c (`,k)∈Uδ,N

We aim to prove that the asymptotic behavior of K(N) is given by K1(δ, N), while K2(δ, N) gives a sub-dominant contribution. We start by proving the latter statement.

Bound on K2(δ, N). Making use of (5.6.40), we upper bound

1/2 X K2(δ, N) ≤ CN exp {NH(`/N, k/N)} . (5.6.50) c (`,k)∈Uδ,N Deﬁning M(δ) := sup H(s, t) ≥ sup H(s, t) , (5.6.51) |s−s∗|>δ c (s,t)∈Uδ,N |t−t∗|>δ c since the values (`/N, k/N) in (5.6.50) belong to W 1 we can bound δ+ N H(`/N, k/N) ≤ M(δ). We conclude that

1/2 c 5/2 K2(δ, N) ≤ CN exp{NM(δ)}|Uδ,N | ≤ CN exp{NM(δ)}, (5.6.52)

92 5.6 Proofs for CMN (1, 2) which, together with M(δ) < H(s∗, t∗), implies that

∗ ∗ exp{−NH(s , t )} K2(δ, N) → 0, as N → ∞. (5.6.53)

Let us remark that, besides the condition 0 < δ < min{p, (1 − p)/2} (which (N) guarantees that (`, k) and (s, t) are contained in the domains of B`,k (n2) and fN (s, t)), in the previous argument no further condition has been imposed on δ.

Asymptotics of K1(δ, N). Here we consider the sum K1(δ, N) deﬁned in (5.6.48). Choose ε > 0 arbitrary and small. By continuity of C(s, t), we can choose δ > 0 small enough so that, for large N,

∗ ∗ ∗ ∗ C(s , t ) − ε ≤ C(s, t) ≤ C(s , t ) + ε, for all (s, t) ∈ W 1 (5.6.54) δ+ N Then using (5.6.37), we obtain

C(s∗, t∗) + ε X ` k K (δ, N) ≤ exp NH , (1 + o(1)), (5.6.55) 1 N N N (`,k)∈Uδ,N and a similar lower bound with C(s∗, t∗) + ε replaced by C(s∗, t∗) − ε. Recalling that Q(s, t) is the Hessian matrix of H(s, t), by Taylor expanding up to second order and using that (s∗, t∗) is the maximum, we can write

∗ ∗ 1 ∗ ∗ 2 H(s, t) − H(s , t ) ≤ x · Q(s , t )x + cδkxk /2, for all (s, t) ∈ Wδ+ 1 2 N where x = (s − s∗, t − t∗), and a similar lower bound with cδ replaced by −cδ. By multiplying (5.6.55) by exp[−NH(s∗, t∗)] and applying the previous inequality, we obtain C(s∗, t∗) + ε exp[−NH(s∗, t∗)]K (δ, N) ≤ (5.6.56) 1 N X N · exp xT · Q(s∗, t∗)x + cδNkxk2/2 , 2 (`,k)∈Uδ,N

(where x is computed with s = `/N and t = k/N) and a similar lower bound with C(s∗, t∗) + ε replaced with C(s∗, t∗) − ε and +cδNkxk2 replaced with −cδNkxk2. The last step is bounding the sum

X N K˜ (δ, N) := exp xT · (Q(s∗, t∗) ± cδI)x , (5.6.57) 1 2 (`,k)∈Uδ,N Now we can substitute the finite sum in the previous display with the infinite one, since the difference is exponentially small [13]. It is known that (as can be seen by extending [13, (3.9.4)] to two-dimensional sums) that

X T 2πN e−j Aj/(2N) = (1 + o(1)), (5.6.58) det(A)1/2 j∈Z2

93 Annealed measure: uniqueness regime

Therefore, 2π K˜ 1(δ, N) = N(1 + o(1)). (5.6.59) det(Q(s∗, t∗) ± cδI)1/2 From the previous equation, recalling (5.6.56), we obtain ∗ ∗ ∗ ∗ 2π(C(s , t ) + ε) exp[−NH(s , t )]K1(δ, N) ≤ (1 + o(1)), (5.6.60) det(Q(s∗, t∗) − cδI)1/2 and a similar lower bound with C(s∗, t∗) + ε replaced with C(s∗, t∗) − ε and Q(s∗, t∗) − cδI with Q(s∗, t∗) + cδI. Since ε is arbitrary, the previous inequality implies 2π C(s∗, t∗) lim exp[−NH(s∗, t∗)]K(N) = , (5.6.61) N→∞ det(Q(s∗, t∗))1/2 which proves the claim. Next, we want to generalize the previous result by computing the asymptotics of (5.6.32) in the case m 6= 0. We start by rewriting (5.6.32) in the following fashion:

∞ n1/2 n2−m Y Nl X X (N) (N) QN [ cl(B) | MN = m] = G`,k (m; n2)B`,k (n2) , (5.6.62) l=2 `=0 k=0 where B(N)(n − m) Qm n1 Qm−1 (N) `,k 2 j=1 2 + n2 − j j=0 (n2 − k − j) G (m; n2) := = , `,k (N) Qm−1 Qm n1 B`,k (n2) j=0 (n2 − j) j=1 2 + n2 − ` − k − j ◦ for m = 0, 1, ... , n2. By deﬁning the function F (s, t; m) on Dp given by !m 1 + pm p − t F (s, t; m) = , (5.6.63) 2p 1+p 2 − s − t we obtain that ` k G(N)(m; n ) = F , ; m (1 + o(1)) (5.6.64) `,k 2 N N as N → ∞. Then the proof is obtained from that for m = 0 by replacing C(s, t) by C(s, t)F (s, t). ∗ ∗ ∗ ◦ Remark 5.6.1 (Bound on b ). Since (s , t ) ∈ Dp, p − t∗ p − t∗ 1 + p = < 1, so that b∗ < . 1+p ∗ ∗ 1−p ∗ ∗ 2p 2 − s − t ( 2 − s ) + (p − t ) This will allow us to use in the following the moment generating function ∗ M QN [(b ) ] deﬁned in (5.6.29).

94 5.6 Proofs for CMN (1, 2)

Boundary contribution. Lemma 5.6.4 proves, for any fixed 0 ≤ m < ∞, Q∞ Nl the asymptotic exponential growth of QN [ l=2 cl(B) | MN = m] as N → ∞. However in formula (5.6.24) we need to sum over a range of values of m that increases with the volume N. In order to overcome this problem, in the proof of Proposition 5.6.1 we introduce a cut-off in the sum over m (and then send the cut-off to infinity at the end). In doing so we need to exclude the contribution (N) arising from B`,k for ` close to the boundary n1/2. This is achieved in the following lemma: Lemma 5.6.5 (Boundary contribution). For every ε > 0 sufficiently small, as N → ∞,

n2 n1/2 n2−m X ¯ (2) X X (N) NH(s∗,t∗) Em[Zm (B)] B`,k (n2 − m)QN (MN = m) = o(e ). m=0 n1 k=0 `>(1−ε) 2

∗ 1−p Proof. By Lemma 5.6.3, we know that s < 2 . Deﬁning

H(s∗, t∗) := sup H(s, t), (5.6.65) (s,t) 1−p s>(1−ε) 2 it follows that H(s∗, t∗) < H(s∗, t∗). Further, we deﬁne

(N) (N) D`,k (n1, n2, m) = B`,k (n2 − m)QN (MN = m), (5.6.66) and using (5.6.26) and the following bound

2(m + 1) 2m 2−2(m+1) ≤ 2−2m , (5.6.67) m + 1 m we obtain

(N) D (n1, n2, m + 1) n − m − k n `,k ≤ 2 ≤ 2 (N) n /2 − ` + n − m − k − 1 n /2 − ` + n − 1 D`,k (n1, n2, m) 1 2 1 2 n ≤ 2 . (5.6.68) n2 − 1 As a consequence, using (5.6.40), m (N) (N) n2 B`,k (n2 − m)QN (MN = m) ≤ B`,k (n2)QN (MN = 0) n2 − 1 ≤ aN 1/2 exp{H(`/N, k/N)},

m n2 since n2 ≤ n2 ≤ a for m ≤ n and some a > e. Therefore, using n2−1 n2−1 2 this inequality together with Lemma 5.5.1, we obtain that

95 Annealed measure: uniqueness regime

n2 n1/2 n2−m −NH(s∗,t∗) X ¯ (2) X X (N) e Em[Zm (B)] B`,k (n2 − m)QN (MN = m) m=0 n1 k=0 `>(1−ε) 2

n1/2 n2 1/2 −NH(s∗,t∗) X X ≤ aAn2N e exp{H(`/N, k/N)} n1 k=0 `>(1−ε) 2 ∗ ∗ ∗ ∗ ≤ aAN 7/2eN(H(s ,t )−H(s ,t )) N−→→∞ 0.

Now we are finally ready for the proof of Proposition 5.6.1. We treat first the case in the presence of an external field B and then the case without field.

Proof of Proposition 5.6.1 (a). We ﬁx µ ∈ {0, ... , n2} and ε > 0 suﬃciently small. Using (5.6.24) and Lemma 5.6.2 we write

∞ (2) Y Nl (1) (2) (3) Q Z¯ (B) c (B) =X n (µ) + X n (µ) + X n , N MN l 1 1 1 N,`≤(1−ε) 2 N,`≤(1−ε) 2 N,`>(1−ε) 2 l=2 where

n1 µ (1−ε) 2 n2−m (1) X ¯ (2) X X (N) X n1 (µ) = Em[Zm (B)] B`,k (n2 − m)QN (MN = m), N,`≤(1−ε) 2 m=0 `=0 k=0 (5.6.69)

n1 n2 (1−ε) 2 n2−m (2) X ¯ (2) X X (N) X n1 (µ) = Em[Zm (B)] B`,k (n2 − m)QN (MN = m), N,`≤(1−ε) 2 m=µ+1 `=0 k=0 (5.6.70)

n2 n1/2 n2−m (3) X ¯ (2) X X (N) X n1 = Em[Zm (B)] B`,k (n2 − m)QN (MN = m). N,`>(1−ε) 2 m=0 n1 k=0 `>(1−ε) 2 (5.6.71)

We analyze the three pieces separately, showing that only the ﬁrst of them con- (2) tributes to the exponential growth of Q Z¯ (B) Q∞ c (B)Nl . By Lemmas N MN l=2 l 5.6.1 and 5.6.4

(1) 1 ∗ ∗ ∗ ∗ − 2 ∗ ∗ X n1 (µ) = 2π C(s , t )(det Q(s , t )) exp{NH(s , t )} N,`≤(1−ε) 2 µ X (2) · Em[Z¯m (B)]QN (M = m)(1 + o(1)). (5.6.72) m=0

96 5.6 Proofs for CMN (1, 2)

The expression in (5.6.70) can be rewritten as

n1 n2 (1−ε) 2 n2−m (N) B (n2 − m)QN (MN = m) (2) X ¯ (2) X X `,k X n1 (µ) = Em[Zm (B)] (N) N,`≤(1−ε) 2 m=µ+1 `=0 k=0 B`,k (n2)QN (MN = 0) (N) · B`,k (n2)QN (MN = 0).

(N) B`,k (n2−m)QN (MN =m) Now, by (5.6.68), (N) is uniformly bounded by 1 − δ B`,k (n2−(m−1))QN (MN =m−1) for δ > 0 suﬃciently small, because ` ≤ (1 − ε)n1/2. Using this bound together with Lemma 5.5.1 yields

n1/2 n2 n2 (2) X X (N) X m X n1 (µ) ≤ A B`,k (n2) (1 − δ) . (5.6.73) N,`≤(1−ε) 2 `=0 k=0 m=µ+1

Thus there exist ε(µ) (with ε(µ) → 0 as µ → ∞) such that, uniformly in N,

(2) ∗ ∗ X n1 (µ) ≤ ε(µ) exp{NH(s , t )}(1 + o(1)). (5.6.74) N,`≤(1−ε) 2

(3) ∗ ∗ Finally, by Lemma 5.6.5, X n1 = o(exp{NH(s , t )}). Thus, from for- N,`>(1−ε) 2 mula (5.6.72) we can identify I = H(s∗, t∗) and

∞ 1 ∗ ∗ ∗ ∗ − X (2) ∗ m J = 2π C(s , t )(det Q(s , t )) 2 Em[Z¯m (B)](b ) P(M = m). m=0 We remark that the previous expression is well deﬁned since, from Lemma 5.5.1, P∞ ¯ (2) ∗ m ∗ M m=0 Em[Zm (B)](b ) P(M = m) ≤ A E [( b ) ], which is ﬁnite because ∗ 1+p b < 2p (see (5.6.29) in Lemma 5.6.1 and Remark 5.6.1).

Proof of Proposition 5.6.1 (b). In this case we work with a vanishing t external ﬁeld. Deﬁning tN := √ and by Taylor expanding (5.6.15) around 0, N we have 2 A− (tN ) t a (tN ) := = C (1 + o(1)), as N → ∞, (5.6.75) A+ (tN ) N where C is a constant (whose value actually depends on β). Thus,

∞ ∞ Nl Y Nl Y l QN [ cl(tN ) | MN = m] = QN [ 1 + a(tN )r (tN ) | MN = m] l=2 l=2 h P∞ l i a(tN )r (tN )Nl = QN e l=2 (1 + o(1)) | MN = m

2 P∞ l (N) Ct r (tN )pl = QN e l=2 | MN = m (1 + o(1)).

97 Annealed measure: uniqueness regime

By writing

2 P∞ l (N) Ct r (tN )pl QN e l=2 | MN = m

∞ ∞ Ct2 P rl(t )Q (p(N)) Ct2 P rl(t )Q (p(N)) = e l=2 N N l + e l=2 N N l

2 P∞ l (N) (N) Ct r (tN )(pl −QN (pl )) · QN e l=2 − 1 | MN = m , (5.6.76) and using formula (5.6.24), we can rewrite

∞ (2) Y Q Z¯ (t ) c (t )Nl = S (N) + S (N) (5.6.77) N MN N l N 1 2 l=2 i.e., as the sum of two contributions, due to the two terms in (5.6.76). Now we analyze S1(N) and S2(N) as N → ∞. First, we remark that the sum in the exponential factor converges in this limit. This can be shown by observing that r(B) < 1 (see (4.5.27)). Therefore, calling r∗ = r(0) and given any ε > 0 such ∗ ∗ that r + ε < 1, thanks to the convergence of r(tN ) to r , we have that for all N suﬃciently large,

∞ N N X l (N) X l (N) X ∗ l r (tN )QN (pl ) ≡ r (tN )QN (pl ) ≤ (r + ε) l=2 l=2 l=2

(N) (N) where we used the fact that pl ≤ 1 for l ≤ N and pl = 0 for all l > N. Since the geometric sum in the r.h.s. of the previous display is convergent, the positive series in the l.h.s. is also convergent to some positive value I¯0. Thus, by inserting the ﬁrst term of the r.h.s. of (5.6.76) in (5.6.24) and applying bounded convergence, we obtain (5.6.23) with

2 I¯ (t) = I¯0Ct . and ∞ X (2) J¯ = Em[Z¯m (0)]P(M = m). (5.6.78) m=0 Further, by Lemma 5.5.1 and the law of total expectation,

∞ (N) (N) ∞ (N) Ct2 P rl(t ) p −Q (p ) Ct2 P rl(t )Q (p ) h l=2 N l N l i l=2 N N l S2(N) ≤ Ae QN e − 1 . We use the Cauchy-Schwarz inequality to bound 2 P∞ l (N) (N) h Ct r (tN ) p −QN (p ) i l=2 l l QN e − 1

∞ Ct2 P rl(t ) p(N)−Q (p(N)) 2 1/2 h l=2 N l N l i ≤ QN e − 1 (5.6.79)

98 5.6 Proofs for CMN (1, 2) and by Jensen’s inequality and Hölder inequality we have

" ∞ (N) (N) # Ct2 P rl(t ) p −Q (p ) l=2 N l N l 1 ≤ QN e

√1 " √ ∞ (N) (N) #! Ct2 N P rl(t ) p −Q (p ) N l=2 N l N l ≤ QN e ≤ 1 + o(1),

√ ∞ Ct2 N P rl(t ) p(N)−Q (p(N)) h l=2 N l N l i due to the existence of the ﬁnite limit of QN e by Lemma 4.6.3 from Chapter 4, Therefore, also the term in (5.6.79) converges to 0, showing that S2(N) gives a vanishing contribution. This completes the proof.

5.6.2 Annealed thermodynamic limits: Proof of Theorem 5.3.1

We are now ready to prove the existence of the thermodynamic limits. The thermodynamic limit of the annealed pressure is given by 1 ψe(β, B) = lim ψeN (β, B) = lim log (QN (ZN (β, B))) . (5.6.80) N→∞ N→∞ N From (5.6.20) we can rewrite ∞ h (2) Y i Q (Z (β, B)) = λN (B)An1/2(B)Q Z¯ (B) c (B)Nl , (5.6.81) N N + + N MN l l=2 and then ∞ 1 − p 1 n h io (2) Y Nl ψe(β, B) = log λ+ + log A+ + lim log QN Z¯M (B) cl(B) . 2 N→∞ N N l=2 Finally, using Proposition 5.6.1 we ﬁnd

1 − p ∗ ∗ ψe(β, B) = log λ+(β, B) + log A+(β, B) + H(s , t ). (5.6.82) 2 To prove the existence of the thermodynamic limit of the magnetization, we use Lemma 4.4.1 and the existence of the pressure in the thermodynamic limit. Then, remembering that (s∗, t∗) is the maximum point of the function H(s, t), we compute ∂ ∂ 1 − p ∂ Mf(β, B) = ψe(β, B) = log λ+(β, B) + log A+(β, B) ∂B ∂B 2 ∂B ∂ + s∗(β, B) log a(β, B)r2(β, B) ∂B ∂ + t∗(β, B) log r(β, B) (5.6.83) ∂B

99 Annealed measure: uniqueness regime

In the limit of small external field B by Taylor expanding (5.6.15) one has a(β, B) = O(B2). Also, from the fixed point equations (5.6.42) one can check ∗ 2 that s (β, B) = O(B ). As a consequence limB→0+ Mf(β, B) = 0 for all β > 0, and therefore, by the definition in (2.4.1), we conclude that there is no phase transition for CMN (1, 2).

5.6.3 Annealed SLLN: Proof of Theorem 5.3.3

As for the proofs of Theorem 5.1.2 and Theorem 5.2.2, the annealed SLLN for

CMN (1, 2) follows immediately from the existence of the annealed pressure in the thermodynamic limit and its diﬀerentiability with respect to B, following the argument in Section 3.2.2.

5.6.4 Annealed CLT: Proof of Theorem 5.3.4

In order to prove the CLT in the annealed setting, we will show that

h t i 2 2 lim PeN exp √ SN − PeN (SN ) = exp(σ2t /2), t ∈ R. (5.6.84) N→∞ N From now on, to alleviate notation we will omit the dependence on β and abbreviate B = B + √t . We ﬁrst rewrite N N

h t i QN [ZN (BN )] PeN exp √ SN = , (5.6.85) N QN [ZN (B)] and then, using formula (5.6.20),

NF (p(N)) (2) BN Q [Z (B )] QN [e Z¯M (BN )] N N N = N (5.6.86) (N) (2) QN [ZN (B)] Q [eNFB (p )Z¯ (B)] N MN

N n1/2 (2) Q∞ Nl λ (BN )A (BN )QN Z¯ (BN ) cl(BN ) = + + MN l=2 . n1/2 (2) ∞ λN (B)A (B)Q Z¯ (B) Q c (B)Nl + + N MN l=2 l Now we proceed in the proof of the theorem using Proposition 5.6.1. We start proving the theorem for B 6= 0. We substitute (5.6.22) into (5.6.86) to arrive at

n 2 Q [Z (B )] λN (B )A 1/ (B )J(B )eI(BN )N N N N = (1 + o(1)) + N + N N (5.6.87) Q [Z (B)] N n1/2 I(B)N N N λ+ (B)A+ (B)J(B)e N n 2 λ+(BN ) A+(BN ) 1/ = (1 + o(1)) eN(I(BN )−I(B)), λ+(B) A+(B)

100 5.6 Proofs for CMN (1, 2)

where we use the fact that B 7→ J(B) is continuous to obtain that J(BN ) = (1 + o(1))J(B). We√ can next use the diﬀerentiability of B 7→ I(B) and the fact that BN = B + t/ N to expand out

∂ n1 ∂ ∂ PeN (SN )=N log λ+(B + t)|t=0 + log A+(B + t)|t=0 + I(B + t)|t=0 ∂t√ 2N ∂t ∂t + o( N) ,

(5.6.88) implies (5.6.84), thus proving the theorem in the case B 6= 0. For B = 0, in a similar way now using (5.6.23), we get √ QN [ZN (t/ N)] σ2t2 2 = (1 + o(1)) e ¯ 2 / QN [ZN (0)] √ n t N ∂ log λ (t)| + 1 ∂ log A (t)| + ∂ I¯(t)| · e ∂t + t=0 2N ∂t + t=0 ∂t t=0

t σ2t2 2 √ PN (SN ) = (1 + o(1)) e ¯ 2 / e N e , where ∂2 (1 − p) ∂2 ∂2 σ¯ 2 = log λ (t)| + log A (t)| + I¯(t)| . (5.6.90) 2 ∂t2 + t=0 2 ∂t2 + t=0 ∂t2 t=0 Then √ t QN [ZN (t/ N)] − √ PN (SN ) σ2t2 2 lim e N e = e ¯ 2 / . N→∞ QN [ZN (0)]

5.6.5 Annealed pressure (way 2): Proof of Theorem 5.3.2

To compute the annealed pressure in a different way, we start by recalling a result by Broutin and de Panafieu [14] about properties of simple graphs with degrees 1 and 2. Let SN (1, 2) denote the collection of simple graphs with n1 T vertices of degree 1 and n2 vertices of degree 2, and, for c¯ = (ct)t=2 for some T ∈ N, define ∞ X Y Nl(G) Gn1,n2 (c¯) = cl , (5.6.91) G∈SN (1,2) l=2

101 Annealed measure: uniqueness regime

where Nl(G) is the number of lines of length l in G. In order to analyse the annealed pressure, we make essential use of results by Broutin and de Panafieu [14] that we explain now. To state their main result, we start by introducing some notation. Define (n + 2n )! a = 1 2 , (5.6.92) n1,n2 n 2 (n1/2)!2 1/ which is independent of c¯. Further, define the functions

T 1 X f(y) = f(y, c¯) = log + (c − 1)yt−2 , (5.6.93) 1 − y t t=2 and 2 −y/2−y /4 T e P (c −1)yt/(2t) g(y) = g(y, c¯) = √ e t=3 t . (5.6.94) 1 − y Finally, we deﬁne

2 PT t−3 ∂ 1 + (1 − y) (t − 2)(ct − 1)y Φ(y) = Φ(y, c¯) = y f(y, c¯) = y t=2 . ∂y 2 PT t−2 1 − y + (1 − y) t=2(t − 2)(ct − 1)y

Then, the asymptotics of Gn1,n2 (c¯) for c¯ having a ﬁnite number of elements is given in the following lemma:

Lemma 5.6.6 (Generating function of number of lines in CMN (1, 2) from [14, Lemma 2]). For every T ≥ 1 ﬁxed, c¯ ∈ RT , as N → ∞,

N 1−p (f(ζ)−α log(ζ)) g(ζ) Gn ,n (c¯) = an ,n e 2 (1 + O(1/n1)), (5.6.95) 1 2 1 2 q ∂ πn1ζ ∂y Φ(y) where α = 2n1 , where ζ is the unique solution in (0, 1) of the equation n2

Φ(ζ) = ζf 0(ζ) = α. (5.6.96)

When c¯ = 1, we can explicitly solve the equation for ζ = ζ(1¯) as ζ Φ(ζ) = = α, (5.6.97) 1 − ζ so that ζ(1¯) = α/(1 + α). In this case, f(y) = f(y, 1¯) = log 1/(1 − y), so that f(ζ(1¯), 1¯) = log(1 + α), (5.6.98) and

f(ζ(1¯), 1¯) − α log(ζ(1¯)) = (1 + α) log(1 + α) − α log(α). (5.6.99)

These identiﬁcations are crucial in dealing with Gn1,n2 (1¯).

102 5.6 Proofs for CMN (1, 2)

We next apply Lemma 5.6.6 to our situation to identify the annealed pressure. For this, we deﬁne as in previous sections λ (β, B) A (β, B) r = r(β, B) = − ∈ (0, 1), a = a(β, B) = − . (5.6.100) λ+(β, B) A+(β, B) We use the joint generating function of the number of lines in the graph, deﬁned as " T # Y Nl Mn1,n2 (c¯) = QN cl , (5.6.101) l=2 T where c¯ = (cl)l=2 and Nl is the number of lines of length l. We can express

Hn1,n2 (c¯) Mn1,n2 (c¯) = , (5.6.102) Hn1,n2 (1¯) where, denoting CMN (1, 2) for the collection of possible graphs (including self-loops) that CMN (1, 2) can take on with positive probability,

∞ X Y Nl(G) −(S(G)+M(G)) Hn1,n2 (c¯) = cl 2 , (5.6.103) G∈CMN (1,2) l=2 where Nl(G) is the number of lines of length l in G, S(G) is the number of self-loops and M(G) is the number if multiple edges. See, e.g., [64, Proposition 7.6] for how this formula arises.

Then we are now ready to prove Theorem 5.3.2

Proof of Theorem 5.3.2. We recall the deﬁnition of the annealed pressure:

1 1 h (N) i NFβ,B (p )+NEN ψe(β, B) := lim log QN [ZN (β, B)] = lim log QN e , N→∞ N N→∞ N

(N) where Fβ,B p and EN are deﬁned, respectively, in (5.6.11) and (5.6.12). We can write

h (N) i h (N) i NFβ,B (p )+NEN NFβ,B (p ) NEN (N) QN e = QN e QN e p . (5.6.104)

(t) From Lemma 4.6.2, because NEN ≤ (log 2) · KN and because conditionally on (N) (t) t t p , KN KN , where KN , i.e. the number of tori in CMN (2), is deﬁned in (4.5.8), we have

(N) t NEN KN o(N) QN e p ≤ QN 2 = e , (5.6.105) by (5.5.7). As a consequence,

1 h (N) i NFβ,B (p ) ψe(β, B) = lim log QN e . (5.6.106) N→∞ N

103 Annealed measure: uniqueness regime

Now, as in Section 5.6.1, we compute

" ∞ # h (N) i Nl NFβ,B (p ) N Y l QN e = λ+ QN A+ + A−r l=2 " ∞ (N) # Nl N Y Npl Y l = λ+ QN A+ 1 + ar . (5.6.107) l=2 l

N The ﬁrst factor λ+ is explicit, while we can compute the second factor as

∞ Y Nl n1/2 A+ = A+ , (5.6.108) l=2 P since l≥2 Nl = n1/2. Therefore, we arrive at

" ∞ # h (N) i Nl NFβ,B (p ) N n1/2 Y l QN e = λ+ (A+) QN 1 + ar , (5.6.109) l=2

N n1/2 which is, apart from the multiplicative constant λ+ (A+) , the generating l function of the graph deﬁned in (5.6.101), computed in cl = 1 + ar with l l ∈ N \{0, 1}. Thus, with cl = 1 + ar for all l ∈ N \{0, 1},

h (N) i H (c) NFβ,B (p ) N n1/2 n1,n2 ¯ QN e = λ+ (A+) . (5.6.110) Hn1,n2 (1¯)

0 We next relate Hn1,n2 (c¯) to Gn1,n2 (c¯). For this, we note that with N (G) = 0 N − S(G) − 2M(G) and n2 = n2 − S(G) − 2M(G), X Hn ,n (c¯) = G 0 (c¯)P S(CMN (1, 2)) = s, M(CMN (1, 2)) = m . 1 2 n1,n2 s,m (5.6.111)

By [64, Proposition 7.12], (S(CMN (1, 2)), M(CMN (1, 2))) converges in distribution to two independent Poisson random variables with parameters ν/2 and ν2/4, respectively, where ν = E[D(D − 1)]/E[D] = 2p/(1 + p). In particular, S(CMN (1, 2)) and M(CMN (1, 2)) are tight sequences of random variables, 0 0 0 P so that n2 = n2(1 + oP(1)). Therefore, α = n1/(2n2) −→ 2p/(1 + p) when α = n1/(2n2) → 2p/(1 + p). Since the asymptotics in Lemma 5.6.6 only depends on n1 (which is unchanged) and on n2 only through α, the asymptotics of G 0 (c¯) is the same as that of Gn ,n (c¯). n1,n2 1 2 The problem in applying Lemma 5.6.6, however, is that c¯ has inﬁnitely many cl 6= 1. However, since cl converges to 1 suﬃciently rapidly when l grows large, we can overcome this problem by a suitable truncation argument. We perform this truncation by proving matching upper and lower bounds on the magnetization. We start with the lower bound, which is the simplest.

104 5.6 Proofs for CMN (1, 2)

Lower bound on the annealed pressure of CMN (1, 2). We use that l cl = 1 + ar ≥ 1, so that, with c¯T = (c3, c4, ... , cT ),

h (N) i H (c ) NFβ,B (p ) N n1/2 n1,n2 ¯T QN e ≥ λ+ (A+) . (5.6.112) Hn1,n2 (1¯) Using (5.6.111), we can now apply Lemma 5.6.6 to both sides of the fraction, to obtain

h (N) i NFβ,B (p ) N n1/2 QN e ≥λ+ (A+) ·e(n1/2)[(f(ζ(c¯T ),c¯T )−α log(ζ(c¯T )))−(f(ζ(1¯),1¯)−α log(ζ(1¯)))]eo(N).

Therefore, to identify the thermodynamic limit, we need to evaluate f(y, c¯T ). For this, we compute that

T T 1 X t−2 1 X t t−2 f(y, c¯T ) = log + (c − 1)y = log + ar y 1 − y t 1 − y t=2 t=2 1 1 − (ry)T −1 = log + ar2 = log(1 + arB (y)) − log(1 − y), 1 − y 1 − ry T where 1 − y B (y) = r 1 − (ry)T −1. T 1 − ry

When T → ∞, BT (y) → B(y) exponentially fast, uniformly for y ∈ (0, 1), since λ−(β,B) r = r(β, B) = ∈ (0, 1). Therefore, also ζ(c¯T ) → ζ(c¯) and we obtain λ+(β,B) that

1 h (N) i NFβ,B (p ) lim inf log QN e N→∞ N (1 − p) ≥ log λ + α log(α) − (1 + α) log(1 + α) + log(A ) + 2 + (1 − p) h 2p i + log (1 + arB(ζ)) − log(1 − ζ) − log(ζ) . 2 1 − p

Since f(y, c¯T ) = log(1 + arB(y)) − log(1 − y), we also obtain that ζ = ζ(c¯) satisﬁes

r(1−ζ) 0 r 1 arB (ζ) ζ − 1−rζ + (1−rζ)2 ζ + = 1 + 1 − ζ 1 + arB(ζ) 1 − ζ 1 + arB(ζ) ζ 1 + arB(ζ)2 2p = = α = . (5.6.113) 1 − ζ 1 + arB(ζ) 1 − p

This proves the lower bound on the annealed pressure.

105 Annealed measure: uniqueness regime

Upper bound on the annealed pressure of CMN (1, 2). For the upper bound of the pressure, we follow a similar path as for the lower bound. We use l l∧T that cl = 1 + ar ≤ 1 + ar . Thus, again deﬁne c¯T = (c3, c4, ... , cT ), and P (N) since l>T Npl ≤ n1/2,

h (N) i H (c ) NFβ,B (p ) N n1/2 T n1/2 n1,n2 ¯T QN e ≤ λ+ (A+) (1 + ar ) . (5.6.114) Hn1,n2 (1¯) Note that 1 (1 − p) lim sup log (1 + arT )n1/2) = log(1 + arT ), N→∞ N 2 which tends to zero when T → ∞. The remaining terms in (5.6.114) are the same as for the lower bound. This completes the proof of Theorem 5.3.2.

106 ANNEALEDMEASURE:ATCRITICALITY 6

In this chapter we study the critical behavior for the Ising model on the generalized random graph under the annealed measure. As shown in Chapter 5, this model reduces to an inhomogeneous versions of the Curie–Weiss model (introduced in Definition 2.6.1), with coupling constant J = {Jij (β)}i,j∈[N] with P Jij (β) = βwiwj /( k∈[N] wk). We identify the critical exponents for these models and a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie–Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent τ with τ ∈ (3, 5), then the critical exponents depend sensitively on τ. In addition, at criticality, the total spin (τ−1)/(τ−2) SN satisfies that SN /N converges in law to some limiting random variable whose distribution we explicitly characterize.

6.1 Weight sequences conditions

Before stating the main results, we establish some conditions for the weight sequences of our models. We study sequences of inhomogeneous Curie–Weiss models and generalized random graphs as N → ∞. For this, we need to assume that the vertex weight sequences w = (wi)i∈[N] are sufficiently nicely behaved, i.e., they satisfy Condi- tion 2.1.2. Furthermore our results depend sensitively on whether the fourth moment of W is finite. When this is not the case, then we will assume a power-law bound on the tail of the asymptotic weight: Condition 6.1.1 (Tail of W ). The random variable W satisfies either of the following: (i) E[W 4] < ∞, (ii) W obeys a power law with exponent τ ∈ (3, 5], i.e., there exist constants

CW > cW > 0 and w0 > 1 such that −(τ−1) −(τ−1) cW w ≤ P(W > w) ≤ CW w , ∀w > w0. (6.1.1)

To prove the results on the scaling limit at criticality we will strengthen our assumptions as follows:

107 Annealed measure: at criticality

Condition 6.1.2 (Tail of WN and deterministic sequences). The sequence of weights (wi)i∈[N] satisﬁes either of the following:

4 1 P 4 4 (i) E[WN ] = N i∈[N] wi → E[W ] < ∞, (ii) it coincides with the deterministic sequence

N 1/(τ−1) w = c , (6.1.2) i w i

for some constant cw > 0 and τ ∈ (3, 5). We remark that the above deterministic sequence is N-dependent (we do not make this dependence explicit) and its limit W satisﬁes (6.1.1) since wi = −1 −(τ−1) [1 − F ] (i/N), where F (x) = 1 − (cwx) for w ≥ cw.

6.2 Critical exponents for GRGN (w)

We start by proving that the annealed critical exponents for the magnetization and the susceptibility take the values conjectured in [46]. In this chapter, because the connection with an inhomogeneous Curie–Weiss an model, we use βc to denote the annealed critical temperature βc .

Theorem 6.2.1 (Annealed critical exponents). Let (GN )N≥1 be a sequence of GRGN (w) graphs fulfilling Conditions 2.1.2 and 6.1.1. Then, in the annealed setting, the critical exponents defined in Definition 2.4.1 using βc given in (5.1.6) exist and satisfy

τ ∈ (3, 5) E[W 4] < ∞ β 1/(τ − 3) 1/2 δ τ − 2 3 γ = γ0 1 1 For the boundary case τ = 5 there are the following logarithmic corrections for β = 1/2 and δ = 3:

1/2 + β − βc Mf(β, 0 ) for β & βc, (6.2.1) log 1/(β − βc) B 1/3 Mf(βc, B) for B & 0. (6.2.2) log(1/B)

The same results hold for the rank-1 inhomogeneous Curie–Weiss model CWN (J), the critical exponents being now deﬁned using βc = 1/ν.

108 6.3 Non-classical limit theorem for GRGN (w)

Remark 6.2.1 (Comparison to the quenched case). In [21], the authors have shown that the same critical exponents hold for the quenched setting of the Ising model on power-law random graphs, such as GRGN (w), under the assumptions in Conditions 2.1.2 and 6.1.1. In [21], however, the authors only managed to 0 prove a one-sided bound on γ . Thus, our results show that for GRGN (w) both the annealed and quenched Ising model have the same critical exponents, but a different critical value. This is a strong example of universality. Remark 6.2.2 (Comparison to the Curie–Weiss model). For the rank-1 inhomogeneous Curie–Weiss model, we see that the inhomogeneity does not change the critical behavior when the fourth moment of the weight distribution remains finite, but it does when the fourth moment of the weight distribution increases to infinity. In the latter case, we call the inhomogeneity relevant. Remark 6.2.3 (Extension of γ = 1). The result γ = 1 holds under more general conditions, i.e., E[W 2] < ∞. See Theorem 6.4.4 below. From the previous theorem we can also derive the joint scaling of the magnetization as (β, B) & (βc, 0):

Corollary 6.2.1 (Joint scaling in B and (β − βc)). Under the conditions of Theorem 6.2.1, for τ 6= 5,

β 1/δ Mf(β, B) = Θ (β − βc) + B , (6.2.3) where f(β, B) = Θ(g(β, B)) means that there exist constants c1, C1 > 0 such that c1g(β, B) ≤ f(β, B) ≤ C1g(β, B) for all B ∈ (0, ε) and β ∈ (βc, βc + ε) with ε small enough. For τ = 5, instead

β − βc 1/2 B 1/3 Mf(β, B) = Θ + . (6.2.4) log 1/(β − βc) log(1/B)

6.3 Non-classical limit theorem for GRGN (w)

Our second main result concerns the scaling limit at criticality. The next theorem provides the correct scaling and the limit distribution of SN at criticality (for a heuristic derivation of the scaling, see the discussion in Remark 6.3.1). For

GRGN (w), we deﬁne the inverse temperature sequence

an βc,N = βc,N = asinh(1/νN ), (6.3.1) where 2 E[WN ] νN = , (6.3.2) E[WN ] so that βc,N → βc for N → ∞. For rank-1 CWN (J), we replace β by asinh(β), so that βc,N = 1/νN . Our main result is the following:

109 Annealed measure: at criticality

Theorem 6.3.1 (Non-classical limit theorem at criticality). Let (GN )N≥1 be a sequence of GRGN (w) graphs satisfying Conditions 2.1.2 and Condition 6.1.2 and let δ have the respective values stated in Theorem 6.2.1. Then, there exists a random variable X such that S N −→D X, as N → ∞, (6.3.3) N δ/(δ+1) where the convergence is w.r.t. the measure PeN at inverse temperature βc,N = asinh(1/νN ) and external ﬁeld B = 0. The random variable X has a density proportional to exp(−f(x)) with

 1 E[W 4]  x4 if E[W 4] < ∞,  12 E[W ]4 f(x) = 2 X 1τ − 2 − 1 τ − 2 − 1  x i τ−1 − log cosh x i τ−1 if τ ∈ (3, 5).  2 τ − 1 τ − 1  i≥1 (6.3.4) The same result holds for the rank-1 inhomogeneous Curie–Weiss model at its critical value βc,N = 1/νN .

We will see that in both the case where the fourth moment is ﬁnite as well as when it is inﬁnite, f(x) lim = C, (6.3.5) x→∞ x1+δ with  1 E[W 4]  if E[W 4] < ∞,  12 E[W ]4 C = τ−1 Z ∞  τ − 2 1 − 2 − 1  y τ−1 − log cosh y τ−1 dy if τ ∈ (3, 5). τ − 1 0 2 (6.3.6) This result extends the non-classical limit theorem for the Curie–Weiss model to the annealed GRGN (w) and the rank-1 CWN (J).

Remark 6.3.1 (Heuristic for the scaling limit). To obtain a guess for the correct scaling, we can use the standard scaling relation between δ and η as in [26]. On a box in the d-dimensional lattice with side lengths n, [n]d ⊂ Zd, the exponent η satisﬁes (d) 2 d+2−η Pn (Sn) ∼ n , (6.3.7) (d) where Pn is the expectation w.r.t. the Ising measure on this box and Sn is the sum of all spins inside the box, where it should be noted that there are nd sites in the box. Hence, to compare this with our setting, we take N = nd and, with an abuse of notation, let Sn = SN . If there is an exponent λ such λ that SN /N converges in distribution to a non-trivial limit, then it must also λ 2 2 2dλ 2 2dλ hold that PeN (Sn/N ) = PeN Sn/n ) converges. Hence Sn ∼ n , so that

110 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

δ−1 d + 2 − η = 2dλ. The standard scaling relation 2 − η = d δ+1 [26] now suggests that we should choose δ λ = . (6.3.8) δ + 1 We prove that this is indeed the correct scaling and we also show that the tail of the density behaves like exp{−Cxδ+1} as is conjectured on Zd (see [26, Section V.8]). Remark 6.3.2 (Near-critical scaling window). Theorem 6.3.1 is proved along the critical sequence βc,N approaching the critical inverse temperature βc in the limit N → ∞. A diﬀerent scaling limit may be obtained by working with a sequence near the critical one, the so-called near-critical window, i.e., βc,N + ∆N with ∆N → 0 at an appropriate rate. As is argued in Section 6.5.5, it turns out that for the annealed Ising model the width ∆N of the scaling window is N −(δ−1)/(δ+1) and the scaling limit diﬀers by a quadratic term that appears in δ/(δ+1) in the function f(x) describing the density of SN /N in (6.3.4). Remark 6.3.3 (At criticality). As a consequence of the previous discussion, we also infer that if one works at the critical inverse temperature βc, the scaling limit will be seen to depend on the speed at which νN approaches ν. Indeed, from

(5.1.6) and (6.3.1), one has βc − βc,N = O(ν − νN ). For a natural example given by the deterministic sequence in Condition 6.1.2 (ii) one has that when τ > 5 1/2 then ν − νN = o(1/N ) and thus the limiting distribution does not change; on −(δ−1)/(δ+1) the contrary when τ ∈ (3, 5] then ν − νN = ζN (1 + o(1)) for some ζ 6= 0, and thus the distribution changes since we are shifted in the near-critical window. See again Section 6.5.5 for more details. Remark 6.3.4 (Comparison to other models). The rank-1 inhomogeneous Curie–Weiss model is similar to some other models, like the spin-glass model studied in [66] and the Hopﬁeld model, in which both the ferromagnetic and anti-ferromagnetic interactions are present. In these models the inhomogeneity is described through the interaction term Ji,j that is a product of two random variables. In particular, in [34] the authors analyze the Hopﬁeld model at the critical point. Results similar to our E[W ]4 < ∞ case appear there as well, showing a non-classical limit theorem.

6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

We follow a strategy similar to that in [21], although the proof in our case is a bit easier since the annealed magnetization is expressed in terms of the deterministic fixed point z∗ in (5.1.5), whereas in the quenched setting the magnetization is expressed in terms of a fixed point of a distributional recursion. The proof of Theorem 6.2.1 is split into Theorem 6.4.3 dealing with the exponents β and δ, Theorem 6.4.4 for the exponent γ and Theorem 6.4.5 for the exponent γ0. We next present a couple of lemmas that underline properties of the fixed point z∗ in (5.1.5), useful to prove our results.

111 Annealed measure: at criticality

Lemma 6.4.1. Referring to (5.1.5), let us consider the function f(z, w, β, B) := h q sinh(β) q sinh(β) i tanh E[w] wz + B E[W ] w deﬁned on R × [0, +∞) × [0, +∞) × [0, +∞) and the average Fβ,B (z) = E[f(z, W , β, B)]. Then, the derivatives of Fβ,B (z) exist and can be computed under the average ∂Fβ,B (z) h ∂f(z,W ,β,B) i sign, e.g. ∂z = E ∂z .

Proof. The function f(z, w, β, B) is diﬀerentiable with respect to z, B and ∂f sinh(β) 2 ∂f cosh(β) 2 β 6= 0. Moreover, for any β > 0, we have 2| ∂z | ≤ E[W ] w , | ∂β | ≤ E[W ] zw + q sinh(β) ∂f q sinh(β) E[W ] w and | ∂B | ≤ 2 E[W ] w on R × [0, +∞) × (0, +∞) × [0, +∞). There- fore, since E[W ] < ∞ and E[W 2] < ∞ hold by Condition 2.1.2, the expectations ∂f ∂f ∂f of the derivatives exist and are ﬁnite: E[ ∂z ] < ∞, E[ ∂β ] < ∞ and E[ ∂B ] < ∞. Form this fact it follows the thesis.

Lemma 6.4.2. For β 6= βc and B > 0 we have ∂z∗(β, B) ∂z∗(β, B) > 0, > 0. (6.4.1) ∂B ∂β

Proof. For β < βc and any B the ﬁxed point equation (5.1.5) has a unique solution z(β, B), which is 0 for B = 0. For β > βc and B 6= 0, there is a unique solution with the same sign as B, while for B = 0 there are three solutions ∗ ∗ ∗ ∗ ∗ z−(β, 0) < z0 (β, 0) = 0 < z+(β, 0), with z−(β, 0) = −z+(β, 0), see Section 5.4. We observe also that for B > 0 the solution z∗(β, B) > 0 is increasing as a function of B and β. Indeed, Fβ,B (z), the function deﬁned by the right hand side of (5.1.5), is increasing in B (for all z) and in β for (z > 0). Thus, taking the derivatives of both sides of (5.1.5), we can see that when β 6= βc and B > 0 ∂z∗(β,B) ∂z∗(β,B) ∗ we have ∂B > 0 and ∂β > 0, since z (β, B) > 0 for B > 0.

6.4.1 Magnetization: critical exponents β and δ

Some lemmas and propositions containing preliminary results are ﬁrst proved in the following. We start by showing that the phase transition is continuous.

Lemma 6.4.3 (Continuous phase transition). Let ((β`, B`))`≥1 be a sequence with β` and B` non-increasing, β` ≥ βc and B` > 0, and β` & βc and B` & 0 as ` → ∞. Then, the solution of (5.1.5) satisﬁes

∗ lim z (β`, B`) = 0. (6.4.2) `→∞ In particular,

∗ ∗ + lim z (βc, B) = 0, and lim z (β, 0 ) = 0. (6.4.3) B&0 β&βc

112 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

Proof. The existence of the limit (6.4.2) is a consequence of the monotonicity of z∗(β, B), see Lemma 6.4.2, and the fact that z∗(β, B) ≥ 0 for B ≥ 0. Suppose ∗ that lim`→∞ z (β`, B`) = c > 0. Then, it follows from (5.1.5) and dominated convergence that

" s ! s # ∗ sinh (βc) sinh (βc) c = lim z (β`, B`) = E tanh W c W `→∞ E [W ] E [W ]

< c sinh(βc)ν = c, (6.4.4) where we used that tanh(x) < x for x > 0 and βc = asinh(1/ν). This contradic- tion proves the lemma.

We next show that z∗ has the same scaling as we want to prove for Mf(β, B) by proving the upper and lower bounds in Propositions 6.4.1 and 6.4.2 below. These then allow us to obtain the theorem. But ﬁrst we state some properties for truncated moments of W in the following lemma: Lemma 6.4.4 (Truncated moments of W ). Assume that W obeys a power law for some τ > 1, see item (ii) in Condition 6.1.1. Then there exist constants ca,τ , Ca,τ > 0 such that, as ` → ∞,   a−(τ−1) a−(τ−1) ca,τ ` a Ca,τ ` for a > τ − 1, ≤ E W 1l{W ≤`} ≤ (6.4.5) cτ−1,τ log ` Cτ−1,τ log ` for a = τ − 1. and, when a < τ − 1,

a a−(τ−1) E W 1l{W >`} ≤ Ca,τ ` . (6.4.6)

Proof. The proof is similar to that of [21, Lemma 3.4]. We start by proving the upper bounds. Recall that P(W ≥ 0) = 1 and denote FW (x) = P(W ≤ x). Then, using integration by parts, we obtain

Z ` Z ` a a a E[W 1l{W ≤`}] = x dFW (x) = − x d(1 − FW (x)) 0 0 Z ` a a−1 = −` (1 − FW (`)) + a x (1 − FW (x))dx . (6.4.7) 0

Taking ` > x0 large enough and using (6.1.1), we can bound the tail of the integral in the r.h.s. of the previous display as  Z ` a CW a−(τ−1) a−1 a − τ + 1` for a > τ − 1, a x (1 − FW (x))dx ≤ (6.4.8) x0 a CW log ` for a = τ − 1.

113 Annealed measure: at criticality

R x0 a−1 This with (6.4.7) and the fact that 0 x (1 − FW (x))dx < ∞ proves the upper bounds in (6.4.5). In order to prove (6.4.6) we proceed as in (6.4.7). Let us take a b > `, then

Z b Z b a a a a−1 x dFW (x) = −b (1 − FW (b)) + ` (1 − FW (`)) + a x (1 − FW (x))dx ` `

a a−(τ−1) The power law (6.1.1) implies that x (1 − FW (`)) ≤ CW x for x > x0. Again by (6.1.1) and recalling that in this case a < τ − 1 and taking the limit b → ∞ in the last integral, we have Z ∞ a a a−1 E W 1l{W >`} = ` (1 − FW (`)) + a x (1 − FW (x))dx ` a ≤ C + 1 `a−(τ−1), (6.4.9) W (τ − 1) − a for any ` > x0, concluding the proof of (6.4.6).

In the following we write ci, Ci, i ≥ 1 for constants that only depend on β and on moments of W and satisfy

0 < lim inf ci(β) ≤ lim sup ci(β) < ∞, (6.4.10) β&βc β&βc and the same holds for Ci. The constants Ci appear in upper bounds and ci in lower bounds. Furthermore, we write ei, i ≥ 1 for error functions that depend on β, B and on moments of W , and satisfy

lim sup ei(β, B) < ∞ and lim ei(βc, B) = 0. (6.4.11) B&0 B&0

Here, the subscript i is just a label for constants and error functions. Further, we introduce the following notation that will be used extensively in the following: s sinh(β) α(β) := . (6.4.12) E[W ] ∗ Proposition 6.4.1 (Upper bound on z ). Let β ≥ βc and B > 0. Then, there exists a C1 > 0 such that

∗ q ∗ ∗δ z ≤ E[W ] sinh(β)B + sinh(β)νz − C1z , (6.4.13) where δ takes the values as stated in Theorem 6.2.1. For τ = 5,

∗ q ∗ ∗3 ∗ z ≤ E[W ] sinh(β)B + sinh(β)νz − C1z log (1/z ) . (6.4.14)

114 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

Proof. We frequently use that tanh(B) ≤ B. A Taylor expansion around x = 0 gives that, for some ζ ∈ (0, x), tanh(x + B) = tanh(B) + (1 − tanh2(B))x − tanh(B)(1 − tanh2(B))x2 1 − (1 − tanh2(ζ + B))x3 + tanh(ζ + B)(1 − tanh2(ζ + B))x3 3 1 4 ≤ B + x − x3 + tanh(x + B)x3, (6.4.15) 3 3 1 where we also used that tanh(x) ≤ 1. If we now assume that x + B ≤ atanh 8 , then 1 tanh(x + B) ≤ B + x − x3. (6.4.16) 6 We apply this result to (5.1.5) where x = α(β)W z∗, which we force to be at 1 most atanh 8 by introducing an indicator function as follows: z∗ ≤ E (B + α(β)W z∗) α(β) W + E {tanh (α(β)W z∗ + B) − (B + α(β)W z∗)} · α(β)W 1l ∗ 1 , {α(β)W z +B≤atanh 8 } since tanh(B + x) ≤ B + x. Hence, using (6.4.16),

∗ q ∗ 1 4 h 4 i ∗3 z ≤ E[W ] sinh(β)B + sinh(β)νz − α(β) E W 1l{α(β)W z∗+B≤atanh 1 } z . 6 8 (6.4.17) For E[W 4] < ∞, this is indeed of the form (6.4.13) and we are done. If τ ∈ (3, 5), then it follows from Lemma 6.4.4 that τ−5 h 4 i α(β) ∗ E W 1l ∗ 1 ≥ c4,τ z , (6.4.18) {α(β)W z +B≤atanh 8 } 1 (atanh 8 − B) which proves the proposition for τ ∈ (3, 5). The proof for τ = 5 is similar and we omit it.

We now proceed with the lower bound: ∗ Proposition 6.4.2 (Lower bound on z ). Let β ≥ βc and B > 0. Then, there exist a c1 > 0 such that

∗ q ∗ ∗δ z ≥ E[W ] sinh(β)B + sinh(β)ν z − c1z − Be1, (6.4.19) where δ takes the values as stated in Theorem 6.2.1. For τ = 5,

∗ q ∗ ∗3 ∗ z ≥ E[W ] sinh(β)B + sinh(β)νz − c1z log (1/z ) − Be1. (6.4.20)

Proof. As in (6.4.15) we can expand tanh(x + B) and then bound in the following way 1 tanh(x + B) ≥ B + x − x3 − B(B + Bx + x2), (6.4.21) 3

115 Annealed measure: at criticality where we have used that B − B2 ≤ tanh(B) ≤ B. For E[W 4] < ∞, we can immediately use this to obtain

∗ q ∗ ∗3 z ≥ E[W ] sinh(β)B + sinh(β)νz − c1z − Be1, (6.4.22) where 1 E[W 4] c = sinh2(β) , (6.4.23) 1 3 E[W ]2 and

q sinh (β) 3/2 e = B E [W ] sinh (β) + B sinh(β)νz∗ + E[W 3]z∗2. (6.4.24) 1 E [W ]

All terms in e1 indeed converge to 0 in the appropriate limit, because of Lemma 6.4.3. For τ ∈ (3, 5), we rewrite z∗ as q z∗ = E[W ] sinh(β)B + sinh(β)νz∗ ∗ ∗ + E {tanh (α(β)W z + B) − (B + α(β)W z )} α(β) W 1l{W ≤1/z∗} . ∗ ∗ + E {tanh (α(β)W z + B) − (B + α(β)W z )} α(β) W 1l{W >1/z∗} .

The case where W ≤ 1/z∗ can be treated as above. This gives ∗ ∗ E tanh (α(β)W z + B) − (B + α(β)W z ) α(β) W 1l{W ≤1/z∗}

4 1 E[W 1l{W ≤1 z∗}] ≥ − sinh2(β) / z∗3 − Be , (6.4.25) 3 E[W ]2 2 where

2 E[W 1l{W ≤1/z∗}] ∗ e = Bα(β)E[W 1l ∗ ] + B sinh(β) z 2 {W ≤1/z } E[W ] 3 3 ∗2 + α(β) E[W 1l{W ≤1/z∗}]z . (6.4.26)

By Lemma 6.4.4, 4 ∗τ−5 E[W 1l{W ≤1/z∗}] ≤ C4,τ z , (6.4.27) so that indeed (6.4.25) is bounded from below by

∗τ−2 −c2z − Be2. (6.4.28) with 1 C c = sinh2(β) 4,τ . (6.4.29) 2 3 E[W ]2

Using Lemma’s 6.4.3 and 6.4.4, one can also show that all terms in e2 indeed converge to 0 in the appropriate limit.

116 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

It remains to bound the term where W > 1/z∗. For this we use that tanh(x + B) ≥ 0: ∗ ∗ E tanh (α(β)W z + B) − (B + α(β)W z ) α(β) W 1l{W >1/z∗}

2 E[W 1l{W >1 z∗}] ≥ − sinh(β) / z∗ − Be , (6.4.30) E[W ] 3 where e3 = α(β)E[W 1l{W >1/z∗}]. (6.4.31) By Lemma 6.4.4, 2 ∗τ−3 E[W 1l{W >1/z∗}] ≤ C2,τ z , (6.4.32) again giving the right scaling. As a consequence (6.4.30) is bounded from below ∗τ−2 by −c3z − Be3 with 1 C c = sinh(β) 2,τ . (6.4.33) 3 3 E[W ]2

Similarly, ∗τ−2 e3 ≤ α(β)C1,τ z , (6.4.34) which indeed converges to 0. We conclude that (6.4.19) holds with c1 = c2 + c3 and e1 = e2 + e3.

The upper and lower bounds on z∗ in the previous two propositions allow us to prove that the critical exponents take the values stated in Theorem 6.2.1: Theorem 6.4.3 (Values of β and δ). The critical exponents β and δ equal the values as stated in Theorem 6.2.1 when E[W 2] < ∞ and τ ∈ (3, 5). Furthermore, for τ = 5 (2.5.12) holds.

Proof. Proof for exponent β. We start by giving upper bounds on the magnetization. From (5.1.4) it follows that q Mf(β, B) = E [tanh (α(β)W z∗ + B)] ≤ B + E [W ] sinh (β)z∗. (6.4.35)

We ﬁrst analyze β and hence take the limit B & 0 for β > βc. This gives

+ q ∗ Mf(β, 0 ) ≤ E [W ] sinh (β)z0, (6.4.36)

∗ ∗ + where we write z0 = limB&0 z . Since Mf(β, 0 ) > 0 by the deﬁnition of βc, ∗ the same must be true for z0. We will deal ﬁrst with the cases τ ∈ (3, 5) and 4 ∗ E[W ] < ∞. Taking the limit B & 0 in (6.4.13) and dividing by z0, we get for τ 6= 5 ∗δ−1 C1z0 ≤ sinh(β)ν − 1, (6.4.37)

117 Annealed measure: at criticality and hence, observing that β = 1/(δ − 1),

∗ −β β z0 ≤ C1 (sinh(β)ν − 1) . (6.4.38) 2 From a Taylor expansion of sinh(β) around βc = asinh(E[W ]/E[W ]) it follows that sinh(β)ν − 1 ≤ cosh(β)ν(β − βc). (6.4.39) Hence,

+ q −β β β Mf(β, 0 ) ≤ E [W ] sinh (β)C1 (cosh(β)ν) (β − βc) , (6.4.40) so that it is easy to see that

Mf(β, 0+) lim sup β < ∞. (6.4.41) β&βc (β − βc) The lower bound can be obtained in a similar fashion. Starting from tanh x ≥ 2 x − x and taking the limit B & 0 for β > βc in (5.1.4), we obtain

+ q ∗ ∗2 Mf(β, 0 ) ≥ E [W ] sinh (β)z0 − sinh(β)νz0 . (6.4.42)

∗ Again, starting from the lower bound (6.4.19), taking B & 0 and dividing by z0

∗ −β β z0 ≥ c1 (sinh(β)ν − 1) , (6.4.43) and, by a Taylor expansion around βc,

2 sinh(β)ν − 1 = cosh(βc)ν(β − βc) + O((β − βc) ). (6.4.44)

Using (6.4.38), (6.4.43) and (6.4.44) in (6.4.42) we obtain:

+ q −β 2 β Mf(β, 0 ) ≥ E [W ] sinh (β)c1 cosh(βc)ν(β − βc) + O((β − βc) ) −2β 2 2β − sinh(β)νC1 cosh(βc)ν(β − βc) + O((β − βc) ) , which shows that also Mf(β, 0+) 0 < lim inf β , (6.4.45) β&βc (β − βc) concluding the proof for the exponent β in the cases τ ∈ (3, 5) and E[W 4] < ∞. In the case τ = 5 we can prove the upper bound for Mf(β, 0+) in a similar ∗ fashion, i.e., taking the limit B & 0 for β > βc of (6.4.14) and dividing by z0. This yields to

∗2 sinh(β)ν − 1 cosh(β)ν(β − βc) (β − βc) z0 ≤ ∗ ≤ ∗ ≤ Ce ∗ , (6.4.46) C1 log(1/z0 ) C1 log(1/z0 ) log(1/z0 ) where (6.4.39) has been used in order to obtain the second inequality and cosh(β) has been bounded in a right neighborhood of βc to obtain the third inequality.

118 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

∗ 1 Since x 7→ 1/ log(1/x) is increasing in (0,1) and z0 ≤ C(β − βc) 2 for some C > 0,1 form (6.4.46) we obtain:

Ce(β − βc) z∗2 ≤ . (6.4.47) 0 1 2 C1 log(1/[C(β − βc) / ]) The previous inequality together with (6.4.36), proves the upper bound

Mf(β, 0+) lim sup < ∞. (6.4.48) 1 2 β&β / c β − βc log(1/(β − βc))

The lower bound can be obtained in the same way. Indeed, from (6.4.20) in the limit B → 0, we obtain, for some positive constants Ce and Cb

∗2 sinh(β)ν − 1 (β − βc) (β − βc) z0 ≥ ∗ ≥ Ce ∗ ≥ Cb , (6.4.49) C1 log(1/z0 ) log(1/z0 ) log(1/(β − βc)) where, once more, we have used that x 7→ 1/ log(1/x) is increasing in (0,1) and ∗ 1/(2−ε) 2 the bound z0 ≥ C(β − βc) for some C > 0 and any 0 < ε < 2. The previous inequality plugged in (6.4.42) gives

Mf(β, 0+) lim inf > 0, (6.4.50) 1 2 β&βc / β − βc log(1/(β − βc)) concluding the proof for τ = 5.

Proof for exponent δ. We continue with the analysis for δ. Setting β = βc in (6.4.13), we obtain q ∗ − 1 1/δ z (βc, B) ≤ (C1 E[W ]) δ B . (6.4.51)

Using this inequality in (6.4.35) with β = βc, we obtain

1 − δ E[W ]C1 1/δ Mf(βc, B) ≤ B + B , (6.4.52) p 1+ 1 ( E[W 2]) δ which proves that Mf(βc, B) lim sup < ∞ (6.4.53) 1 δ B&0 B /

1 ∗ 2 ∗2 ∗2 1 The proof of z0 ≤ C(β − βc) can be obtained by rewriting (6.4.46) as −z0 log z0 ≤ k(β − βc), for some k > 0. Since w < −w log w for w < 1/e, we conclude that for β − βc > 0 ∗2 small enough, the previous inequality gives z0 < k(β − βc). ∗ 1/(2−ε) 2 The proof of the inequality z0 ≥ C(β − βc) , for 0 < ε < 2 can be obtained starting −ε from the rightmost inequality of (6.4.49) combined with the fact that log 1/x ≤ Aεx for all x ∈ (0, 1) and any ε > 0.

119 Annealed measure: at criticality

since δ > 1. Inequality (6.4.19) with β = βc gives 1/δ ∗ −1/δ 1 1/δ z (βc, B) ≥ c − e1(βc, B) B (6.4.54) 1 pE[W 2] This estimate, along with (6.4.51), will be used in the lower bound of the 2 magnetization at β = βc obtained by tanh x ≥ x − x :

E[W ] ∗ ∗ 2 2 Mf(βc, B) ≥ B + (1 − 2B)z (βc, B) − z (βc, B) − B , (6.4.55) pE[W 2] giving, for B > 0 small, 1/δ E[W ] −1/δ 1 1/δ Mf(βc, B) ≥ B + (1 − 2B)c − e1(βc, B) B pE[W 2] 1 pE[W 2] q −2/δ 2/δ 2 − (C1 E[W ]) B − B . (6.4.56)

Recalling that limB&0 e1(βc, B) = 0 and δ > 1, the previous bound gives q q Mf(βc, B) 2 −1/δ lim inf ≥ E[W ](c1 E[W ]) > 0, (6.4.57) B&0 B1/δ which concludes the proof for δ in the cases τ ∈ (3, 5) and E[W 2] < ∞. The analysis for τ = 5 can be performed in a similar way as for β.

Proof of Corollary 6.2.1. The proof can be simply adapted as in [21, Corollary 2.9].

6.4.2 Susceptibility: critical exponents γ and γ0

We now analyze the susceptibility and compute the critical exponents γ and γ0. We start by computing the former under more general condition than those of Theorem 6.2.1. Theorem 6.4.4 (Value of γ). For E[W 2] < ∞,

2 + E[W ] lim χe(β, 0 )(βc − β) = 2 tanh(βc), (6.4.58) β%βc E[W ] so that γ = 1.

Proof. From Theorem 5.1.1 it follows that in the one-phase region, i.e., for β < βc or B 6= 0, ∂ χ(β, B) = Mf(β, B) e ∂B ∂z∗ = E 1 + α(β)W 1 − tanh2 (α(β)W z∗ + B) . (6.4.59) ∂B

120 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

We can also compute the derivative of z∗ by taking the derivative of (5.1.5):

∂z∗ ∂z∗ = E α(β)W + α(β)2W 2 1 − tanh2 (α(β)W z∗ + B) . (6.4.60) ∂B ∂B

2 If we take the limit B & 0 for β < βc, then the tanh (·) term vanishes, since by ∗ ∗ deﬁnition of βc it holds that z0 ≡ limB&0 z = 0. Hence, if we write ∂z∗ ∂ 0 = lim z∗(β, B), (6.4.61) ∂B B&0 ∂B then (6.4.60) simpliﬁes to

∂z∗ q ∂z∗ 0 = E[W ] sinh(β) + sinh(β)ν 0 . (6.4.62) ∂B ∂B

∗ ∂z0 Solving for ∂B gives ∂z∗ pE[W ] sinh(β) 0 = . (6.4.63) ∂B 1 − sinh(β)ν Also taking the limit B & 0 in (6.4.59) and using the above gives

E[W ] sinh(β) χ(β, 0+) = 1 + . (6.4.64) e 1 − sinh(β)ν

From a Taylor expansion around βc, we get that

sinh(βc) − cosh(βc)(βc − β) ≤ sinh(β) ≤ sinh(βc) − cosh(β)(βc − β), (6.4.65) so that

2 2 E[W ] sinh(β) + E[W ] sinh(β) 1 + 2 ≤ χe(β, 0 ) ≤ 1 + 2 , E[W ] cosh(βc)(βc − β) E[W ] cosh(β)(βc − β) since sinh(βc)ν = 1. Hence, (6.4.58) follows. We now analyze γ0: Theorem 6.4.5 (Value of γ0). For W satisfying Condition 6.1.1 with E[W 4] < ∞ or with τ ∈ (3, 5), γ0 = 1. (6.4.66)

Proof. We split the proof into the two cases that cover the hypotheses of the theorem. (a) Proof under the assumption E[W 4] < ∞. We are now in the regime ∗ where β > βc, so that z0 > 0. We start from (6.4.60), take the limit B & 0 and linearize the hyperbolic tangent. In order to control this approximation, we deﬁne g(x) = x2 − tanh2(x) and remark that on the basis of our assumption on W , we

121 Annealed measure: at criticality have that E[(W 2 ∨ 1)g(W )] < ∞. It will be useful also to factorize g(x) = x4k(x) with k(x) = O(1) as x → 0, so that we also have E[W 6k(W )] < ∞. This gives

∂z∗ ∂z∗ 0 = E α(β)W + α(β)2W 2 0 1 − tanh2 (α(β)W z∗) ∂B ∂B 0 q ∂z∗ = E[W ] sinh(β) − e + 0 sinh(β)ν − α(β)4 E[W 4]z∗2 0 ∂B 0 ∂z∗ + E 0 α(β)2W 2 + α(β)W g (α(β)W z∗) , (6.4.67) ∂B 0 where sinh(β) 3/2 e = E W 3 z∗2. (6.4.68) 0 E [W ] 0 ∗ ∂z0 Solving (6.4.67) for ∂B gives

∂z∗ pE[W ] sinh(β) − e − E [α(β)W g (α(β)W z∗)] 0 = 0 0 . (6.4.69) 4 4 ∗2 2 2 ∗ ∂B 1 − sinh(β)ν + α(β) E[W ]z0 − E [α(β) W g (α(β)W z0 )] To analyze (6.4.69) we use the lower and upper bounds in Propositions 6.4.2 and 6.4.1. Taking the limit B & 0 in (6.4.19) with δ = 3, c1 given in (4.5.26) ∗ and dividing by z0, we obtain E[W ]2 1 z∗2 ≥ 3 (sinh(β)ν − 1) . (6.4.70) 0 E[W 4] sinh2(β)

∗ Taking the same limit B & 0 in (6.4.17) and dividing by z0 we obtain also

2 ∗2 E[W ] 1 z0 ≤ 6 4 2 (sinh(β)ν − 1) . (6.4.71) E[W 1l ∗ 1 ] sinh (β) {α(β)W z ≤atanh 8 } By Taylor expansion,

2 sinh(β)ν − 1 = cosh(βc)ν(β − βc) + O((β − βc) ), (6.4.72)

4 we conclude, from (6.4.70), (6.4.71), and the fact that E[W 1l ∗ 1 ] {α(β)W z ≤atanh 8 } 4 ∗2 → E[W ] as β → βc, that z0 = O(β − βc). Using this, we can now evaluate the terms in numerator and denominator of (6.4.69) as β → βc. The ﬁrst term in the numerator has a non vanishing ﬁnite limit, while e0 = O(β − βc). The third ∗ term (ignoring the irrelevant multiplicative factor α(β) ) is E [W g (α(β)W z0 )] = 4 ∗4 5 ∗ 1 α(β) z0 E W k (α(β)W z0 ) = O((β − βc)). Indeed, since k(x) ≤ x2 ,

4 ∗4 5 ∗ 2 3 ∗2 α(β) z0 E W k (α(β)W z0 ) ≤ α(β) E W z0 = O(β − βc). (6.4.73)

Let us now consider the denominator and deﬁne

4 4 ∗2 D(β) := 1 − sinh(β)ν + α(β) E[W ]z0 . (6.4.74)

122 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

By (6.4.70), (6.4.71) and (6.4.72),

2 2 cosh(βc)ν(β − βc) + O((β − βc) ) 2 ≤ D(β) ≤ (a(β) − 1) cosh(βc)ν(β − βc) + O((β − βc) ), (6.4.75) where a(β) is a function that converges to 6 as β → βc. Thus, from the previous display we obtain D(β) = O(β − βc). The fourth term in the denominator of (6.4.69), again discarding an irrelevant factor and arguing as before, is 2 ∗ 4 ∗4 6 ∗ 2 E W g (α(β)W z0 ) = α(β) z0 E W k (α(β)W z0 ) = O((β − βc) ). There- fore, summarizing our ﬁndings, ∂z∗ 0 = O((β − β )−1). (6.4.76) ∂B c From (6.4.59), the upper bound follows using (6.4.76):

∂z∗ χ(β, 0) ≤ E 1 + α(β)W 0 e ∂B q −1 ≤ 1 + sinh(β)E[W ]O((β − βc) ). (6.4.77)

Similarly, for the lower bound we use that 1 − tanh2(x) ≥ 1 − x2 for every x, we obtain ∂z∗ χ(β, 0) ≥ E 1 + α(β)W 0 1 − α(β)2W 2z∗2 e ∂B 0 ∂z∗ ∂z∗ = 1 + E α(β)W 0 − E α(β)2W 2z∗2 − E α(β)3W 3z∗2 0 ∂B 0 0 ∂B q −1 = 1 + sinh(β)E[W ]O((β − βc) ) − sinh(β)νO(β − βc) − α(β)3E[W 3]O(1), (6.4.78)

∗2 again starting from (6.4.59), using (6.4.76) and z0 = O(β − βc). From (6.4.77) and (6.4.78) we obtain

+ + 0 < lim inf χe(β, 0 )(β − βc) ≤ lim sup χe(β, 0 )(β − βc) < ∞, (6.4.79) β&βc β&βc proving the theorem in the case that E[W 4] < ∞. (b) Proof for W satisfying Condition 6.1.1 (ii). Now we generalize the previous proof in order to encompass also the case of those W whose distribution function F (w) = 1 − P(W > w) satisﬁes Condition 6.1.1(ii). We start by deﬁning

∗ 2 2 ∗ hβ,B,z∗ (w) = tanh (αwz + B) α w − α w z , (6.4.80) where the dependence of α on β has been dropped, and rewrite (5.1.5) as

∗ 2 ∗ 2 z = E hβ,B,z∗ (W ) + α z E[W ]. (6.4.81)

123 Annealed measure: at criticality

Using integration by parts,

Z +∞ Z +∞ E hβ,B,z∗ (W ) = hβ,B,z∗ (w)dF (w) = − hβ,B,z∗ (w)d(1 − F (w)) 0 0 = − lim [hβ,B,z∗ (w)(1 − F (w))] + hβ,B,z∗ (0)(1 − F (0)) w→+∞ Z +∞ 0 + hβ,B,z∗ (w)(1 − F (w))dw. (6.4.82) 0 The boundary terms in the previous display vanish and therefore

Z +∞ 0 E hβ,B,z∗ (W ) = hβ,B,z∗ (w)(1 − F (w))dw. (6.4.83) 0

Taking into account that the power law of Condition 6.1.1(ii) holds for w > w0, we write the previous integral as

∗ ∗ E hβ,B,z∗ (W ) = G(β, B, z ) + J(β, B, z ), (6.4.84) where

Z w0 ∗ 0 G(β, B, z ) := hβ,B,z∗ (w)(1 − F (w))dw, (6.4.85) 0 Z +∞ ∗ 0 J(β, B, z ) := hβ,B,z∗ (w)(1 − F (w))dw. (6.4.86) w0 Therefore, (6.4.81) can be rewritten as

z∗ = G(β, B, z∗) + J(β, B, z∗) + α2z∗E[W 2]. (6.4.87)

∗ Now we take the limit B & 0 in the previous equation. Recalling that z0 := ∗ limB&0 z > 0, and since the following limits exist:

∗ ∗ ∗ ∗ lim G(β, B, z ) = G(β, z0 ), lim J(β, B, z ) = J(β, z0 ) (6.4.88) B&0 B&0 by bounded convergence, then we arrive to

∗ ∗ ∗ 2 ∗ 2 z0 = G(β, z0 ) + J(β, z0 ) + α z0E[W ]. (6.4.89)

∗ ∗ In the next step we bound J(β, z0 ). From the deﬁnition of J(β, B, z ) in (6.4.86), and Condition 6.1.1(ii),

Z +∞ 0 −(τ−1) ∗ cW hβ,B,z∗ (w)w dw ≤ J(β, B, z ) w0 Z +∞ 0 −(τ−1) ≤ CW hβ,B,z∗ (w)w dw. (6.4.90) w0

124 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

Applying the change of variable y = αz∗w leads to Z +∞ 0 −(τ−1) hβ,B,z∗ (w)w dw w0 Z +∞ = ατ−1z∗τ−2 tanh(y + B) − y tanh2(y + B) − y y−(τ−1)dy. (6.4.91) ∗ αw0z Therefore, denoting Z +∞ I(β, B, z∗) := tanh(y + B) − y tanh2(y + B) − y y−(τ−1)dy, (6.4.92) ∗ αw0z we can rewrite (6.4.90) as follows: τ−1 ∗τ−2 ∗ ∗ τ−1 ∗τ−2 ∗ cW α z I(β, B, z ) ≤ J(β, B, z ) ≤ CW α z I(β, B, z ). (6.4.93) Since, again by bounded convergence, Z +∞ ∗ 2 −(τ−1) ∗ lim I(β, B, z ) = tanh(y) − y tanh (y) − y y dy =: I(β, z0 ), B&0 ∗ αw0z0 we obtain from (6.4.90) that τ−1 ∗τ−2 ∗ ∗ τ−1 ∗τ−2 ∗ cW α z0 I(β, z0 ) ≤ J(β, z0 ) ≤ CW α z0 I(β, z0 ). (6.4.94) On the other hand, since tanh(y) − y tanh2(y) − y < 0 for y > 0, we also have Z +∞ 2 −(τ−1) ∗ k(τ) := [y tanh (y) + y − tanh(y)]y dy ≤ −I(β, z0 ) 1 Z +∞ ≤ [y tanh2(y) + y − tanh(y)]y−(τ−1)dy =: K(τ). (6.4.95) 0 Therefore, from (6.4.89), (6.4.94) and (6.4.95), ∗ ∗ τ−1 ∗τ−2 2 ∗ 2 z0 ≥ G(β, z0 ) − cW α z0 K(τ) + α z0E[W ] (6.4.96) and ∗ ∗ τ−1 ∗τ−2 2 ∗ 2 z0 ≤ G(β, z0 ) − CW α z0 k(τ) + α z0E[W ]. (6.4.97) ∗ The next step is to control the behavior of G(β, z0 ) as β → βc. We start by ∗ ∗3 ∗ showing that G(β, z0 ) is O(z0 ) as β → βc. From the deﬁnition of G(β, z0 ), Z w0 ∗ 2 ∗ 2 ∗ 2 ∗ ∗ G(β, z0 ) = −α z0w − tanh (αz0w)α z0w + α tanh(αz0w) (1 − F (w))dw. 0 Since the function between the square brackets is negative for y > 0 and decreasing, we have ∗ 0 ≥ G(β, z0 ) Z w0 2 ∗ 2 ∗ 2 ∗ ∗ ≥ [−α w0z0 − α w0z0 tanh (αw0z0 ) + α tanh(αw0z0 )] (1 − F (w))dw 0 2 ∗ 2 ∗ 2 ∗ ∗ ≥ [−α w0z0 − α w0z0 tanh (αw0z0 ) + α tanh(αw0z0 )] 4 = − α4w3z∗3 + O(z∗5) (6.4.98) 3 0 0

125 Annealed measure: at criticality where the last equality is obtained by Taylor expansion. ∗ ∗3 Thus, the previous inequality implies that G(β, z0 ) = O(z0 ). Again, from ∗ (6.4.96) and (6.4.97) dividing by z0, 2 2 ∗τ−3 ∗ ∗2−τ τ−1 1 − α E[W ] ≥ z0 G(β, z0 )z0 − cW α K(τ) , (6.4.99) and

2 2 ∗τ−3 ∗ ∗2−τ τ−1 1 − α E[W ] ≤ z0 G(β, z0 )z0 − CW α k(τ) . (6.4.100)

∗ ∗2−τ ∗5−τ Since G(β, z0 )z0 = O(z0 ) and τ ∈ (3, 5), the previous inequalities to- ∗τ−3 gether with (6.4.72) imply that z0 = O(β − βc) as β & βc. ∗ Next, we consider the derivative of z0. Again, taking the limit B & 0 for β > βc of (6.4.60) we obtain

∂z∗ αE[W ] − αE[W tanh2(αW z∗)] 0 = 0 . (6.4.101) 2 2 2 2 2 ∗ ∂B 1 − α E[W ] + α E[W tanh (αW z0 )]

Since the numerator has a ﬁnite positive limit as β & βc (in particular, the second term is vanishing), we will focus on the denominator

2 2 2 2 2 ∗ D2(β) := 1 − α E[W ] + α E[W tanh (αW z0 )]. (6.4.102) We start by decomposing the average

2 2 ∗ 2 2 ∗ E[W tanh (αW z0 )] = E[W tanh (αW z0 )1l{W ≤w0}] 2 2 ∗ + E[W tanh (αW z0 )1l{W >w0}], (6.4.103) and analyze the two terms separately. The ﬁrst one can be bounded as follows

2 2 ∗ 2 4 ∗2 0 ≤ E[W tanh (αW z0 )1l{W ≤w0}] ≤ α w0z0 , (6.4.104) showing that

2 2 2 ∗ ∗2 τ−3 E[W tanh (αW z0 )1l{W ≤w0}] = O(z0 ) = O((β − βc) ), (6.4.105) with the exponent satisfying 2/(τ − 3) > 1 since τ ∈ (3, 5). The second term can be treated with the integration by parts formula

2 2 ∗ E[W tanh (αW z0 )1l{W >w0}] 2 2 ∗ 2 2 ∗ = − lim [w tanh (αwz0 )(1 − F (w))] + w0 tanh (αw0z0 )(1 − F (w0)) w→+∞ Z +∞ ∂ 2 2 ∗ + [w tanh (αwz0 )](1 − F (w))dw. (6.4.106) w0 ∂w Since τ > 3, from Condition 6.1.1 we conclude that the limit in the previous display vanishes. It is also simple to see that

2 2 2 ∗ ∗2 τ−3 w0 tanh (αw0z0 )(1 − F (w0)) = O(z0 ) = O((β − βc) ). (6.4.107)

126 6.4 Proofs of Theorem 6.2.1: Annealed critical exponents

From (6.4.102) and using (6.4.103), (6.4.105), (6.4.106), (6.4.107), we can write

2 D2(β) = D2(β) + O((β − βc) τ−3 ), (6.4.108) with Z +∞ 2 2 2 ∂ 2 2 ∗ D2(β) := (1 − α E[W ]) + α [w tanh (αwz0 )](1 − F (w))dw. w0 ∂w s The second term in the r.h.s. of (6.4.108) is O((β − βc) ) with s > 1, therefore we can forget it since the first term of D2(β) is O(β − βc). Now we focus on the second term of D2(β). By using (6.1.1) and applying the change of variable y = αz∗w, we can bound the integral in the last display as Z +∞ ∂ 2 2 ∗ τ−3 ∗τ−3 [w tanh (αwz0 )](1 − F (w))dw ≤ CW α z0 M(τ), (6.4.109) w0 ∂w where Z +∞ M(τ) := 2y tanh2(y) + 2y2 tanh(y)(1 − tanh2(y)) y−(τ−1)dy, 0 and the bound in (6.4.109) is obtained thanks to the positivity of the integrand. The convergence of the integral is ensured by the fact that this function is O(y4−τ ) close to y = 0 with 1 > 4 − τ > −1 and is O(y−τ+2) as y → ∞ with −τ + 2 < −1. In a similar fashion, we can also obtain Z +∞ ∂ 2 2 ∗ τ−3 ∗τ−3 [w tanh (αwz0 )](1 − F (w))dw ≥ cW α z0 m(τ), (6.4.110) w0 ∂w with Z +∞ m(τ) := 2y tanh2(y) + 2y2 tanh(y)(1 − tanh2(y)) y−(τ−1)dy, ε (6.4.111) for β sufficiently close to βc. At this stage ε > 0 is an arbitrary fixed quantity that will be chosen later (but independently of β). By (6.4.96) and (6.4.97),

∗ ∗−1 2 2 ∗ ∗−1 2 2 G(β, z0 )z0 − (1 − α E[W ]) ∗τ−3 G(β, z0 )z0 − (1 − α E[W ]) τ−1 ≤ z0 ≤ τ−1 , cW α K(τ) CW α k(τ) which, substituted in (6.4.109) and (6.4.110), gives m(τ) m(τ) G(β, z∗)z∗−1 − (1 − α2E[W 2]) 0 0 K(τ) K(τ) Z +∞ 2 ∂ 2 2 ∗ ≤ α [w tanh (αwz0 )](1 − F (w))dw w0 ∂w M(τ) M(τ) ≤ G(β, z∗)z∗−1 − (1 − α2E[W 2]) . (6.4.112) 0 0 k(τ) k(τ)

127 Annealed measure: at criticality

By deﬁnition of D2(β),

m(τ) m(τ) G(β, z∗)z∗−1 + (1 − α2E[W 2]) 1 − 0 0 K(τ) K(τ) M(τ) M(τ) ≤ D (β) ≤ G(β, z∗)z∗−1 + (1 − α2E[W 2]) 1 − . (6.4.113) 2 0 0 k(τ) k(τ)

m(τ) In the last step of the proof, we show that K(τ) > 1. This can be done by properly choosing the arbitrary quantity ε in (6.4.111). We will prove the ﬁrst inequality, the second one can be obtained in the same way. Starting from (6.4.95) and (6.4.111), we introduce the functions Kb(τ) and ma(τ) for a ≥ 0, b ≥ 0 as Z +∞ d 2 −(τ−1) Kb(τ) := y − y tanh(y) y dy, b dy Z +∞ d 2 2 −(τ−1) ma(τ) := y tanh (y) y dy, a dy which coincide with K(τ) and m(τ) for b = 0 and a = ε, respectively. By applying the integration by parts formula, the two functions can be written as

Z +∞ 3−τ 2−τ 2 −τ Kb(τ) = −b + b tanh(b) + (τ − 1) (y − y tanh(y))y dy, b Z +∞ 2 2 −τ ma(τ) = −a tanh(a) + (τ − 1) y tanh (y)y dy. a Since Z +∞ y2 tanh2(y) y−τ dy m (τ) m (τ) lim a = 0 = 0 > 1, (6.4.114) + Z +∞ a→0 Kb(τ) K(τ) 2 −τ b→0+ y − y tanh(y) y dy 0 where the inequality can be proved by observing that y2 tanh2(y) > y2 − y tanh(y) for all y > 0, we obtain that for any ε > 0 suﬃciently small,

m(τ) > 1. (6.4.115) K(τ)

∗ ∗−1 ∗2 s 2 Since G(β, z0 )z0 = O(z0 ) = O((β − βc) ) with s = τ−3 > 1 and (1 − 2 2 2 2 α E[W ]) = O(β − βc), with 1 − α E[W ] < 0 for β > βc and close to βc (see (6.4.72)), we conclude that 0 < D2(β) = O(β − βc), for the same values of β. This proves that ∂z∗ 0 < 0 = O((β − β )−1). (6.4.116) ∂B c ∗τ−3 The previous equation together with z0 = O(β − βc) allows us to conclude the proof along the same lines of the case with E[W 4] < ∞. Indeed, the

128 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem upper bound (6.4.77) is still valid in the present case, since only the ﬁrst moment of W is involved. For the lower bound we argue as follows. Since, 1 − tanh2(x) > 1 − tanh(x) > 1 − x for x > 0, we have

∂z∗ χ(β, 0) ≥ E 1 + α(β)W 0 (1 − α(β)W z∗) e ∂B 0 q q ∂z∗ ∂z∗ = 1 − sinh(β)E[W ]z∗ + sinh(β)E[W ] 0 − sinh(β)νz∗ 0 0 ∂B 0 ∂B q 1/(τ−3) = 1 − sinh(β)E[W ]O((β − βc) )

q −1 + sinh(β)E[W ]O((β − βc) ) 1/(τ−3) −1 − sinh(β)νO((β − βc) )O((β − βc) ). (6.4.117)

The inequalities (6.4.77) and (6.4.117) imply (6.4.79) concluding the proof of the theorem.

6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

In this section we prove Theorem 6.3.1. For this, we follow the strategy of the proof for the Curie–Weiss model (see e.g. [26, Theorem V.9.5]). It suﬃces to prove that for any real number r R ∞ SN −∞ exp (rz − f(z)) dz lim PeN exp r = ∞ , (6.5.1) N→∞ δ/(δ+1) R N −∞ exp (−f(z)) dz where f(z) is the density stated in (6.3.4).

As observed in Section 5.4, the measure PeN is approximately equal to the inhomogeneous Curie–Weiss measure

w w 1 1 sinh β P i j σ σ X 2 i,j∈[N] ` i j PbN (g) = g(σ)e N ZbN σ∈ΩN 2 1 sinh β P 1 X 2 ` wiσi = g(σ)e N i∈[N] , (6.5.2) ZbN σ∈ΩN where g(σ) is any bounded function deﬁned in ΩN and ZbN is the associated normalization factor, i.e., 2 1 sinh β P wiσi X 2 `N i∈[N] ZbN = e . (6.5.3) σ∈ΩN

We ﬁrst prove the theorem for this measure PbN , which is the rank-1 inhomogeneous Curie–Weiss model with β replaced with sinh(β).

129 Annealed measure: at criticality

For this, we use the Hubbard–Stratonovich identity to rewrite P exp r SN bN N λ as a fraction of two integrals of an exponential function in Lemma 6.5.1. We next split the analysis into the cases E[W 4] < ∞ and τ ∈ (3, 5) in Sections 6.5.2 and 6.5.3, respectively. For both these cases we analyze the exponents in the integrals and use Taylor expansions to show that they converge in Lemmas 6.5.2 and 6.5.4, respectively. We then use dominated convergence to show that the integrals also converge in Lemmas 6.5.3 and 6.5.5, respectively. The tail behavior of f(x) for τ ∈ (3, 5) is analyzed in Lemma 6.5.6. Combining these results we conclude the proof of Theorem 6.3.1 in Section 6.5.4: we ﬁrst prove the theorem for PbN and then we show that the theorem also holds for PeN in Lemma 6.5.7. Finally, in Section 6.5.5, we discuss how to adapt the proof to obtain the results on the scaling window.

6.5.1 Rewrite of the moment generating function

λ To ease notation we ﬁrst rescale SN by N and later set λ = δ/(δ + 1). We rewrite P exp r SN in the following lemma: bN N λ λ Lemma 6.5.1 (Moment generating function of SN /N ). For B = 0,

R ∞ −NGN (z;r) SN −∞ e dz PbN exp r = ∞ , (6.5.4) N λ R −NGN (z;0) −∞ e dz where 1 h r i G (z; r) = z2 − E log cosh α (β)W z + , (6.5.5) N 2 N N N λ with s sinh β αN (β) = . (6.5.6) E[WN ]

2 Proof. We use the Hubbard–Stratonovich identity, i.e., we write et /2 = E etZ , with Z standard Gaussian, to obtain

2 r P 1 sinh β P w σ SN X σi 2 ` i i Nλ i∈[N] N i∈[N] ZbN PbN exp r = e e N λ σ∈ΩN q r P sinh β P w σ Z X σi h ` i∈[N] i i i = e Nλ i∈[N] E e N

σ∈ΩN r N h Y sinh β r i = 2 E cosh wiZ + λ `N N i∈[N] q P sinh β r h log cosh wiZ+ i = 2N E e i∈[N] `N Nλ . (6.5.7)

130 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

We rewrite the sum in the exponential, using the fact that WN = wUN , where UN is a uniformly chosen vertex in [N], as

SN ZbN PbN exp r N λ s N h n h sinh β r ioi = 2 E exp NE log cosh WN Z + λ | Z NE[WN ] N N ∞ 2 Z 2 n h z r io √ −z /2 √ = e exp NE log cosh αN (β)WN + λ dz. 2π −∞ N N √ By substituting z/ N for z, we get

SN ZbN PbN exp r N λ r Z ∞ N N −Nz2/2 n h r io = 2 e exp NE log cosh αN (β)WN z + λ dz 2π −∞ N r N Z ∞ = 2N e−NGN (z;r)dz. (6.5.8) 2π −∞ For r = 0, this gives r N Z ∞ N −NGN (z;0) ZbN = 2 e dz, (6.5.9) 2π −∞ so that the lemma follows.

6.5.2 Convergence for E[W 4] < ∞

We ﬁrst analyze the asymptotics of the function GN (z; r): 4 Lemma 6.5.2 (Asymptotics of GN for E[W ] < ∞). For β = βc,N , B = 0 and E[W 4] < ∞,

r 4 1/4 E[W ] 1 E[W ] 4 lim NGN (z/N ; r) = −zr + z . (6.5.10) N→∞ ν 12 E[W 2]2

Proof. Taylor expanding log cosh(x) about x = 0 gives that

x2 1 log cosh(x) = − x4 + O(x6). (6.5.11) 2 12

1/4 We want to use this to analyze NGN (z/N ; r) and hence need to analyze the q sinh βc,N z r second, fourth and sixth moment of WN 1 4 + λ . E[WN ] N / N

131 Annealed measure: at criticality

The second moment equals, using that λ = δ/(δ + 1) = 3/4, h z r 2i E αN (βc,N )WN + N 1/4 N λ z2 q zr r2 = νN sinh βc,N √ + 2 sinh βc,N E[WN ] + N N N 6/4 s z2 zr E[W ] = √ + 2 N + o(1/N), (6.5.12) N N νN where we have used that sinh βc,N = 1/νN in the second equality.

For the fourth moment we use that by assumption the ﬁrst four moments of WN are O(1). Hence, for all r, h z r 4i E αN (βc,N )WN + N 1/4 N λ sinh2 β z4 1 1 1 1 = c,N E[W 4 ] + O + + + 2 N 3 4+λ 2 4+2λ 1 4+3λ 4λ E[WN ] N N / N / N / N E[W 4 ] z4 = N + o(1 N). (6.5.13) 2 2 / E[WN ] N For the sixth moment, we have to be a bit more careful since E[W 6] is potentially inﬁnite. We can, however, use that N N 6 1 X 6 2 1 X 4 2 4 E[WN ] = wi ≤ (max wi) wi = (max wi) E[WN ]. (6.5.14) N i N i i=1 i=1

N 1/4 D 4 It can easily be seen that maxi=1 wi = o(N ) when WN −→ W and E[WN ] → E[W 4] < ∞. Hence, h z 6i sinh3 β z6 o(N 1/2)E[W 4 ] z6 E α (β )W = c,N E[W 6 ] = N N c,N N 1/4 3 N 6/4 2 3 6/4 N E[WN ] N E[WN ] N = o(1/N). In a similar way, it can be shown that h z r 6i E αN (βc,N )WN + = o(1/N). (6.5.15) N 1/4 N λ

Putting everything together and using that the ﬁrst four moments of WN converge by assumption, 1/4 lim NGN (z/N ; r) N→∞ √ N 2 h z r i = lim z − NE log cosh αN (βc,N )WN + N→∞ 2 N 1/4 N λ r E[W ] 1 E[W 4] = −zr + z4. (6.5.16) ν 12 E[W 2]2

132 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

From Lemma 6.5.2 it also follows that the integral converges as we show next: 4 Lemma 6.5.3 (Convergence of the integral for E[W ] < ∞). For β = βc,N , B = 0 and E[W 4] < ∞,

4 ∞ ∞ zr √E[W ] − 1 E[W ] z4 Z 1/4 Z 12 E[W 2]2 lim e−NGN (z/N ;r)dz = e E[W 2] dz. (6.5.17) N→∞ −∞ −∞ Proof. We prove this lemma using dominated convergence. Hence, we need to 1/4 ﬁnd a lower bound on NGN (z/N ; r). We ﬁrst rewrite this function by using that 2 2 1 2 2 z 1 z E αN (βc,N ) W = . (6.5.18) 2 N N 1/4 2 N 1/4 Hence,

1/4 GN (z/N ; r) (6.5.19) 2 1 2 2 z z r = E αN (βc,N ) W − log cosh αN (βc,N )WN + 2 N N 1/4 N 1/4 N λ 1 z 2 z = E αN (βc,N )WN − log cosh αN (βc,N )WN 2 N 1/4 N 1/4 z r z − E log cosh αN (βc,N )WN + − log cosh αN (βc,N )WN N 1/4 N λ N 1/4 Since d2 1 x2 − log cosh x = 1 − (1 − tanh2(x)) = tanh2(x) ≥ 0, (6.5.20) dx2 2 1 2 the function 2 x − log cosh x is convex and we can use Jensen’s inequality to bound

h1 z 2 z i E αN (βc,N )WN − log cosh αN (βc,N )WN 2 N 1/4 N 1/4 1 z 2 z ≥ αN (βc,N )E[WN ] − log cosh αN (βc,N )E[WN ] 2 N 1/4 N 1/4 s s 1 E[W ] z 2 E[W ] z = N − log cosh N . (6.5.21) 1 4 1 4 2 νN N / νN N /

As observed in the proof of [26, Theorem V.9.5], there exist positive constants A and ε so that ( 1 εx4, for |x| ≤ A, x2 − log cosh x ≥ d(x) := (6.5.22) 2 εx2, for |x| > A.

To bound the second term in (6.5.19), we can use the Taylor expansion

log cosh(a + x) = log cosh(a) + tanh(ξ)x, (6.5.23)

133 Annealed measure: at criticality for some ξ ∈ (a, a + x), and that | tanh(ξ)| ≤ |ξ| ≤ |a| + |x| to obtain h z r z i E log cosh αN (βc,N )WN + − log cosh αN (βc,N )WN N 1/4 N λ N 1/4 h z r z i ≤ E log cosh αN (βc,N )WN + − log cosh αN (βc,N )WN N 1/4 N λ N 1/4 h z |r| |r| i ≤ E αN (βc,N )WN + N 1/4 N λ N λ s |zr| r2 E[W ] |zr| r2 = α (β )E[W ] + = N + . N c,N N 1 4+λ 2λ 3 2 N / N νN N N / Hence,

s 2 s 1/4 n E[W ] r E[W ] z o e−NGN (z/N ;r) ≤ exp N |zr| + − Nd N , 1 2 1 4 νN N / νN N / which we use as the dominating function. Hence, we need to prove that the integral of this function over z ∈ R is uniformly bounded. We split the integral as

s s Z ∞ n E[W ] r2 E[W ] z o exp N |zr| + − Nd N dz 1/2 1/4 −∞ νN N νN N s s Z n E[W ] r2 E[W ] z o = exp N |zr| + − Nd N dz q E[W ] 1/2 1/4 N z ≤A νN N νN N νN N1/4 s s Z n E[W ] r2 E[W ] z o + exp N |zr| + − Nd N dz q E[W ] 1/2 1/4 N z >A νN N νN N νN N1/4 The ﬁrst integral equals s Z 2 4 n E[WN ] r E[WN ] 4o q exp |zr| + − ε z dz, (6.5.24) E[W ] z 1/2 2 2 N ≤A νN N E[WN ] νN N1/4 which clearly is uniformly bounded. The second integral equals s Z n E[W ] r2 E[W ] √ o exp N |zr| + − ε N z2 N dz q E[W ] 1/2 N z >A νN N νN νN N1/4 s 1 Z n E[W ] |yr| r2 E[W ] o = exp N + − ε N y2 dy, 1/4 q E[W ] 1/4 1/2 N N y >A νN N N νN νN N1/2 where we have substituted y = zN 1/4. This converges to zero for N → ∞, because the integral is uniformly bounded. Together with the pointwise convergence proved in Lemma 6.5.2, this proves Lemma 6.5.3.

134 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

6.5.3 Convergence for τ ∈ (3, 5)

We next analyze GN (z; r) for τ ∈ (3, 5), assuming Condition 6.1.2.

Lemma 6.5.4 (Asymptotics of GN for τ ∈ (3, 5)). Assume that Condition 6.1.2(ii) holds. For β = βc,N , B = 0 and τ ∈ (3, 5), r r 1/(τ−1) E[W ] E[W ] lim NGN (z/N ; r) = −zr + f z , (6.5.25) N→∞ ν ν where f(z) is deﬁned in (6.3.4). Proof. Deﬁne the function 1 g(w, z) = (α (β )wz)2 − log cosh (α (β )wz) , (6.5.26) 2 N c,N N c,N so that we can rewrite, in a similar way as in (6.5.19), 1 τ−1 NGN z/N ; r (6.5.27)

1 τ−1 = NE[g(WN , z/N )] W z r W z − NE log cosh N + − log cosh N . p 2 1 N λ p 2 1 E[WN ] N τ−1 E[WN ] N τ−1

By the deﬁnition of WN , we can rewrite N 1/(τ−1) 1 X 1/(τ−1) E[g(WN , z/N )] = g(w , z/N ). (6.5.28) N i i=1 With the deterministic choice of the weights as in (6.1.2), 2 1/(τ−1) 1 wiz wiz g(wi, z/N ) = αN (βc,N ) − log cosh αN (βc,N ) 2 N 1/(τ−1) N 1/(τ−1) 1 1 c z 2 1 c z = w − log cosh w . 2 p 2 1/(τ−1) p 2 1/(τ−1) E[WN ] i E[WN ] i From this it clearly follows that, for all i ≥ 1, 1/(τ−1) lim g(wi, z/N ) N→∞ 1 1 c z 2 1 c z = w − log cosh w . (6.5.29) 2 pE[W 2] i1/(τ−1) pE[W 2] i1/(τ−1) It remains to show that also the sum converges, which we do using dominated convergence. For this, we use a Taylor expansion of log cosh(x) about x = 0 up to the fourth order x2 x4 log cosh(x) = + −2 + 2 tanh2(ξ) + 6 tanh2(ξ)(1 − tanh2(ξ)) 2 4! x2 x4 ≥ − , (6.5.30) 2 12

135 Annealed measure: at criticality for some ξ ∈ (0, x). Hence,

4 4 1/(τ−1) 1 1 cwz 1 1 (cwz) g(wi, z/N ) ≤ = . 12 p 2 1/(τ−1) 12 E[W 2]2 + o(1) 4/(τ−1) E[WN ] i i

Since τ ∈ (3, 5), it holds that 4/(τ − 1) > 1, so that

N 4 X 1 1 (cwz) lim < ∞. (6.5.31) N→∞ 12 E[W 2]2 + o(1) i4/(τ−1) i=1 We conclude that

N X 1/(τ−1) lim g(wi, z/N ) N→∞ i=1 ∞ 2 X 1 1 cwz 1 cwz = − log cosh 2 p 2 i1/(τ−1) p 2 i1/(τ−1) i=1 E[W ] E[W ] r E[W ] = f z , (6.5.32) ν

τ−1 where in the last equality we have used that E[W ] = cw τ−2 . This is in turn a consequence of the following explicit computation giving an upper and lower bound on E[WN ] matching in the limit N → ∞. An upper bound on the ﬁrst moment is given by

N 1/(τ−1) N 1 X N − τ−2 X E[W ] = c = c N τ−1 c i−1/(τ−1) N N w i w w i=1 i=1

τ−2 Z N τ−2 − −1/(τ−1) τ − 1 1 − ≤ cwN τ−1 1 + i di = cw − cw N τ−1 , 1 τ − 2 τ − 2 and a lower bound by

τ−2 Z N τ − 1 τ − 1 τ−2 − τ−1 −1/(τ−1) − τ−1 E[WN ] ≥ cwN i di = cw − cw N . 1 τ − 2 τ − 2 From this it indeed follows that τ − 1 E[W ] = lim E[WN ] = cw . (6.5.33) N→∞ τ − 2 To analyze the second term in (6.5.27), we can use the Taylor expansions

log cosh(a + x) = log cosh(a) + tanh(a)x + (1 − tanh2(ξ))x2 = log cosh(a) + (a − tanh ζ(1 − tanh2 ζ)a2)x + (1 − tanh2(ξ))x2, (6.5.34)

136 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem for some ξ ∈ (a, a + x) and ζ ∈ (0, a). This gives

1 z r 1 z NE log cosh W + − log cosh W p 2 N 1 N λ p 2 N 1 E[WN ] N τ−1 E[WN ] N τ−1 s 2 E[WN ] z r h 2 2 i 1 z r = N 1 λ − NE tanh ζ(1 − tanh ζ)WN 2 2 λ νN N τ−1 N E[WN ] N τ−1 N h i r2 + NE (1 − tanh2(ξ)) N 2λ s E[W ] = N zr + o(1), (6.5.35) νN

τ−2 where the last equality follows from λ = τ−1 and τ ∈ (3, 5).

Again it follows that also the integral converges:

Lemma 6.5.5 (Convergence of the integral for τ ∈ (3, 5)). For β = βc,N , B = 0 and τ ∈ (3, 5), q q ∞ ∞ E[W ] E[W ] Z 1/(τ−1) Z zr −f z lim e−NGN (z/N ;r)dz = e ν ν dz. (6.5.36) N→∞ −∞ −∞

Proof. We again start from the rewrite of GN in (6.5.19). As before,

1 τ−1 NE[g(WN , z/N )] N X h1 1 cwz 2 1 cwz i = − log cosh , 2 p 2 i1/(τ−1) p 2 i1/(τ−1) i=1 E[WN ] E[WN ] where it is easy to see that the summands are positive and decreasing in i. Hence,

1 τ−1 NE[g(WN , z/N )] Z N h1 1 c z 2 1 c z i ≥ w − log cosh w dy. 2 p 2 1/(τ−1) p 2 1/(τ−1) 1 E[WN ] y E[WN ] y We want to use (6.5.22), and hence split the integral in the region where

1 cwz is bigger or smaller than A. This gives p 2 y1/(τ−1) E[WN ]

1 τ−1 NE[g(WN , z/N )]

ApE[W 2 ] τ−1 2 2 Z N 4 4 Z N cwz cw|z| 1 cwz 1 τ−1 ≥ ε 2 dy + ε ApE[W 2 ] 4 dy 2 2 N E[WN ] 1 τ−1 E[WN ] τ−1 y cw|z| y

137 Annealed measure: at criticality

2 2 p 2 τ−3 cwz τ − 1 A E[WN ] = ε 2 − 1 E[WN ] τ − 3 cw|z| p c4 z4 τ − 1 5−τ A E[W 2 ] −(5−τ) w − τ−1 N − ε 2 N − E[WN ] 5 − τ cw|z| −(τ+1) 2 4 9−τ = k1|z| − k2z − o(1)z + k3|z| , for the proper constants k1, k2, k3 > 0. Since 9 − τ > 4, ∞ Z −(τ+1) 2 4 9−τ e−k1|z| +k2z +o(1)z −k3|z| dz < ∞. (6.5.37) −∞ Together with the pointwise convergence in the previous lemma, this proves this lemma for r = 0. For r 6= 0, the proof can be adapted as for the case E[W 4] < ∞. We next analyze the large x behavior of f(x) arising in the density of the limiting random variable: Lemma 6.5.6 (Asymptotics of f for τ ∈ (3, 5)). For τ ∈ (3, 5), f(x) τ − 2τ−1 Z ∞ 1 1 lim = − log cosh dy < ∞. τ−1 2 (τ−1) 1 (τ−1) x→∞ x τ − 1 0 2y / y / Proof. We first prove that the integral is finite. For this, define 1 h(y) = y2 − log cosh y, (6.5.38) 2 so that h(y) ≥ 0. Then, Z ∞ 1 1 Z ∞ 1 − log cosh dy = h dy. (6.5.39) 2 (τ−1) 1 (τ−1) 1 (τ−1) 0 2y / y / 0 y / 1 2 Since log cosh y ≥ 0, we have h(y) ≤ 2 y , and hence 1 1 h ≤ . (6.5.40) y1/(τ−1) 2y2/(τ−1) This is integrable for y → 0, because 2/(τ − 1) < 1 for τ ∈ (3, 5). Using (6.5.30), for y large, 1 1 1 h ≤ . (6.5.41) y1/(τ−1) 12 y4/(τ−1) This is integrable for y → ∞, because 4/(τ − 1) > 1 for τ ∈ (3, 5). To prove that f(x)/xτ−1 converges to the integral as x → ∞ we rewrite, with a = (τ − 2)/(τ − 1), ∞ ∞ 1/(τ−1) f(x) 1 X x 1 X 1 = h a = aτ−1 h xτ−1 xτ−1 i1/(τ−1) τ−1 τ−1 i=1 (ax) i=1 i/ (ax) Z ∞ 1 = aτ−1 h dy (1 + o(1)) . (6.5.42) 1 (τ−1) 0 y /

138 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

6.5.4 Proof of Theorem 6.3.1

We can now prove Theorem 6.3.1 for the measure PbN :

Proof of Theorem 6.3.1 for the measure PbN . We can do a change of variables so that ∞ ∞ Z Z 1/(δ+1) e−NGN (z;r)dz = N 1/(δ+1) e−NGN (z/N ;r)dz. (6.5.43) −∞ −∞ Hence, using Lemma 6.5.1 ∞ Z 1/(δ+1) e−NGN (z/N ;r)dz SN −∞ PbN exp r = Z ∞ (6.5.44) N δ/(δ+1) 1/(δ+1) e−NGN (z/N ;0)dz −∞ It follows from Lemma 6.5.3 for E[W 4] < ∞ and from Lemma 6.5.5 for τ ∈ (3, 5) that q q Z ∞ zr E[W ] −f E[W ] z e ν ν dz SN −∞ lim PbN exp r = N→∞ δ/(δ+1) q N Z ∞ −f E[W ] z e ν dz −∞ Z ∞ exr−f(x)dx −∞ = Z ∞ , (6.5.45) e−f(x)dx −∞

q E[W ] where we made the change of variables x = ν z in both integrals to obtain the last equality. As mentioned, this is suﬃcient to prove the convergence in distribution of SN to the random variable X (see [26, Theorem A.8.7(a)]). N δ/(δ+1) For the case E[W 4] < ∞,

E[W 4] 1 4 4 f(x) 12 E[W ]4 x 1 E[W ] lim = lim = . (6.5.46) x→∞ x1+δ x→∞ x4 12 E[W ]4

For τ ∈ (3, 5), the proof that lim f(x) = C is given in Lemma 6.5.6. x→∞ x1+δ It remains to show that the statement of Theorem 6.3.1 also holds for the measure PeN . This follows from the following lemma: Lemma 6.5.7. For E[W 4] < ∞ and τ ∈ (3, 5),

SN SN lim PeN exp r − PbN exp r = 0. (6.5.47) N→∞ N λ N λ

139 Annealed measure: at criticality

Proof. As shown in Section 5.4,

1 P J σ σ P g(σ)e 2 i,j∈[N] ij i j σ∈ΩN PeN g(σ) = , (6.5.48) 1 P J σ σ P e 2 i,j∈[N] ij i j σ∈ΩN where β 1 e pij + (1 − pij ) 2 3 Jij = log −β = pij sinh β − pij sinh β(cosh β − 1) + O(pij ), 2 e pij + (1 − pij ) where we have used the Taylor expansion of log(1 + x) about x = 0 in the last equality. Hence, using (2.1.8),

1 P J σ σ e 2 i,j∈[N] ij i j 1 P wiwj wiwj 2 3 − sinh β−p sinh β(cosh β−1)+O(p ) σiσj = e 2 i,j∈[N] `N +wiwj `N ij ij 1 P wiwj sinh β σiσj · e 2 i,j∈[N] `N 1 P wiwj E (σ) sinh β σiσj =: e N e 2 i,j∈[N] `N . (6.5.49)

Hence, we can rewrite (6.5.48) as

1 P wiwj sinh β σiσj P EN (σ) 2 i,j∈[N] `N EN (σ) g(σ)e e PbN g(σ)e P g(σ) = σ∈ΩN = . eN 1 P wiwj E σ sinh β σiσj P e N ( ) P eEN (σ)e 2 i,j∈[N] `N bN σ∈ΩN

Combining this with the Cauchy-Schwarz inequality gives

SN SN PeN exp r − PbN exp r N λ N λ P exp r SN eEN (σ) − P eEN (σ) bN N λ bN = EN (σ) PbN e s r 2 S P exp 2r N P eEN (σ) − P eEN (σ) bN N λ bN bN ≤ EN (σ) PbN e r r S 2 P exp 2r N P e2EN (σ) − P eEN (σ) bN N λ bN bN = (6.5.50) EN (σ) PbN e

From (6.5.45), it follows that the ﬁrst square root converges as N → ∞.

140 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

We next analyze EN (σ) and show that EN (σ) → 0 in probability w.r.t. PbN . We also show that EN (σ) is uniformly bounded from above, so that the lemma follows by dominated convergence.

3 We ﬁrst analyze the contribution of the O(pij ) terms in EN (σ). Note that

3 3 2 X 3 X wiwj X wiwj 1 X 3 pij = ≤ = 3 wi . `N + wiwj `N ` i,j∈[N] i,j∈[N] i,j∈[N] N i∈[N]

√ 2 2 For E[WN ] → E[W ] < ∞ it holds that maxi wi = o( N). Hence, 2 2 X 3 2 X 2 3 2 2 3 wi ≤ (max wi) wi = o(N )E[WN ] = o(`N ), (6.5.51) i i∈[N] i∈[N]

because `N = O(N). Hence,

EN (σ) 1 X wiwj wiwj 2 3 = − sinh β − pij sinh β(cosh β − 1) + O(pij ) σiσj 2 `N + wiwj `N i,j∈[N] 2 2 2 1 X wi wj wiwj = − sinh β + sinh β(cosh β − 1) σiσj 2 `N (`N + wiwj ) `N + wiwj i,j∈[N] + o(1) 2 2 1 X wi wj = − sinh β cosh β σ σ + o(1) 2 `2 i j i,j∈[N] N 2 2 1 X wi = − sinh β cosh β σi + o(1), (6.5.52) 2 `N i∈[N]

3 where the third equality can be proved as in the analysis of pij. Hence, EN (σ) is indeed uniformly bounded from above, so that eEN (σ) is uniformly bounded.

It remains to prove that EN (σ) → 0 in probability w.r.t. PbN . We deﬁne YN = 2 P wi σi, so that i∈[N] `N 1 E = − sinh β cosh β Y 2 + o(1). (6.5.53) N 2 N

λ We analyze the moment generating function of YN the same way as SN /N . That is, we use the Hubbard–Stratonovich identity to rewrite

141 Annealed measure: at criticality

2 1 sinh β P P rY 2 ` wiσi e N e N i∈[N] rYN σ∈ΩN PbN e = 2 1 sinh β P wiσi P e 2 `N i∈[N] σ∈ΩN w2 q P i sinh β P h r σi+ wiσiZ i P i∈[N] NE[WN ] NE[WN ] i∈[N] σ∈Ω E e = N q sinh β P h wiσiZ i P E e NE[WN ] i∈[N] σ∈ΩN

W 2 N q sinh β h NE log cosh r + WN Z Z i E e NE[WN ] NE[WN ] = q sinh β h NE log cosh WN Z Z i E e NE[WN ]

W 2 q ∞ 2 N sinh β Z −z /2+NE log cosh r + WN z e NE[WN ] NE[WN ] dz −∞ = q . (6.5.54) ∞ 2 sinh β Z −z /2+NE log cosh WN z e NE[WN ] dz −∞ √ We do a change of variables replacing z/ N by z, so that, calling again αN (β) := q sinh β , E[WN ]

rYN PbN e (6.5.55)

h W 2 i ∞ 2 N Z −Nz /2+NE log cosh r +αN (β)WN z e NE[WN ] dz −∞ = h i ∞ 2 Z −Nz /2+NE log cosh αN (β)WN z e dz −∞ h W 2 i ∞ N Z −NGN (z;0)+NE log cosh r +αN (β)WN z −log cosh αN (β)WN z e NE[WN ] dz −∞ = Z ∞ e−NGN (z;0)dz −∞ 1 h W 2 i N αN (β)WN z αN (β)WN z Z ∞ −NGN (z/N δ+1 ;0)+NE log cosh r + −log cosh NE[WN ] 1 1 e N δ+1 N δ+1 dz = −∞ , Z ∞ 1 e−NGN (z/N δ+1 ;0)dz −∞ where we did another change of variable in the last equality. 1/(δ+1) In Lemmas 6.5.2 and 6.5.4, we proved that NGN (z/N ; 0) converges for β = βc. We Taylor expand the remaining term,

142 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

2 s s WN sinh β WN z sinh β WN z NE log cosh r + 1 − log cosh 1 NE[WN ] E[WN ] N δ+1 E[WN ] N δ+1 s 2 sinh β WN z WN = E tanh 1 r + o(1) . (6.5.56) E[WN ] N δ+1 E[WN ]

3 3 For E[WN ] → E[W ] < ∞, which includes power-law distributions with τ > 4, we can use that | tanh(x)| ≤ |x|, so that

s 2 s 3 sinh β WN z WN sinh β |zr| E[WN ] E tanh 1 r ≤ 1 = o(1). E[WN ] N δ+1 E[WN ] E[WN ] N δ+1 E[WN ]

For τ ∈ (3, 4] we use the deterministic choice of the weights as in (6.1.2) and δ = τ − 2 to rewrite

s sinh β z W 2 E tanh W r N N 1 (δ+1) E[WN ] N / E[WN ] N s 1 X sinh β z w2 = tanh w r i i 1 (τ−1) N E[WN ] N / E[WN ] i=1 N s 2 2/(τ−1) 1 X sinh β cwz rc N = tanh w 1 (τ−1) N E[WN ] i / E[WN ] i i=1 s N |r|c2 sinh β |rz|c3 X ≤ w N 2/(τ−1)−1 + N 2/(τ−1)−1 w i−3/(τ−1). (6.5.57) E[W ] E[W ] E[W ] N N N i=2 For τ > 3 the ﬁrst term is o(1). For τ ∈ (3, 4), N Z N X − τ−3 N 2/(τ−1)−1 i−3/(τ−1) ≤ N τ−1 i−3/(τ−1)di i=2 1 τ − 1 − τ−3 = N τ−1 − N −1/(τ−1) = o(1), 4 − τ whereas for τ = 4 N Z N X − τ−3 − τ−3 N 2/(τ−1)−1 i−3/(τ−1) ≤ N τ−1 i−3/(τ−1)di = N τ−1 log N = o(1). i=2 1 Hence, in all cases the integrands in the numerator and denominator of (6.5.55) have the same limit. In Lemmas 6.5.3 and 6.5.5 it is proved that the integral in the denominator converges. Since s sinh β z W 2 rE[W 2 ] E tanh W r N ≤ N = O(1), (6.5.58) N 1 (δ+1) E[WN ] N / E[WN ] E[WN ]

143 Annealed measure: at criticality it follows by dominated convergence that the integral in the numerator has the same limit. Hence, rYN lim PbN e = 1, (6.5.59) N→∞ from which it follows that YN → 0 in probability w.r.t. PbN . Hence, also 1 2 − 2 sinh β cosh β YN → 0 in probability w.r.t. PbN . Since o(1) also converges to 0 in probability, so does the sum:

1 2 EN = − sinh β cosh β Y + o(1) −→ 0 in probability w.r.t. PbN . (6.5.60) 2 N

Remark 6.5.1 (Sharp asymptotics of the partition function). It follows from the changes of variables in (6.5.8) and (6.5.43) that

1/2+1/(δ+1) N ZN (βc, 0) = AN 2 (1 + o(1)).

For E[W 4] < ∞, this exponent equals 1/2 + 1/(δ + 1) = 3/4, whereas for τ ∈ (3, 5), it is 1/2 + 1/(δ + 1) = (τ + 1)/(2τ − 2). Thus the partition function has ﬁnite-size power-law corrections (in agreement with [6] where the classical Curie–Weiss model is considered).

6.5.5 Scaling window

Instead of looking at the inverse temperature sequence βN = βc,N we can also δ−1 0 δ+1 look at βN = βc,N + b/N for some constant b. The analysis still works and the limiting density instead becomes

b E[W 2]2 exp cosh(β ) x2 − f(x) . (6.5.61) 2 c E[W ]3

To see why this is correct we look at the following second moment, which shows up in the expansion of GN , see (6.5.12): v u δ−1 2 δ+1 1 usinh(β N + b/N ) z r E t c, W + N 1 (δ+1) δ (δ+1) 2 E[WN ] N / N /

z2 δ−1 E[W 2 ] = sinh(β + b N δ+1 ) N 2 (δ+1) c,N / 2N / E[WN ] q δ−1 zr δ+1 + sinh(β N + b/N )E[WN ] + o(1/N). (6.5.62) c, N

In the ﬁrst term, we Taylor expand the sine hyperbolic about βc,N , which gives

δ−1 δ−1 δ−1 δ+1 δ+1 2 δ+1 sinh(βc,N + b/N ) = sinh(βc,N ) + cosh(βc,N )b/N + O 1/N .

144 6.5 Proofs of Theorem 6.3.1: Non-classical limit theorem

For the other term, and also for the other terms in the expansion of GN , it suﬃces to note that

q δ−1 q δ−1 δ+1 δ+1 sinh(βc,N + b/N ) = sinh(βc,N ) + O 1/N . (6.5.63)

Hence, (6.5.62) equals

2 2 2 z z b E[WN ] zr E[WN ] + cosh(βc,N ) + + o(1/N) 2/(δ+1) 2/(δ+1) δ−1 E[W ] N p 2 2N 2N N δ+1 N E[WN ] 2 2 2 z bz E[WN ] zr E[WN ] = + cosh(βc,N ) + + o(1/N) (6.5.64) 2/(δ+1) 2N E[W ] N p 2 2N N E[WN ]

1/(δ+1) In the expansion of GN (z/N ; r) the ﬁrst term in (6.5.64) drops as usual, whereas the second term in (6.5.64) remains. After multiplication by N (cf. (6.5.16)), one has

2 2 1/(δ+1) bz E[WN ] E[WN ] −NGN (z/N ; r) = cosh(β ) − f z + o(1) 2 c E[W ] p 2 N E[WN ]

Using the substitution x = E[WN ] z the above converges in the limit N → ∞ p 2 E[WN ] to the exponent in (6.5.61), as required.

Limit distribution at βc instead of βc,N . In the above, we look at the inverse temperature sequence βN = βc,N and then take the limit N → ∞. Alternatively, we could immediately start with β = βc. The scaling limit that will be seen depends on the speed at which νN approaches ν. Indeed, from (5.1.6) and (6.3.1), one has βc − βc,N = O(ν − νN ). We investigate this for the deterministic weights according to (6.1.2), and ﬁrst −η −η investigate how close νN is to ν. By [2, Lemma 2.2], νN = ν + ζN + o(N ) with η = (τ − 3)/(τ − 1) and ζ an explicit non-zero constant. Thus, for τ > −1/2 5, νN = ν + o(N ). Hence, the results stay the same (see the previous discussion). δ−1 −η −η − δ+1 When τ ∈ (3, 5), instead, νN = ν + ζN + o(N ) = ν + ζN + − δ−1 o(N δ+1 ), so we are shifted inside the critical window (see the previous discussion). Hence, in this case the limiting distribution changes.

145

ZEROTEMPERATURERECURSION 7

This chapter diversifies a bit from the main topic of this thesis, although it does concern the study of Ising model on random graphs. The idea is to analyze the stochastic recursion equation for the cavity field of the Ising model on locally tree-like random graphs. This equation, already introduced in Section 2.5.3, is crucial in the analysis of the thermodynamic quantities in the random quenched setting. Indeed, as we can see from Proposition 2.5.4, in [16, 20] it is proved that the expressions for the pressure and the magnetization at infinite volume depend explicitly on the fixed point of the recursion equation. Therefore, the investigation on this solution becomes particularly interesting. Our aim is to describe the distribution of the fixed point of the distributional recursion equation. The purpose is to perform this study for different values of the parameters β and B. In this thesis we take into consideration only the simplest case, i.e. the zero temperature setting, where β → ∞, because in this situation we will see that the equation reduces to a linear recursion. Beyond the analytical study, a numerical analysis is provided. We remark that this chapter is quite speculative, since only heuristic results are presented. This is a preliminary investigation and we will not treat the analytical solution in the non-zero temperature case, which is quite challenging, but we supply some numerical simulation in this scenario. Particularly the behavior close to criticality is very interesting.

7.1 Analytical solution

First of all, we recall the expression of the distributional recursion equation and we provide a result that guarantees the existence of a fixed point. This result was achieved first by [16] and then generalized by [20] for locally random graphs where the degree distribution has infinite variance.

Lemma 7.1.1 (Tree recursion [16, 20]). Let B > 0 and let (Kt)t≥0 be i.i.d. random variables with distribution ρ and assume that K0 < ∞ a.s. Consider the (t) (0) sequence of random variables {h }t≥0 deﬁned by h ≡ B and, for t ≥ 0, by

Kt (t+1) d X (t) h = B + atanh(tanh(β) tanh(hi )), (7.1.1) i=1

(t) (t) where (hi )i≥1 are i.i.d. copies of h independent of Kt. Then, the distributions of h(t) are stochastically monotone and h(t) converges in distribution to the unique ﬁxed point h∗ of the recursion (7.1.1) that is supported in [0, ∞).

147 Zero temperature recursion

With this result in our hands, we can proceed to the analysis of the distribution of the ﬁxed point at zero temperature.

7.1.1 Fixed point at zero temperature and zero external ﬁeld

We start by analyzing the easiest case, i.e., we consider the case of zero temperature and zero external ﬁeld. When we take the recursion equation (7.1.1) at β = ∞ and B = 0 we obtain the following linear recursion

Kt (t+1) d X (t) h = hi . (7.1.2) i=1 We proceed initializing h(0) with a positive value A > 0:

h(0) ≡ A, (7.1.3) then, at the ﬁrst step we obtain

K0 (1) d X h = A = K0A. (7.1.4) i=1

d We introduce a new random variable Z1 such that K0 = Z1 and then we go ahead in the iteration rewriting

K1 (2) d X d h = A Z1,i = AZ2, (7.1.5) i=1

K2 (3) d X d h = A Z2,i = AZ3. (7.1.6) i=1 After n steps, we obtain the following equation

(n) d h = AZn, (7.1.7)

th where we observe that Zn is the number of individuals in the n generation of a branching process, with Z0 = 1:

Kn−1 d X Zn = Zn−1,i. (7.1.8) i=1 Thus, we have reduced the study of the recursion equation (7.1.2) to the analysis of a branching process. If we denote by η the extinction probability of this branching process, i.e.,

η = P (∃n : Zn = 0) , (7.1.9)

148 7.1 Analytical solution from equation (7.1.7), we ﬁnally get the distribution for the ﬁxed point h∗: ( P(h∗ = ∞) = 1 − η, (7.1.10) P(h∗ = 0) = η.

We can evaluate the extinction probability η using classical results on branching processes (see [64]). For example, when we consider the Ising model on the Erdős–Rényi random ER graph GN (p) with p = c/N, the (Kt)t≥1 are i.i.d. random variables with distribution P oiss(c), and we have ( η = 1, if E[K ] = c ≤ 1, 1 (7.1.11) η < 1, if E[K1] = c > 1.

In particular, η is the smallest solution in [0, 1] of the equation η = E[ηK1 ] that, in this case, can be rewritten as

η = e−c(1−η). (7.1.12)

Remark 7.1.1. Now that we know the distribution of the ﬁxed point h∗, stated in (7.1.10), we can calculate explicitly the magnetization for the Ising model on ER the Erdős–Rényi random graph GN (p) with p = c/N. Indeed, using Proposition 2.5.4 for β = ∞ and B = 0, we obtain

M(∞, 0+) = E[tanh(h∗)] = 1 − η. (7.1.13)

ER This result comes from the fact that in the Erdős–Rényi random graph GN (p) the size of the giant component is approximately (1 − η)N. Then, at infinite volume, if we consider the limit for β to infinity and B to 0+, we find that the spontaneous magnetization is 1 − η. The non-giant components, instead, do not contribute, since their size is finite. See also Remark 7.1.2 below.

7.1.2 Fixed point at zero temperature and positive external ﬁeld

Now we take into account the presence of a positive external ﬁeld. When β = ∞ and B > 0 the equation (7.1.1) becomes

Kt (t+1) d X (t) h = B + hi . (7.1.14) i=1 Proceeding similarly to the previous case, we have

h(0) ≡ B > 0, (7.1.15)

(1) d d h = B + K0B = B(1 + K0), (7.1.16)

149 Zero temperature recursion

d We consider again a random variable Z1 such that K0 = Z1 and we write

K1 (2) X d h ≥ B Z1,i = BZ2, (7.1.17) i=1

K2 (3) X d h ≥ B Z2,i = BZ3. (7.1.18) i=1 Then (n) h ≥ BZn (7.1.19) th where, as before, Zn is the number of individuals in the n generation of a branching process, with Z0 = 1:

Kn−1 d X Zn = Zn−1,i. (7.1.20) i=1 Therefore, considering (7.1.19), we again obtain

P(h∗ = ∞) = 1 − η, (7.1.21) where we remember that η is the extinction probability of the branching process, as in Section 7.1.1. We next investigate the probability that h∗ takes values different from infinity. We start introducing the total progeny T of the branching process defined as

∞ X T = Zn. (7.1.22) n=0

(t) th (t) Pt We call T the total progeny at t generation, i.e. T = n=0 Zn. We set T (0) = 1, then h(0) ≡ B = BT (0), (7.1.23) and

K0 K0 ! (1) d X (0) d X (0) d (1) h = B + BTi = B 1 + Ti = BT . (7.1.24) i=1 i=1 Iterating, we have

Kt−1  Kt−1  (t) d X (t−1) d X (t−1) d (t) h = B + BTi = B 1 + Ti  = BT , (7.1.25) i=1 i=1 and taking the limit t → ∞ we ﬁnally obtain

d h∗ = BT . (7.1.26)

150 7.1 Analytical solution

When (Kt)t≥1 are i.i.d. random variables with distribution P oiss(c), using [64, Theorem 3.16] for each n ∈ N, with n ≥ 1, we can calculate

(cn)n−1 P(h∗ = B n) = P(T = n) = e−cn. (7.1.27) n! Remark 7.1.2. As in the previous case, we can provide the explicit expression for the magnetization of the Ising model on the Erdős–Rényi random graph ER GN (p) with p = c/N. For β = ∞ and B > 0, using Proposition 2.5.4 together with (7.1.21) and (7.1.27), it results

X (cn)n−1 M(∞, B) = E[tanh(h∗)] = 1 − η + tanh(Bn) e−cn. (7.1.28) n! n≥1 When we consider the limits N → ∞ and β → ∞, we have that the in each connected component all the spins are aligned in the same direction. We remember that the giant component has size approximately equal to (1 − η)N and from this the term 1 − η in formula (7.1.28) rises. The non-giant components are finite clusters of size n. In these cases the probability to have a configuration with all spins equal to +1 is proportional to eBn and, similarly, the probability to have a configuration with all spins equal to −1 is proportional to e−Bn. This gives rise to the second term in (7.1.28).

7.1.3 Fixed point approaching zero temperature

In this case we analyze the situation when the ﬁxed point of the recursion equation is equal to inﬁnity, i.e., h∗ = ∞ and we consider β → ∞. We suppose that there exists a random variable X∗ such that, for β → ∞, h∗ −→d X∗. (7.1.29) β Now, our aim is to study the distribution of X∗. Referring to equation (7.1.1) we call, for sake of notation,

(t) (t) ξ(hi ) := atanh(tanh(β) tanh(hi )). (7.1.30)

Then, taking x ∈ R+, and using tanh(β) ≤ 1, we have

ξ(βx) atanh(tanh(β) tanh(βx)) βx = ≤ = x. (7.1.31) β β β Considering the approximation for β large, we obtain

eβ − e−β 1 − e−2β tanh(β) = = = 1 − 2e−2β + o(e−2β ) (7.1.32) eβ + e−β 1 + e−2β and similarly tanh(βx) = 1 − 2e−2βx + o(e−2βx). (7.1.33)

151 Zero temperature recursion

Then tanh(β) tanh(βx) = 1 − 2e−2β + o(e−2β 1 − 2e−2βx + o(e−2βx)

= 1 − 2e−2βx − 2e−2β + 4e−2β(x+1) + o(e−2β(x∧1)), (7.1.34) where a ∧ b = min {a, b}. Now, because for any y ∈ (−1, 1), 1 (1 + y) atanh(y) = ln , (7.1.35) 2 (1 − y) we can write 1 (1 + tanh(β) tanh(βx)) ξ(βx) = ln 2 (1 − tanh(β) tanh(βx)) ! 1 2 − 2e−2βx − 2e−2β + 4e−2β(x+1) + o(e−2β(x∧1)) = ln 2 2e−2βx + 2e−2β − 4e−2β(x+1) − o(e−2β(x∧1)) 1 1 = ln 2 − 2e−2βx − 2e−2β + 4e−2β(x+1) + o(e−2β(x∧1)) − ln 2 2 2 1 − ln e−2β 1 − 2e−2βx + e−2βx − o(e−2β(x∧1)) . 2 Finally, we can consider the limit for β → ∞: ξ(βx) 1 lim = − lim ln e−2β 1 + 2e−2βx + e−2βx · e2β − o(e−2β(x∧1)) β→∞ β β→∞ 2β 1 = 1 − lim ln 1 + 2e−2βx + e−2β(x−1) − o(e−2β(x∧1)) , β→∞ 2β and then we obtain ( ξ(βx) x if x < 1, g(x) := lim = (7.1.36) β→∞ β 1 if x ≥ 1. We rewrite now the recursion equation (7.1.1) for the ﬁxed point h∗ over β,

h∗ ∗ K ∗ K ξ β i h d B X ξ (h ) d B X β = + i = + . (7.1.37) β β β β β i=1 i=1

d Recalling that for β → ∞ we have that h∗/β −→ X∗ and using (7.1.36) we obtain K ∗ K d X ξ (βX ) d X X∗ = i = g(X∗), (7.1.38) β i i=1 i=1 where the contribution given by the external ﬁeld B vanishes in the limit. If K has distribution P oiss(c), one solution of equation (7.1.38), is

d X∗ = P oiss(c θ), with θ = 1 − η. (7.1.39)

152 7.2 Numerical analysis

∗ ∗ Indeed, if Xi ≥ 1 with P(X1 ≥ 1) = θ, where θ = 1 − η is the survival probability of a branching process with number of oﬀspring distributed as K, it satisﬁes K ∗ d X X = 1 ∗ . (7.1.40) {Xi ≥1} i=1 d Note that, if K has generic distribution, the solution becomes X∗ = Bin(K, θ), again with θ the survival probability of a branching process. Remark 7.1.3. We calculate now the explicit expression of the magnetization ER for the Ising model on the Erdős–Rényi random graph GN (p) with p = c/N in this scenario. We remember that K has distribution P oiss(c). Then, for β large, from Proposition 2.5.4 and the previous analysis, we have, heuristically,

M(β, B) = E[tanh(h∗)] = E[tanh(βX∗)] = E[tanh(βP oiss(c θ))] X = tanh(β)P[P oiss(c θ) = 1] + tanh(βk)P[P oiss(c θ) = k] k>1 = 1 − 2e−2β + o(e−2β )P[P oiss(c θ) = 1] X + 1 − 2e−2βk + o(e−2βk)P[P oiss(c θ) = k] k>1 = 1 − P[P oiss(c θ) = 0] − 2e−2βP[P oiss(c θ) = 1] + o(e−2β ) = 1 − η − 2cθe−2βe−cθ + o(e−2β ), (7.1.41) because P[P oiss(c θ) = 0] = e−cθ = e−c(1−η) = η, from equation (7.1.12).

7.2 Numerical analysis

To support the previous analysis, we also provide an investigation on our problem from a numerical point of view. In this section we present, in particular, some ﬁgures, obtained through numerical simulations, that show the distribution for the ﬁxed point of our random recursion. We study equations (7.1.2) and (7.1.14) approximating them by the following random iteration algorithm called Population Dynamics (see [49]):

Kt (t+1) X (t) Xi = B + Xνj , i = 1, 2, ... , M (7.2.1) j=1

(t) where Xi ∈ R, (Kt)t≥1, are i.i.d. with distribution P oiss(c), (νi)i≥1, are i.i.d. uniformly distributed in the set {1, 2, ... , M} with M ∈ N. The meaning of (7.2.1) is the following: at the iteration step t the random ﬁeld h(t) is approximated by the vector X(t) ∈ RM (to be considered as a sample (t+1) of hi); in the next step the ﬁeld is represented by a vector X whose M components are sums of a random number Kt of uniformly chosen components

153 Zero temperature recursion

(t) (t+1) of X . Each component Xi is constructed by (7.2.1) independently from the others. Obviously, a starting vector X(0) is needed. From a practical point of view, we take M large enough, such that P(Kt ≥ M) ≈ 0. This is equivalent to a random graph for which degrees are less or equal to M with probability 1. The data obtained by the implementation of such algorithm are then collected in the ﬁgures displayed in the following. We remark that we perform an approximate analysis working on Erdős–Rényi ER random graphs GN (p), with p = c/N.

7.2.1 Zero temperature and zero external ﬁeld case

From the analysis in Section 7.1.1 we know that the ﬁxed point of equation (7.1.2) is h∗ = 0 with probability η. We show now how this probability η changes with respect to the value c representing the connectivity of the Erdős–Rényi random graph. We compare numerical data, obtained using the Population Dynamics algorithm, and the analytical solution of the equation η = e−c(1−η).

Figure 1: Plot of η versus c at T = 1/β = 0 and B = 0. The blue circles represent the analytical solution of (7.1.12) and the full red dots are the numerical data.

Figure 1 shows that the probability η decreases when the connectivity c of the graph increases. Furthermore, we note that there is a perfect match between the data obtained from the numerics and the analytical analysis.

154 7.2 Numerical analysis

7.2.2 Zero temperature and positive external ﬁeld case

The following figures display results obtained by numerical simulation to estimate the distribution for fixed points different from infinity of the equation (7.1.14). We present some histograms, based on a sample of 1000000 data points obtained using the Population Dynamics algorithm, for different values of the field B and the connectivity c.

(a) c = 0.5 (b) c = 1

(e) c = 4 (f) c = 5

Figure 2: Distribution of h∗ at T = 0, B = 0.1, c = 0.5, 1, 2, 3, 4, 5.

155 Zero temperature recursion

(a) c = 0.5 (b) c = 1

(e) c = 4 (f) c = 5

Figure 3: Distribution of h∗ at T = 0, B = 1, c = 0.5, 1, 2, 3, 4, 5.

156 7.2 Numerical analysis

(a) c = 0.5 (b) c = 1

(e) c = 4 (f) c = 5

Figure 4: Distribution of h∗ at T = 0, B = 2, c = 0.5, 1, 2, 3, 4, 5,

From the figures we note that the fixed point h∗ takes values on integer multiples of the external field B and that the shape of the distributions is similar for the same values of c, regardless of B. Moreover, we can observe that the distribution originating from the numerical analysis agrees with the law of the distribution in equation (7.1.27), i.e., P(h∗ = (cn)n−1 −cn B n) = n! e . Normalizing the data from the numerical simulation, we can compare the two distributions. For example, in figures 5 and 6 below, we can see this resemblance for B = 0.1 and for c = 2, 3. Some tables that report the probabilities for the numerical and analytical data are provided.

157 Zero temperature recursion

Figure 5: Plot of distributions of h∗ at T = 0, B = 0.1, c = 2. The red bars represent the numerical data and the blue bars are the analytical data.

c = 2 P(h∗ = ∞) P(h∗ = 0.1) P(h∗ = 0.2) P(h∗ = 0.3) P(h∗ = 0.4)

Num. data 0.7967 0.1353 0.0367 0.0150 0.0073 An. data 0.7968 0.1353 0.0366 0.0149 0.0072

c = 2 P(h∗ = 0.5) P(h∗ = 0.6) P(h∗ = 0.7) P(h∗ = 0.8) P(h∗ = 0.9)

Num. data 0.0038 0.0021 0.0011 0.0008 0.0005 An. data 0.0038 0.0021 0.0012 0.0007 0.0005

Figure 6: Plot of distributions of h∗ at T = 0, B = 0.1, c = 3. The red bars represent the numerical data and the blue bars are the analytical data.

c = 3 P(h∗ = ∞) P(h∗ = 0.1) P(h∗ = 0.2) P(h∗ = 0.3) P(h∗ = 0.4)

Num. data 0.94047 0.0499 0.0074 0.0016 0.0004 An. data 0.94048 0.0498 0.0074 0.0017 0.0004

158 7.2 Numerical analysis

7.2.3 Non-zero temperature case

We present next some figures that represent the distribution of the fixed point h∗ at non-zero temperature. We set c = 2, where we remember that the value c represents the connectivity of the Erdős–Rényi random graph. Then, the critical temperature for the Ising model is Tc = 1/atanh(1/c) ≈ 1.82. The following histograms, always obtained using the Population Dynamics algorithm, display the situation for three different values of the temperature T , below the critical temperature. The picture on the right side show an enlargement of the segment (0, 1].

(a) T = 1 (b) T = 1 zoom

(e) T = 1.2 (f) T = 1.2 zoom

Figure 7: Distribution of h∗ at T = 1, 1.1, 1.2, c = 2,

159 Zero temperature recursion

In the absence of a rigorous analytical study, this numerical analysis is useful to provide an idea on the shape of the distribution at non-zero temperature. In all the pictures we can note a high peak of the distribution of h∗ on the value 0, plus a self-similar behavior on the segment (0, 1]. This scenario is quite interesting and it deserves a more detailed investigation in the future.

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165

SUMMARY

The Ising model is one of the most classical models studied in physics. Histori- cally, the model was introduced by Lenz and Ising to understand ferromagnets, investigating interactions between atoms of a magnetic material. Despite its simple definition, the model presents a rich behavior and interesting features, such as the presence of a phase transition for the spontaneous magnetization as a function of the temperature. Today there is a rising interest in the Ising model on random graphs, mainly due to applications in fields as diverse as social science, biology, neuroscience and opinion formation. Indeed, this model is a paradigm model in statistical physics to analyze the effect of cooperative phenomena on networks. The Ising model on random graphs is also an ideal model in probability to study asymptotic results in the presence of two sources of randomness: a dependence structure between the Ising random variables and an extra level of randomness given by the random graphs. Taking this double randomness into account in various ways, three different measures arise. For a given assignment of the graph the Random Quenched measure coincides with the random Boltzmann–Gibbs distribution on that random graph. Averaging over all possible random graphs in two different ways we obtain the Averaged Quenched and the Annealed measures. The aim of this thesis is to derive asymptotic results, like the law of large numbers and the central limit theorem, for the sum of the spin variables of the Ising model on random graphs with respect to these three different measures. In the random quenched setting our results apply to the class of locally tree-like random graphs in the uniqueness regime outside of the critical region. To prove the random quenched central limit theorem (CLT) on locally tree-like random graphs we adapt Ellis’ proof for the CLT in the Zd case. As in the case of the Ising model on lattice the variance appearing in the central limit theorem is given by the susceptibility of the model. Unfortunately, the general strategy used to prove the CLT for the local tree- like random graphs in the random quenched setting does not work for the averaged quenched and annealed measures, since the fluctuations due to the spatial structure become relevant there. Therefore we study the Ising model on particular examples of random graphs. In the average quenched setting, while the law of large numbers can be formulated for the entire class of local tree-like random graph, for the study of the central limit theorem we restrict ourselves to the 2-regular configuration model and the configuration model with degrees 1 and 2. Our proofs are based on explicit computations relying on the solution of the classical one-dimensional Ising model.

167 Bibliography

Regarding the annealed measure, besides the two configuration models afore- mentioned, we investigate the generalized random graph, where a phase transition is present. Our proofs are again based on explicit computations since the Ising model on the annealed generalized random graph reduces to an inhomogeneous Curie–Weiss model. In the uniqueness regime, we first compute the limit of thermodynamic quantities, identifying a critical annealed temperature which is larger than the quenched one, thus showing the relevance of the double randomness. Then we prove the law of large numbers and the central limit theorem. Our analysis shows that fluctuations of the random graphs play a crucial role in the averaged quenched and annealed settings. In particular, the variance of the Gaussian limiting law of the central limit theorem is in general affected by the graph fluctuations. Finally we analyze what happens at the critical point. Here the variance diverges, so the CLT breaks down and a different scaling of the total spin is needed. We investigate this problem for the Ising model on annealed generalized random graphs. We show that the critical exponents for this model match those of an Ising model on locally tree-like random graphs in the random quenched setting, thus showing a remarkable universality. Further, we prove a non-classical limit theorem at the critical point. Interestingly, the proof reveals that we need different scalings for the total spin according to whether the degree distribution has a finite fourth moment or not.

168 CURRICULUMVITAE

Maria Luisa Prioriello was born on February 1, 1986 in Isernia, Italy. After finishing the secondary school at Liceo Scientifico I.S.I.S.S. in Bojano (CB), in 2005 she began her studies in Mathematics at the University of Perugia. She graduated cum laude for her Master degree in 2011. During these years she worked as a tutor and as a librarian at the Department of Mathematics and Computer Science. Moreover she worked as a scientific animator in school laboratories, scientific festivals and exhibitions. In January 2012 she started a PhD research project at University of Modena and Reggio Emilia, under the supervision of Prof. Cristian Giardinà and Prof. Claudio Giberti. The topic was the study of asymptotic theorems for Ising models on random graphs. This project was a Double Doctoral Degree Program between the University of Modena and Reggio Emilia and the Eindhoven University of Technology and Maria Luisa spent several periods at the Eindhoven University of Technology. In January 2015 she moved to Eindhoven to complete her PhD under the supervision of Prof. Remco van der Hofstad. The results obtained during these years of research are presented in this dissertation and collected in several papers. During her PhD Maria Luisa was involved in teaching activities and she participated in various conferences, schools and workshops in Modena, Eindhoven, Oberwolfach, Rome, Ravello, Marseille, Paris and Montecatini.

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