Graphical representations of Ising and Potts models Stochastic geometry of the quantum Ising model and the space–time Potts model JAKOB ERIK BJÖRNBERG Doctoral Thesis Stockholm, Sweden 2009 TRITA-MAT-09-MA-13 ISSN 1401-2278 KTH Matematik ISRN KTH/MAT/DA 09/09-SE SE-100 44 Stockholm ISBN 978-91-7415-460-3 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik onsdagen den 18 november 2009 klockan 13.30 i Kollegiesalen, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm. © Jakob Erik Björnberg, oktober 2009 Tryck: Universitetsservice US AB iii Abstract Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to math- ematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in Z. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This repre- sentation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential in- equalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in Z and in ‘star-like’ geometries. iv Sammanfattning Statistisk fysik syftar till att förklara ett materials makroskopiska egen- skaper i termer av dess mikroskopiska struktur. En särskilt intressant egen- skap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två mo- deller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två mo- deller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har se- dermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Ising- modellen med transverst fält, vilken är en utveckling av den klassiska Ising- modellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontakt- modellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggan- de egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad kom- ponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteore- tiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet myc- ket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på Z, samt i ‘stjärnliknankde’ geo- metrier. Acknowledgements I have had the great fortune to be able to share my time as a PhD student between KTH and Cambridge University, UK. I would like to thank my advisor Professor Anders Björner for making this possible, as well as for extremely generous guidance and advice throughout my studies. I would also like to thank my Cambridge advisor Professor Geoffrey Grimmett for providing me with many interesting mathematical problems and stimulating discussions, as well as invaluable advice and suggestions. Chapter 3 and Section 4.1 were done in collaboration with Geoffrey Grimmett, and has appeared in a journal as a joint publication [15]. Section 4.2 has been published in a journal [14]. Riddarhuset (the House of Knights) in Stockholm, Sweden, has supported me extremely generously throughout my studies. I have received further generous sup- port from the Engineering and Physical Sciences Research Council under a Doc- toral Training Award to the University of Cambridge. Thanks to grants from Rid- darhuset I was able to spend a month at UCLA, and to attend two workshops at the Oberwolfach Institute. Grants from the Department of Pure Mathematics and Mathematical Statistics in Cambridge and from the Park City Mathematics Institute made it possible for me to attend a summer school in Park City; grants from Gonville & Caius College and the Institut Henri Poincaré in Paris made it possible for me to attend a workshop at the latter. Large parts of the writing of this thesis took place during a very stimulating stay at the Mittag-Leffler Institute for Research in Mathematics, Djursholm, Sweden, during the spring of 2009. I am very grateful to the those who have supported me. Finally I would like to express my deep appreciation of my family, my friends, and my colleagues at KTH and Cambridge for their support and encouragement. Professor Moritz Diehl helped kickstart my research career. Becky was always there to scatter my doubts. When I haven’t been working, I have mostly been rowing; while I have been working, I have constantly been listening to music. Also thanks, therefore, to all my rowing buddies, and to all the artists on my playlist. v Contents Acknowledgements v Contents vi 1 Introduction and background 1 1.1 Classicalmodels ............................. 2 1.2 Quantummodelsandspace–timemodels. 6 1.3 Outline .................................. 8 2 Space–time models 11 2.1 Definitionsandbasicfacts . 11 2.2 Stochasticcomparison . 24 2.3 Infinite-volume random-cluster measures . .... 39 2.4 Duality in Z R ............................. 52 2.5 Infinite-volumePottsmeasures× . 56 3 The quantum Ising model 65 3.1 ClassicalandquantumIsingmodels . 65 3.2 Therandom-parityrepresentation . 70 3.3 Theswitchinglemma........................... 80 3.4 Proof of the main differential inequality . ... 95 3.5 Consequencesoftheinequalities . 100 4 Applications and extensions 107 4.1 Inonedimension ............................. 107 4.2 Onstar-likegraphs. .. .. .. .. .. .. .. .. .. .. .. 110 4.3 Reflectionpositivity . 118 4.4 RandomcurrentsinthePottsmodel . 122 A The Skorokhod metric and tightness 129 B Proof of Proposition 2.1.4 133 Bibliography 135 vi List of notation Conjugate transpose, page 7 h·| Expectation under Ising measure, page 23 h·i ± Ising measure with boundary condition, page 59 h·i ± Basis of C2, page 7 |±i σ Basis vector in , page 6 | i H α Part of Potts boundary condition, page 20 B Edge set of H, page 111 (K) Borel σ-algebra, page 13 B B Process of bridges, page 15 b Boundary condition, page 17 χ Magnetic susceptibility, page 101 ∂ˆΛ (Inner)boundary,page13 ∆ Processofcuts,page81 δ Intensity of D, page 14 d(v) Number of deaths in Kv, page 76 Dv Deaths in Kv, page 76 ∂Λ Outerboundary,page13 ∂ψ Weight of colouring ψ, page 72 D Process of deaths, page 15 E Edge set of L, page 11 vii viii List of notation ev(ψ) Set of ‘even’ points in ψ, page 72 E Edge set of L, page 12 E(D) Edgesetofthegraph G(D), page 73 Skorokhod σ-algebra on Ω, page 15 F Restricted