DOCSLIB.ORG

Dissipation in the Abelian Sandpile Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dissipation in the Abelian Sandpile Model Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Faculteit Technische Natuurwetenschappen Delft Institute of Applied Mathematics Dissipation in the Abelian Sandpile Model Verslag ten behoeve van het Delft Institute of Applied Mathematics als onderdeel ter verkrijging van de graad van BACHELOR OF SCIENCE in Technische Wiskunde en Technische Natuurkunde door HENK JONGBLOED Delft, Nederland Augustus 2016 Copyright c 2016 door Henk Jongbloed. Alle rechten voorbehouden. BSc verslag TECHNISCHE WISKUNDE en TECHNISCHE NATUURKUNDE \Dissipation in the Abelian Sandpile Model" HENK JONGBLOED Technische Universiteit Delft Begeleiders Prof. dr.ir. F.H.J. Redig Dr.ir. J.M. Thijssen Overige commissieleden Dr. T. Idema Dr. J.L.A. Dubbeldam Dr. J.A.M. de Groot Augustus, 2016 Delft Abstract The Abelian Sandpile model was originally introduced by Bak, Tang and Wiesenfeld in 1987 as a paradigm for self-organized criticality. In this thesis, we study a variant of this model, both from the point of view of mathematics, as well as from the point of view of physics. The effect of dissipation and creation of mass is investigated. By linking the avalanche dynamics of the infinite-volume sandpile model to random walks, we derive some criteria on the amount of dissipation and creation of mass in order for the model to be critical or non-critical. As an example we prove that a finite amount of conservative sites on a totally dissipative lattice is not critical, and more generally, if the distance to a dissipative site is uniformly bounded from above, then the model is not critical. We apply also applied a renormalisation method to the model in order to deduce its critical exponents and to determine whether a constant bulk dissipation destroys critical behaviour. Numerical simulations and a statistical analysis are performed to estimate critical exponents. Finally, we give a short discussion on self-organized criticality. Preface Sitting here in the pittoresque Zillertal in Austria, the beauty of nature intrigues me. Moreover, the symphony of light, sound and movement has triggered existential sensations in many people in history. No wonder that the great Greek philosopher Aristotle has said `Wonder is the beginning of all philosophy'. The research documented in this report forms my Bachelor Project at the TU Delft for the pro- grammes Applied Mathematics and Applied Physics. When choosing a project, I wanted to study a subject contained a robust mathematical aspect as well as an interesting physical interpretation. Also, numerical simulation was a preference. These properties are all found in the subject of this research, which is the Abelian Sandpile Model. Proposed for the first time in 1987 by Bak, Tang and Wiesenfeld in a physical context, it served as an illustration of `self-organized criticality'. Since then, the model and related concepts have been studied both in the mathematics and physics literature. Although much of the model has been understood thus far, there remain open questions as to how complex structure can arise out of some elementary local interaction rules. This touches on the philosophical concept of emergence. These three disciplines combined -physics, mathematics and philosophy- make the Abelian Sandpile model a beautiful subject for me to study. Some personal notes are in place here. During this project, I had a lot of other activities, like three courses, extracurricular activities and preparations for next year, which will be a year on the board of a student association. All of this made it difficult to really focus on the project, especially since it is on my own. In fact, I have come to better know myself, my interests and my weaknesses in studying. Projects are different from courses in a lot of ways and require a great deal of independence. To double-degree bachelor students, the difference can be even greater due to the relative small number of projects we have had before and necessity of ‘efficient course studying' in order to graduate in time. In a project, efficiency is a lot harder to achieve, at least in my case. Delay of thesis defence was one of the consequences of my behaviour. I do not know exactly how to say it, but the last year of my bachelor was a lot harder that expected. This project was a real challenge for me; I have learned a lot but also disappointed myself due to my bad skills in planning. These are all lessons I will take with me in the future. I would like to thank my supervisors Frank Redig and Jos Thijssen for the time and effort they offered to help me in my project. Making appointments always went smoothly and I have learned a lot from them. The mathematical background of this project I discussed with Frank, and many times during these meetings we ran into problems that I could not even begin to solve with my relative small experience in the field of probability. Luckily, Frank always found a way to tackle a specific problem. With Jos, I mostly discussed numerical simulation, renormalisation theory and general theory of critical phenomena. These meetings were always very informative and relaxed, for which I thank him greatly. I also thank dr. Johan Dubbeldam, dr. Timon Idema and dr. Joost de Groot for taking place in my thesis committee. Finally, I would like to thank my family, roommates and close friends for giving me advice and helping me. Henk Jongbloed, August 15th, 2016 Contents 1 Introduction 1 2 The classical Abelian Sandpile Model 3 3 Introduction to mixed dissipative/source systems 7 4 Markov Processes, semigroups and generators 11 4.1 Definitions . 11 4.2 Deriving the generator of Markov Processes . 13 4.3 The Feynman-Kac formula for countable state space Markov Processes . 16 5 Toppling numbers, Avalanches and Random Walks 18 5.1 Linking avalanche dynamics to random walks . 20 5.2 Towards infinite volume . 23 5.3 Estimating critical behaviour . 26 5.4 From a CRW to a DRW . 26 6 Mathematical results 29 6.1 Conditions on D to obtain non-criticality . 29 6.2 Adding sources . 32 6.3 Finitely many sources . 32 7 A Renormalisation Approach to the BTW model 35 7.1 Introduction: Critical phenomena and renormalisation . 35 7.2 General remarks . 36 7.3 Renormalisation equations . 37 7.4 Fixed points . 43 7.5 Critical exponents . 43 7.6 Introducing dissipation . 45 8 Numerical simulation 46 8.1 Simulating the ASM . 46 9 Critical avalanche data analysis 53 9.1 The BTW approach . 53 9.2 Likelihood estimation of τ ................................... 55 9.3 Truncated power law MLE estimation . 58 9.4 Our recommendation . 60 10 Self-Organized Criticality 61 11 Conclusions, Discussion, Recommendations and personal notes 64 Appendices 68 A Project Description: Dissipation in the Abelian Sandpile Model 68 B Code 69 1 Introduction It has been almost 30 years since Per Bak, Chao Tang and Kurt Wiesenfeld (BTW) proposed the sandpile model as a paradigm of self-organized criticality (SOC) [1]. It serves as the simplest and best- studied example of a non-equilibrium system, driven at a slow steady rate by adding particles, with local threshold relaxation rules, which in its critical state shows power-law behaviour obtained without fine-tuning of any control parameters. BTW claimed that the concept of SOC is an explanation of many different physical phenomena: from the formation of the Earth's crust to the dynamics of solar flares to the distribution of skyscrapers in the world's biggest cities [2]. Figure 1: Per Bak, Chao Tang and Kurt Wiesenfeld.1 This immediately explains the relevance of studying `toy models' such as the sandpile model. Through detailed understanding of these relatively easy models, we may be able to make predictions about certain phenomena in the real world. The model was originally defined as follows. In one dimension (d = 1), consider a connected subset of Z, which we will call Vn. Without loss of generality we can take Vn = [−n; n] \ Z, and thus 2n + 1 is the size of the system. We denote by x 2 Vn a site, and we define a height function η on Vn. Choosing a site x at random, we consequently keep adding particles to the system, increasing local heights by one unit. η(x) ! η(x) + 1 Given the fixed threshold value 2, when a local height becomes greater or equal to 1, one particle tumbles, which is referred to as a toppling event. η(x) ! η(x) − 2 η(x ± 1) ! η(n ± 1) + 1 if η(x) ≥ 2 Boundaries can be either absorbing or periodic. With absorbing boundaries, the one-dimensional BTW model naturally evolves to a minimally stable state, where all but one site have a height of 1. The toppling rules in higher dimensions are essentially the same as in one dimension. In most papers, a d sandpile on a simply connected subset of Z is defined as a height function η with a critical height of 2d determines the toppling condition. Upon toppling, a site distributes its height to its 2d nearest- neighbours. On boundary sites, particles are lost. Thereafter, if another site becomes unstable from a previous toppling, it topples, until the time that no unstable sites exist in the system anymore. At this point, a particle is again added to a random location in Vn. A series of connected toppling events is called an avalanche. However, the dynamics of higher-dimensional BTW models is fundamentally 1Sources: http://www.eoht.info/page/Per+Bak, https://en.wikipedia.org/wiki/Chao_Tang, https://www. physics.gatech.edu/user/kurt-wiesenfeld 1 INTRODUCTION different from the one-dimensional model. Rather than evolving to its minimally stable configuration, it naturally evolves to a critical state in which avalanches of all sizes, up to the size of the system itself, occur.
Load more

Recommended publications
  • Abelian Sandpile Model on Symmetric Graphs
    Abelian Sandpile Model on Symmetric Graphs Natalie J. Durgin Francis Edward Su, Advisor David Perkinson, Reader May, 2009 Department of Mathematics Copyright c 2009 Natalie J. Durgin. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self- organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sand- pile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid. Contents Abstract iii Acknowledgments ix 1 Introduction to the Theory of Sandpiles 1 1.1 Origins of the Model . .1 1.2 Model Structure and Definitions . .3 1.3 Standard Sandpile Theorems . .7 1.4 Sandpile Literature . 13 1.5 Alternate Approaches to Standard Theorems . 14 2 Symmetric Sandpiles 19 2.1 Applications to Tiling Theorems . 19 2.2 Graphs with Symmetry . 21 2.3 Summary of Notation . 25 2.4 The Symmetric Sandpile Subgroup . 26 3 Developing a Quotient Graph 29 3.1 The Row Quotient .
    [Show full text]
  • The Sandpile Load-Balancing Problem for Latency Minimization
    THE SANDPILE LOAD BALANCER AND LATENCY MINIMIZATION PROBLEM BERJ CHILINGIRIAN MASSACHUSETTS INSTITUTE OF TECHNOLOGY Abstract. I present the sandpile load balancer, an application of the abelian sandpile model to dynamic load balancing. The sandpile load balancer exploits the self-organized criticality of the abelian sandpile model to distribute web traffic across a network of servers. I also propose the sandpile load balancer latency minimization problem in which the goal is to minimize the latency of packets across the network of servers. I show the number of possible strategies for solving this problem and explore heuristic-based approaches via simulation. 1. Introduction Bak, Tang, and Wiesenfeld introduced the abelian sandpile model in 1987. Their seminal paper was concerned with modeling the avalanches that occur in growing piles of sand and how the frequency of such avalanches relates to pink noise [1]. Dhar extended their work in 1990 by proving several mathematical properties of the abelian sandpile model [2]. One such property is self-organized criticality: the tendency of the system to converge on a critical state in which it is simultaneously stable and on the edge of chaos. Coincidentally, self-organized criticality has also been observed in a variety of dynamic systems in nature [1]. The notion of self-organized criticality conveniently applies to dynamic load bal- ancing in which a network of servers distribute incoming traffic in order to \balance" the processing burden. This paper is concerned with the application of the abelian sand pile model to dynamic load balancing in order to exploit self-organized criti- cality for a real-world problem.
    [Show full text]
  • Harmonic Dynamics of the Abelian Sandpile
    Harmonic dynamics of the abelian sandpile Moritz Langa,1 and Mikhail Shkolnikova aInstitute of Science and Technology Austria, 3400 Klosterneuburg, Austria Edited by Yuval Peres, Microsoft Research, Redmond, WA, and approved January 3, 2019 (received for review July 12, 2018) The abelian sandpile is a cellular automaton which serves as “avalanche.” Due to the loss of particles at the boundaries of the the archetypical model to study self-organized criticality, a phe- domain, this process eventually terminates (ref. 4, theorem 1), nomenon occurring in various biological, physical, and social pro- and the “relaxed” sandpile reaches a stable configuration which cesses. Its recurrent configurations form an abelian group, whose is independent of the order of topplings (ref. 3, p. 13). The distri- identity is a fractal composed of self-similar patches. Here, we ana- bution of avalanche sizes—the total number of topplings after a lyze the evolution of the sandpile identity under harmonic fields random particle drop—follows a power law (1) and is thus scale of different orders. We show that this evolution corresponds to invariant. However, the critical exponent for this power law is yet periodic cycles through the abelian group characterized by the unknown (5). smooth transformation and apparent conservation of the patches Soon after the introduction of the sandpile model (1), it was constituting the identity. The dynamics induced by second- and observed that the set of stable configurations can be divided third-order harmonics resemble smooth stretchings and transla- into two classes, recurrent and transient ones (6). Thereby, a tions, respectively,while the ones induced by fourth-order harmon- stable configuration is recurrent if it appears infinitely often ics resemble magnifications and rotations.
    [Show full text]
  • Abelian Sandpile Model on Symmetric Graphs Natalie Durgin Harvey Mudd College
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Scholarship@Claremont Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2009 Abelian Sandpile Model on Symmetric Graphs Natalie Durgin Harvey Mudd College Recommended Citation Durgin, Natalie, "Abelian Sandpile Model on Symmetric Graphs" (2009). HMC Senior Theses. 217. https://scholarship.claremont.edu/hmc_theses/217 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Abelian Sandpile Model on Symmetric Graphs Natalie J. Durgin Francis Edward Su, Advisor David Perkinson, Reader May, 2009 Department of Mathematics Copyright c 2009 Natalie J. Durgin. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self- organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sand- pile group is isomorphic to the symmetric subgroup of the original graph.
    [Show full text]
  • A Structure Theory of the Sandpile Monoid for Digraphs (EXTENDED ABSTRACT)
    A Structure Theory of the Sandpile Monoid for Digraphs (EXTENDED ABSTRACT) L. Babai∗ E. Toumpakari† May, 2005 Abstract The Abelian Sandpile Model is a diffusion process on graphs, studied, under various names, in statisti- cal physics, theoretical computer science, and algebraic graph theory. The model takes a rooted directed multigraph X ∗, the ambient space, in which the root is accessible from every vertex, and associates with it a commutative monoid M, a commutative semigroup S, and an abelian group G as follows. For ver- ∗ tices i, j, let aij denote the number of i → j edges and let deg(i) denote the out-degree of i in X . Let V be the set of ordinary (non-root) vertices. With each i ∈ V associate a symbol xi and consider the P relations deg(i)xi = j∈V aij xj . Let M, S, and G be the commutative monoid, semigroup and group, respectively, generated by {xi : i ∈ V } subject to these defining relations. M is the sandpile monoid, S is the sandpile semigroup, and G is the sandpile group associated with X ∗. We observe that G is the unique minimal ideal of M. We establish connections between the algebraic structure of M, S, G, and the combinatorial structure of the underlying ambient space X ∗. M is a distributive lattice of semigroups each of which has a unique idempotent. The distributive lattice in question is the lattice L of idempotents of M; L turns out to be isomorphic to the dual of the lattice of ideals of the poset of normal strong components of X ∗ (strong components which contain a cycle).
    [Show full text]
  • The Computational Complexity of the Abelian Sandpile Model
    This is page i Printer: Opaque this The Computational Complexity of The Abelian Sandpile Model Juan Andres Montoya, Carolina Mejia June 2011 ii ABSTRACT This is the …rst volume of a series of studies related to the complexity theoretical analysis of statisitical mechanics. In this …rst work we have considered some predicting tasks arising in statistical mechanics. Predicting the evolution of dynamical systems is one of the foundational tasks of natural sciences. Statistical mechanics is mainly concerned with …nite dynamical systems, which are more easy (from the conceptual point of view) of analyzing. We have chosen to work with one speci…c system of statisitical mechanics: The Abelian Sandpile Model. This is page iii Printer: Opaque this Contents 0.1 Organization of the work . vi 0.2 Acknowledgements . vi 1 Basics vii 1.1 Lattices . vii 1.2 Complexity theory . viii 1.2.1 Parallel complexity . viii 1.3 Exercises ............................ x 2 The Complexity of Predicting xi 2.1 Prediction problems and Boltzmann systems . xi 2.2 Prediction problems and PSPACE . xii 2.3 Exercises . xvi 3 The Abelian Sandpile Model xvii 3.1 Exercises . xxvi 4 Algorithmic problems xxvii 4.1 The algorithmic problems . xxvii 4.2 The relative hardness of sandpile prediction problems . xxix 4.3 Exercises . xxxiii 5 Statistics of critical avalanches xxxv 5.1 Exercises . xli 6 Dimension 1 xliii iv Contents 6.1 GC [ 1] belongs to logDCF L . xliii L 0 6.2 SP A [ 1] is TC -hard . xlvi 6.3 A longL remark: one-dimensional critical avalanches . xlviii 6.4 Exercises . xlix 7 Dimension 2 li 7.1 The hardness of two-dimensional sandpile prediction problems li 7.2 Two-dimensional critical avalanches .
    [Show full text]
  • Patterned and Disordered Continuous Abelian Sandpile Model
    Patterned and Disordered Continuous Abelian Sandpile Model N. Azimi-Tafreshi,∗ S. Moghimi-Araghi† Department of Physics, Sharif University of Technology, Tehran, P.O.Box: 11365-9161, Iran Abstract We study critical properties of the continuous Abelian sandpile model with anisotropies in toppling rules that produce ordered patterns on it. Also we consider the continuous directed sandpile model perturbed by a weak quenched randomness and study critical behavior of the model using perturbative conformal field theory and show the model has a new random fixed point. PACS: 05.65+b, 89.75.Da Keywords: Self-Organized Criticality, Sandpile models, Conformal field theory. 1 Introduction The idea of self-organized criticality, introduced by Bak, Tang and Wisenberg [1], provides a useful framework for the study of non-equilibrium systems which dynamically evolve into a critical state without tuning of a control parameter. At critical state, these systems show scaling behaviors and this scaling behavior of the system is characterized by critical exponents [2]. The BTW sandpile model, renamed Abelian sandpile model after Dhar’s work [3], is the simplest lattice model that displays self-organized critical behavior. The Abelian structure of the model allows the theoretical determination of many of its properties [4, 5]. This model is usually defined on a square lattice. At each site of the lattice an integer height variable between 1 to 4 is assigned which represents the number of sand grains of that site. The evolution of the model at each time arXiv:0906.0459v1 [cond-mat.stat-mech] 2 Jun 2009 step is simple: a grain of sand is added to a random site.
    [Show full text]
  • Sandpile Groups
    Sandpile Groups Luis Garcia Puente (Sam Houston State University), Dino Lorenzini (University of Georgia), Criel Merino (National Autonomous University of Mexico), David Perkinson (Reed College), Carlos Enrique Valencia Oleta (National Polytechnic Institute) November 15–20, 2015 1 Overview of the Field The physicists Bak, Tang, and Wiesenfeld [5] created an idealized version of a sandpile in which sand is stacked on the vertices of a graph and is subjected to certain avalanching rules. They used the model as an example of what they called self-organized criticality. The abelian sandpile model is a variation, due to the physicist Deepak Dhar in 1990 [24], in which the avalanching obeys a useful commutativity rule. He realized that the model provided an expression of the dynamics inherent in the discrete Laplacian of a graph. The long-term behavior of the abelian sandpile model on a graph is encoded by the critical config- urations. These critical configurations have connections to parking functions [45], to the Tutte poly- nomial [46], and to the lattices of integral flows and cuts of a graph [53]. Among other properties, the critical configurations of the sandpile model have the structure of a group, and this group is our main object of study. It has been discovered in several different contexts and received many names: the sand- pile group for graphs [24] and digraphs [49], the critical group [8], the group of bicycles [7], the group of components [51], and the jacobian of the graph [54]. The abelian sandpile model and its close relative the chip-firing game [44] have become a cross- roads of a wide range of mathematics, physics, and computer science.
    [Show full text]
  • Mathematical Aspects of the Abelian Sandpile Model
    Mathematical aspects of the abelian sandpile model F. Redig∗ June 21, 2005 1 Introduction In 1987, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been stud- ied intensively, both in the physics and in the mathematics literature. This model shows that a simple dynamics can lead to the emergence of very complex structures and drives the system towards a stationary state which shares several properties of equilibrium systems at the critical point, e.g. power law decay of cluster sizes and of correlations of the height-variables. Some years later, Deepak Dhar generalized the model, discovered the “abelian group structure of addition operators” in it and called it “the abelian sandpile model”( abbreviated from now on ASM). He studied the self-organized critical nature of the stationary measure and gave an algorithmic characterization of recurrent configu- rations, the so-called “burning algorithm”. This algorithm gives a one-to one cor- respondence between the recurrent configurations of the ASM and rooted spanning trees. The correspondence with spanning trees allowed Priezzhev to compute the height probabilities in dimension 2 in the infinite volume limit. Probabilities of cer- tain special events -so-called “weakly allowed clusters”- can be computed exactly in the infinite volume limit using the “Bombay-trick”. In the physics literature peo- ple studied critical exponents with scaling arguments, renormalization group method and conformal field theory (in d = 2), and it is argued that the critical dimension of the model is d = 4 [35]. Dhar and Majumdar studied the model on the Bethe lattice where they computed various correlation functions and avalanche cluster-size distributions exactly in the thermodynamic limit, using a transfer matrix approach.
    [Show full text]
  • Abelian Networks IV. Dynamics of Nonhalting Networks
    Abelian networks IV. Dynamics of nonhalting networks Swee Hong Chan Lionel Levine Author address: Swee Hong Chan, Department of Mathematics, Cornell Univer- sity, Ithaca, NY 14853. http://www.math.cornell.edu/~sc2637/ Email address: [email protected] Lionel Levine, Department of Mathematics, Cornell University, Ithaca, NY 14853. http://www.math.cornell.edu/~levine Email address: [email protected] arXiv:1804.03322v2 [cs.FL] 14 Nov 2020 Dedicated to Sui Lien Peo and Bernard Ackerman Contents Chapter 1. Introduction1 1.1. Flashback1 1.2. Atemporal dynamics1 1.3. Relating atemporal dynamics to traditional dynamics2 1.4. Computational questions3 1.5. The torsion group of a nonhalting abelian network4 1.6. Critical networks4 1.7. Example: Rotor networks and abelian mobile agents5 1.8. Proof ideas6 1.9. Summary of notation6 Chapter 2. Commutative Monoid Actions9 2.1. Injective actions and Grothendieck group9 2.2. The case of finite commutative monoids 11 Chapter 3. Review of Abelian Networks 13 3.1. Definition of abelian networks 13 3.2. Legal and complete executions 15 3.3. Locally recurrent states 15 3.4. The production matrix 16 3.5. Subcritical, critical, and supercritical abelian networks 17 3.6. Examples: sandpiles, rotor-routing, toppling, etc 19 Chapter 4. The Torsion Group of an Abelian Network 29 4.1. The removal lemma 29 4.2. Recurrent components 31 4.3. Construction of the torsion group 38 4.4. Relations to the critical group in the halting case 41 Chapter 5. Critical Networks: Recurrence 45 5.1. Recurrent configurations and the burning test 45 5.2.
    [Show full text]
  • Tropical Curves, Convex Domains, Sandpiles and Amoebas
    Universite´ de Geneve` Faculte´ des Sciences Section des Math´ematiques Professeur Grigory Mikhalkin Tropical Curves, Convex Domains, Sandpiles and Amoebas These` pr´esent´ee `ala Facult´edes Sciences de l’Universit´ede Gen`eve pour obtenir le grade de Docteur `essciences, mention math´ematiques par Mikhail Shkolnikov de Saint-P´etersbourg, Russie Th`eseNo. Gen`eve Atelier d’impression ReproMail de l’Universit´ede Gen`eve 2017 Contents 1 Main results 4 2 R´esum´een fran¸cais 12 3 The caustic curve 16 3.1 Q-polygons and Steiner problem . 27 4 The canonical evolution 31 4.1 The canonical Cauchy problem . 34 4.2 Three topological constructions of XΩ .......... 37 5 Tropical Steiner problem and the limit of sandpiles 40 5.1 Symplectic area and complex curves . 41 5.2 Steiner problem . 45 5.3 Scaling limit theorem . 48 6 Integral affine invariants of convex domains 55 6.1 Computations for the disk . 57 6.2 The invariant stratification . 62 7 Panoramic view 69 2 8 Hyperbolic amoebas 78 8.1 Amoebas of lines . 80 8.2 Amoebas of higher degree curves . 86 8.3 Amoebas of Surfaces . 88 8.4 Other . 94 9 Tropical sandpiles 98 Figure 0.1: Illustration of the scaling limit Theorem 1.1. 3 Chapter 1 Main results Consider a lattice polygon ∆: A state φ of the sandpile model is a non-negative integral valued function on ∆ Z2: We think of every lattice point as container or cell. It can contain\ some integer amount of sand, where \sand" is a metaphor for mass-energy.
    [Show full text]
  • Nearly Tight Bounds for Sandpile Transience on the Grid
    Nearly Tight Bounds for Sandpile Transience on the Grid David Durfee Matthew Fahrbach∗ Georgia Institute of Technology Georgia Institute of Technology [email protected] [email protected] Yu Gaoy Tao Xiaoy Georgia Institute of Technology Shanghai Jiao Tong University [email protected] xt [email protected] Abstract We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an O(n4 log4 n) upper bound and an Ω(n4) lower bound for the transience class of the n × n grid. Our methods naturally extend to nd-sized d-dimensional grids to give O(n3d−2 logd+2 n) upper bounds and Ω(n3d−2) lower bounds. arXiv:1704.04830v2 [cs.DS] 15 Nov 2017 ∗Supported in part by a National Science Foundation Graduate Research Fellowship under grant DGE-1650044. yA substantial portion of this work was completed while the author visited the Institute for Theoretical Computer Science at Shanghai University of Finance and Economics.
    [Show full text]