A Macroscopic View of Two Discrete Random Models by Alma Sarai Hernandez Torres B.Sc. Mathematics, Universidad de Guanajuato, 2014 M.Sc. Mathematics, The University of British Columbia, 2016 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the faculty of graduate and postdoctoral studies (Mathematics) The University of British Columbia (Vancouver) August 2020 ⃝c Alma Sarai Hernandez Torres, 2020 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: A Macroscopic View of Two Discrete Random Models submitted by Alma Sarai Hernandez Torres in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics. Examining Committee: Omer Angel, Mathematics Supervisor Martin Barlow, Mathematics Supervisor Mathav Murugan, Mathematics Supervisory Committee Member Lior Silberman, Mathematics University Examiner Nick Harvey, Computer Science University Examiner Russell Lyons, Indiana University External Examiner ii Abstract This thesis investigates the large-scale behaviour emerging in two discrete models: the uniform spanning tree on Z3 and the chase-escape with death process. Uniform spanning trees We consider the uniform spanning tree (UST) on Z3 as a measured, rooted real tree, continuously embedded into Euclidean space. The main result is on the existence of sub-sequential scaling limits and convergence under dyadic scalings. We study properties of the intrinsic distance and the measure of the sub-sequential scaling limits, and the behaviour of the random walk on the UST. An application of Wilson’s algorithm, used in the study of scaling limits, is also instrumental in a related problem. We show that the number of spanning clusters of the three-dimensional UST is tight under scalings of the lattice. Chase-escape with death Chase-escape is a competitive growth process in which red particles spread to adjacent uncoloured sites while blue particles overtake adjacent red particles. We propose a variant of the chase-escape process called chase-escape with death (CED). When the underlying graph of CED is a d-ary tree, we show the existence of critical parameters and characterize the phase transitions. iii Lay Summary Statistical mechanics states that natural phenomena arise as the average behaviour of a large number of particles with random interactions. A central endeavour in probability theory is to establish a mathematical foundation for this paradigm. Our objective is to obtain precise relations between the microscopic and macroscopic descriptions of a phenomenon. This thesis is a contribution to the task. In particular, we are interested in the macroscopic properties emerging in two discrete random models. In this work, we study the “uniform spanning tree” and the “chase-escape with death” process. The first one is a combinatorial model that provides insights into other models in statistical mechanics. In a different setting, “chase-escape with death” mimics the behaviour of predators chasing prey on space, or the spread of a rumor throughout a social network. iv Preface Part I is the introduction for this thesis. Chapter 1 is an overview, while Chapters 2, 3 and 4 are surveys on background material. Part II presents original research on uniform spanning trees. Chap- ter 5 and Chapter 6 are based on the preprints “Scaling limits of the three- dimensional uniform spanning tree and associated random walk” [11] and “The number of spanning clusters of the uniform spanning tree in three di- mensions” [10], respectively. Our work in [10] will appear in the proceedings of “The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) Stochastic Analysis, Random Fields and Integrable Probability”, while [11] is under review for publication. The research leading to these was an equal collaboration between Omer Angel, David Croydon, Daisuke Shiraishi, and myself. The writings of [11] and [10] were done in equal parts between Omer Angel, David Croydon, Daisuke Shiraishi, and myself. Part III is original work on competitive growth processes. The re- search and writing was conducted in equal collaboration with Erin Beckman, Keisha Cook, Nicole Eikmeier and Matthew Junge. Chapter 7 is based on “Chase-escape with death on trees” [25] and has been submitted for publi- cation. v Table of Contents Abstract . iii Lay Summary . iv Preface . v Table of Contents . vi List of Tables . xi List of Figures . xii List of Symbols . xv Acknowledgements . xix Dedication . xxi I Introduction . 1 1 Discrete and Continuous Probability Models . 2 1.1 Scaling limits as weak convergence of probability measures . 3 1.2 Phase transitions . 5 1.3 Structure of this thesis . 8 2 Random Walks . 9 vi 2.1 Notation . 9 2.1.1 Subsets . 9 2.1.2 Paths and curves . 10 2.1.3 Constants and asymptotic notation . 11 2.2 Simple random walk . 12 2.2.1 Recurrence and transience . 13 2.2.2 Harmonic measure and hitting probabilities . 14 2.2.3 Scaling limit of the simple random walk . 17 2.3 Loop-erased random walks . 21 2.3.1 The infinite loop-erased random walk . 21 2.3.2 Growth exponent . 23 2.3.3 Hitting probabilities . 25 2.4 Scaling limits of loop-erased random walks . 26 2.4.1 Two dimensions . 26 2.4.2 Three dimensions . 30 2.4.3 Four and higher dimensions . 31 3 Uniform Spanning Forests . 33 3.1 Definition and basic properties . 33 3.2 Relation to other models . 35 3.2.1 The random-cluster model . 35 3.3 Sampling algorithms . 36 3.3.1 Wilson’s algorithm . 36 3.3.2 Aldous-Broder algorithm . 37 3.3.3 Interlacement Aldous-Broder algorithm . 38 3.4 Encodings of uniform spanning trees . 38 3.4.1 Paths . 39 3.4.2 Graphs as metric spaces . 39 3.5 Scaling limits of uniform spanning trees . 41 3.5.1 Finite graphs . 41 3.5.2 Infinite graphs . 46 vii 4 Competitive Growth Processes . 49 4.1 Markov processes . 51 4.2 Interacting particle systems . 53 4.3 Richardson models . 57 4.3.1 The one-type Richardson model . 58 4.3.2 The two-type Richardson model . 60 4.3.3 Multiple type Richardson model . 62 4.4 First passage percolation . 63 4.4.1 First passage competition models . 65 4.4.2 Geodesics . 66 4.5 Chase-escape process . 69 4.5.1 Phase transitions . 71 II Scaling Limits of Uniform Spanning Trees . 76 5 Scaling Limit of the Three-Dimensional Uniform Spanning Tree................................. 77 5.1 Introduction . 78 5.1.1 Scaling limits of the three-dimensional UST . 79 5.1.2 Properties of the scaling limit . 82 5.1.3 Scaling the random walk on U ............. 85 5.2 Topological framework . 91 5.2.1 Path ensembles . 95 5.3 Loop-erased random walks . 96 5.3.1 Notation for Euclidean subsets . 97 5.3.2 Notation for paths and curves . 98 5.3.3 Definition and parameterization of loop-erased ran- dom walks . 102 5.3.4 Path properties of the infinite loop-erased random walk103 5.3.5 Loop-erased random walks on polyhedrons . 109 5.3.6 Proof of Proposition 5.3.9 ................ 113 5.4 Checking the assumptions sufficient for tightness . 121 viii 5.4.1 Assumption 1 ....................... 121 5.4.2 Assumption 2 ....................... 127 5.4.3 Assumption 3 ....................... 139 5.5 Exponential lower tail bound on the volume . 142 5.6 Exponential upper tail bound on the volume . 152 5.7 Convergence of finite-dimensional distributions . 158 5.7.1 Parameterized trees . 158 5.7.2 The scaling limit of subtrees of the UST . 161 5.7.3 Parameterized trees and random walks . 163 5.7.4 Essential branches of parameterized trees . 164 5.7.5 Proof of Theorem 5.7.2 ................. 167 5.7.6 Proof of Proposition 5.7.9 ................ 169 5.8 Proof of tightness and subsequential scaling limit . 183 5.9 Properties of the limiting space . 188 5.10 Simple random walk and its diffusion limit . 194 6 Spanning Clusters of the Uniform Spanning Tree in Three Dimensions . 198 6.1 Introduction . 199 6.2 Notation . 202 6.3 Proof of the main result . 204 III Competitive Growth Processes . .211 7 Chase-Escape with Death on Trees . 212 7.1 Introduction . 212 7.1.1 Results . 213 7.1.2 Proof methods . 216 7.1.3 History and context . 217 7.2 CED on the line . 218 7.3 Properties of weighted Catalan numbers . 223 7.3.1 Preliminaries . 223 7.3.2 Properties of f and M .................. 225 ix 7.4 M and CED . 230 7.5 Proofs of Theorems 7.1.1, 7.1.2 , and 7.1.3 .......... 232 7.6 Proof of Theorem 7.1.4 ..................... 235 7.7 Proof of Theorem 7.1.5 ..................... 236 λ,ρ 7.7.1 A lower bound on Ck .................. 236 λ,ρ 7.7.2 An upper bound on Ck ................. 237 7.7.3 A finite runtime algorithm . 240 8 Conclusions . 241 8.1 Contributions . 241 8.1.1 Uniform spanning trees . 241 8.1.2 Competitive growth process . 243 8.2 Future directions . 244 8.2.1 Uniform spanning trees . 245 8.2.2 Competitive growth models . 246 Bibliography . 247 x List of Tables Table 5.1 Exponents associated with the LERW and UST in two and three dimensions. The two-dimensional exponents are known rigorously from [20, 21, 24, 95]. The three- dimensional values are based on the results of this study, together with the numerical estimate for the growth expo- nent of the three-dimensional LERW from [170]. 87 xi List of Figures Figure 5.1 A realisation of the UST in a three-dimensional box, as embedded into R3 (left), and drawn as a planar graph tree (right).