SUPERPROBABILITY ON GRAPHS Andrew Charles Kenneth Swan Clare College A thesis submitted for the degree of Doctor of Philosophy October 2020 Abstract The classical random walk isomorphism theorems relate the local times of a continuous- time random walk to the square of a Gaussian free field. The Gaussian free field is a spin system (or sigma model) that takes values in Euclidean space; in this work, we generalise the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas, which give exact random walk representations. The proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case. To illustrate the utility of these new isomorphism theorems, we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the Sabot–Tarrès magic formula for the limiting local time of the vertex- reinforced jump process. The second ingredient is a new Mermin–Wagner theorem for hyperbolic sigma models. This result is of intrinsic interest for the sigma models, and together with the aforementioned isomorphism theorems, implies our main theorem on the VRJP, namely, that it is recurrent in two dimensions for any translation invariant finite-range initial jump rates. We also use supersymmetric hyperbolic sigma models to study the arboreal gas. This is a model of unrooted random forests on a graph, where the probability of a forest F with jF j edges is multiplicatively weighted by a parameter βjF j > 0. In simple terms, it can be defined β to be Bernoulli bond percolation with parameter p = 1+β conditioned to be acyclic, or as the q ! 0 limit with p = βq of the random cluster model. It is known that on the complete graph KN with β = α=N there is a phase transition similar to that of the Erdos–Rényi˝ random graph: a giant tree percolates for α > 1 and all trees have bounded size for α < 1. In contrast to this, by exploiting an exact relationship with the hyperbolic sigma model, we show that the forest constraint is significant in two dimensions: trees do not percolate on Z2 for any finite β > 0. This result is again a consequence of our hyperbolic Mermin–Wagner theorem, and is used in conjunction with a version of the principle of dimensional reduction. To further illustrate our methods, we also give a spin-theoretic proof of the phase transition on the complete graph. Preface This thesis consists five chapters, which I have further grouped into two parts. The second part, entitled superprobability, consists of three papers. These are as follows: • The geometry of random walk isomorphism theorems, Ann. Inst. Henri Poincaré Probab. Stat., 57(1): 408-454, 2021, coauthored with Roland Bauerschmidt and Tyler Helmuth. • Random spanning forests and hyperbolic symmetry, Commun. Math. Phys. 381 1223–1261, 2021, coauthored with Roland Bauerschmidt, Nicholas Crawford, and Tyler Helmuth. • Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and re- currence of the two-dimensional vertex-reinforced jump process, Ann. Probab., 47(5):3375– 3396, 2019, coauthored with Roland Bauerschmidt and Tyler Helmuth. These three can be read in any order, but I would recommend reading “The geometry of random walk isomorphism theorems” before the other two. The first part, entitled superanalysis, contains the necessary background material on supermathematics that is required for the second part. It is an expanded version of the appendix to [8], and as such, it can be used either as an introduction or an appendix to the second part, being referred back to as needed. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared above and specified in the text. It is not substantially the same as any that I have submitted, or is being concurrently submitted, for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. I further state that no substantial part of my dissertation has already been submitted, or is being concurrently submitted, for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution. Acknowledgements Before we dive into mathematical matters, I thought I might take a brief moment to reflect upon my experience at Cambridge over the last five years, and to give my heart felt thanks to all those who have helped me along the way. And what a experience it has been! I have been so fortunate to have met so many wonderful people who have inspired me on both personal and professional levels. Without doubt, I owe my greatest thanks to my supervisor, Roland Bauerschmidt. Roland has been a continuous source of guidance, support, and inspiration, and I am truly grateful for the time that he has devoted towards helping me develop as a mathematician. I could not have asked for a better supervisor. Secondly, I would like to extend a very big thank you to both Tyler Helmuth and Nick Crawford for their enthusiastic collaboration which has ultimately lead to this thesis. These last few years have reinforced my belief that the real joy in mathematics comes from sharing it with others, and on that note, there are two communities to which I would like to express my thanks. The first, of course, is the Cambridge Centre for Analysis: Matthias, Erlend, Kweku, Maxime, Mo Dick, Nicolai, Fritz, Eardi, Lisa, Megan, (and Sam!), thank you all so much for such stimulating company, and for providing just the right amount distraction when necessary. I owe particular thanks to Tessa Blackman for her unending help and support in dealing with my administrative shortfalls. I would also like to thank Arieh Iserles and Perla Sousi for supervising me during my first year at the CCA. The second, is the ‘superprobability’ community (if I may call it that!): I would like to thank David Brydges, Gordon Slade, Tom Spencer, Persi Diaconis, Margarita Disertori, and particularly Christophe Sabot and Pierre Tarrès for sharing their thoughts and insights at various conferences and visits over the years. I would also like to thank Codina Cotar and Sergio Bacallado for taking the time to carefully examine this thesis, and for their helpful comments and suggestions which have surely improved the final product. Before attending Cambridge, I had the fortune of studying under both Geoff Vasil and Sheehan Olver at the University of Sydney, and I would like to thank them both for their significant role in preparing me for graduate school, and for their continued support and interest in my studies. A huge thank you to my siblings, Maddie and Simon, and to my friends, James, Michael, Andy, Jim, Kal, Will, and Camilo, for all the joy you have shared with me over the years. A very special thank you to Miruna for all of her love, support, and encouragement, and for helping me carry on when things were tough. Finally, I would like to thank Mr Ruthven for teaching me how to integrate by parts: despite my initial protests1, it has indeed proven quite useful. BFS–Dynkin SRW GFF Ray–Knight Reinforcement Curvature Hyperbolic BFS–Dynkin 2j2 VRJP H Hyperbolic Ray–Knight t ! 1 Gaussian Lorentz Integration ST Magic Boost L1 GFF Recoupling a t-marginal L1 Sabot–Tarrès Limit Isomorphism To Mum and Dad, of course. 1“Why can’t I just integrate it like normal?” Contents I SUPERANALYSIS 1 1 Superalgebra and Supergeometry 3 1.1 Supervector Spaces . 3 1.2 Superalgebras . 7 1.3 Superfunction algebras . 9 1.4 Further remarks on superspaces . 16 1.5 Supermodules and Supermatrices . 18 1.6 Superspins . 21 2 Supercalculus and Supersymmetries 25 2.1 Derivatives and Derivations . 25 2.2 Berezin Integration . 27 2.3 Supersymmetries and Lie Superalgebras . 30 2.4 Supersymmetric Localisation . 33 II SUPERPROBABILITY 37 3 The geometry of random walk isomorphism theorems 39 3.1 Introduction . 39 3.2 Isomorphism theorems for flat geometry . 42 3.3 Isomorphism theorems for hyperbolic geometry . 49 3.4 Isomorphism theorems for spherical geometry . 54 3.5 Isomorphism theorems for supersymmetric spin models . 58 3.6 Application to limiting local times: the Sabot–Tarrès limit . 66 3.7 Time changes and resolvent formulas . 67 3.8 Application to exponential decay of correlations in spin systems . 70 Appendices 73 3.A Introduction to supersymmetric integration . 73 3.B Further aspects of symmetries and supersymmetry . 78 3.C Sabot–Tarrès limit and the BFS–Dynkin Isomorphism Theorem . 86 4 Random spanning forests and hyperbolic symmetry 93 4.1 The arboreal gas and uniform forest model . 93 4.2 Hyperbolic sigma model representation . 98 4.3 Phase transition on the complete graph . 109 4.4 No percolation in two dimensions . 114 Appendices 119 4.A Percolation properties . 119 4.B Rooted spanning forests and the uniform spanning tree . 121 CONTENTS 5 Recurrence of the two-dimensional VRJP 125 5.1 Introduction and results . 125 5.2 Supersymmetry and horospherical coordinates . 131 5.3 Proof of Theorem 5.1.2 . 135 5.4 Proof of Theorem 5.1.5 . 136 Appendices 139 5.A Horospherical coordinates .