Convergence Rates of Random Discrete Model Curves Approaching SLE Curves in the Scaling Limit by Larissa Richards A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto © Copyright 2021 by Larissa Richards Abstract Convergence Rates of Random Discrete Model Curves Approaching SLE Curves in the Scaling Limit Larissa Richards Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2021 Recently, A. Kempannien and S. Smirnov provided a framework for showing convergence of discrete model interfaces to the corresponding SLE curves. They show that given a uniform bound on specific crossing probabilities one can deduce that the interface has subsequential scaling limits that can be described almost surely by Löwner evolutions. This leads to the natural question to investigate the rate of convergence to the corresponding SLE curves. F. Johansson Viklund has developed a framework for obtaining a power-law convergence rate to an SLE curve from a power-law convergence rate for the driving function provided some additional geometric information along with an estimate on the growth of the derivative of the SLE map. This framework is applied to the case of the loop-erased random walk. In this thesis, we show that if your interface satisfies the uniform annulus condition proposed by Kempannien and Smirnov then one can deduce the geometric information required to apply Viklund’s framework. As an application, we apply the framework to the critical percolation interface. The first step in this direction for critical percolation was done by I. Binder, L. Chayes and H.K. Lei where they proved that the convergence rate of the Cardy-Smirnov observable is polynomial in the size of the lattice. It relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic approximations as well as a Percolation construction of the Harris systems. Further, we exploit the toolbox developed by D. Chelkak for discrete complex analysis on isoradial graphs to show polynomial rate of convergence for the discrete martingale observables for harmonic explorer and the FK Ising model to the corresponding continuum objects. Then, we apply the framework developed above to gain a polynomial convergence rate for the corresponding curves. ii Contents 1 Introduction and Background 1 1.1 The Space of Curves ....................................... 3 1.2 Introduction to the SLE-Quest via Loop-Erased Random Walk ............... 6 1.2.1 Uniform Spanning Tree ................................. 9 1.3 Loewner Evolution and SLE ................................... 11 1.3.1 Loewner Evolution .................................... 12 1.3.2 Schramm Loewner Evolution .............................. 15 1.3.3 Carathéodory convergence ................................ 19 1.4 Approaches to Proving Convergence to SLE and Main Results in the Literature ...... 20 1.5 Main Theorem .......................................... 27 1.5.1 Outline of Thesis ..................................... 32 2 Convergence of Driving Term 36 2.1 Key Estimate ........................................... 37 2.2 Proof of Theorem 2.0.1 ...................................... 40 3 From Convergence of Driving Terms to Convergence of Paths 48 3.1 Main Estimate for the Tip Structure Modulus ......................... 49 3.2 Results from Aizenman and Burchard ............................. 50 3.3 Proof of Main Estimate for Tip Structure Modulus ...................... 52 3.4 Convergence of Discrete Domains ................................ 55 3.5 Proof of Main Theorem ..................................... 57 4 Applications 60 4.1 Percolation ............................................ 60 4.2 Approximate harmonicity .................................... 68 4.3 Harmonic Explorer ........................................ 72 4.3.1 Setup and main results ................................. 72 4.3.2 Definitions and preliminaries .............................. 72 4.3.3 Harmonic explorer satisfies the KS condition ..................... 75 4.3.4 Harmonic explorer observable convergence ....................... 76 4.3.5 Harmonic explorer convergence rate for paths ..................... 77 4.4 Ising Model ............................................ 78 4.4.1 Setup and main results ................................. 78 4.4.2 Review of the Model ................................... 81 4.4.3 FK Ising satisfies KS Condition ............................. 82 4.4.4 FK Ising observable convergence ............................ 84 4.4.5 FK Ising convergence rate for paths .......................... 86 Bibliography 88 iii For Dad and Delmie. iv Acknowledgements Many people have contributed to this thesis. I wish to thank my advisor Ilia Binder for his advice, encouragement, and support throughout these years. I have learned so much from you and thoroughly enjoyed our many discussions. Also, thank you for sending me to all the summer schools, workshops, and conferences. I have met many brilliant people thanks to you. I am also greatly indebted to Michael Kozdron for introducing me to this subject, opening up my mind to mathematics as a career for me, and constant encouragement and support over all the years. Thank you for sharing your passion and excitement towards this subject. Thank you to Dmitry Chelkak for his discussions with us on the Ising model and Fredrick Viklund for spending time listening to me talk on this work and insightful questions. Thanks to Jemima Merisca, Sonja Injac, Ashley Armogan, Patrina Seepersaud and the main office. Things would not have worked without you. My graduate studies at University of Toronto were partially funded by NSERC. During my time as a graduate student, I had the privilege of attending the Masters class in statistical physics at the University of Geneva which provided me not only the opportunity to learn but many friendships as well. I most gratefully acknowledge these institutions. I would also like to give a special thanks to all my family and friends all over the world. I owe a special thank you to Grace, Rachel, Jackie, and Sydney for your continuous, invaluable, and essential support along the way. Sheldon, thank you for encouraging me towards academia and always inspiring me. Mom, Miranda, Chris and John thank you for being the most amazing family, always believing in me and sharing this experience with me from the very beginning. Elif, thank you for sharing this experience with us this past year, opening up your home, and being an inspiration as a successful academic and mother. Özgür, you always believe in me, support me, and love me. Thank you for sharing all the laughs and experiences with Delmie this past year. Delmie, thank you for writing this thesis with me, being the best possible distraction, and bringing so much joy in our life. This past year has made me appreciate more all the friendship, love and support that I have in my life. Thank you to all my family and friends for always contributing to my happiness and encouraging me in everything I attempt. I could not have done this without you. v Chapter 1 Introduction and Background Introduced by Oded Schramm [51], SLE is a one-parameter family of conformally invariant random curves in simply connected planar domains. It is conjectured that these curves are the scaling limits of the various interfaces in critical lattice models. These two-dimensional lattice models describe a variety of physical phenomenon including percolation, the Ising model, loop-erased random walk and the Potts model. Physicists had predicted that conformal invariance would play a key role in understanding the universal behaviour of these two-dimensional systems. Universality essentially means that the global properties of the physical system do not depend on the detailed local description of the model such as the underlying lattice. Recently, there has been a number of remarkable breakthroughs in the study of these models. In fact, Fields Medals were awarded to W. Werner in 2006 and to S. Smirnov in 2010 for their contributions in the study of SLE and critical lattice models. For several two dimensional lattice models at criticality, it has been shown that the discrete interfaces converge in the scaling limit to SLE curves [38, 40, 52, 54, 13, 56, 9]. The proofs of these all begin in the same manner, that is, by describing the scaling limit of some observable related to the interface. The limit is constructed from the interface itself through conformal invariance. Generally, the difficulty in the proof arises in how to deduce the strong convergence of interfaces from some weaker notions resulting in a need to solve some specific technical estimates. The goal of this thesis is to study the rateofthe above-mentioned convergence. In particular, we obtain a power-law convergence rate to an SLE curve from a power-law convergence rate for a martingale observable under suitable conditions on the discrete curves. The Loewner equation is a partial differential equation that produces a Loewner chain which isa family of conformal maps from a reference domain onto a continuously decreasing sequence of simply 1 Chapter 1. Introduction and Background 2 connected domains. A real-valued function called the driving term controls the Loewner evolution. If the driving term satisfies a smoothness assumption, then the Loewner equation generates agrowing continuous curve. Conversely, given a suitable curve, one can define the associated conformal maps