Approximations for finite spin systems and occupancy processes Liam Hodgkinson B.Sc. (Hons) A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2019 School of Mathematics and Physics Abstract Treating a complex network as a large collection of microscopic interacting entities has become a standard modelling paradigm to account for the effects of heterogeneity in real-world systems. In the probabilistic setting, these microscale models are often formulated as binary interacting particle systems: Markov processes on a collection of sites, where each site has two states (occupied and unoccupied). Of these, models comprised of distinct entities that transition individually form the broad class of finite spin systems in continuous time, and occupancy processes in discrete time. This class includes many general and highly detailed models of use in a wide variety of fields, including the stochastic patch occupancy models in ecology, network models and spreading processes in epidemiology, Ising-Kac models in physics, and dynamic random graph models in social and computer science. Unfortunately, the increased precision realised by this microscale approach often comes at the expense of tractability. In cases where these models become sufficiently complex to be meaningful, classical analysis techniques are rendered ineffective, and simulation and Monte Carlo methods become nontrivial. This is true especially for long-range processes, where virtually all of the existing techniques for studying interacting particle systems break down. To help meet these challenges, this thesis develops and analyses three types of practical approximations for finite spin systems and occupancy processes. They prove to be effective proxies for the macroscopic behaviour of many long-range systems, especially those of ‘mean-field type’, as demonstrated through the development of explicit, non-asymptotic error bounds. The first of these approximations is deterministic, estimating the evolution of the probability of occupancy at each site. This is useful for identifying and establishing trends, especially in conjunction with the existing literature in dynamical systems. Error bounds and concentration inequalities are established using the method of bounded differences. The second is a system of Gaussian processes as a distributional approximation, also useful for estimating and visualising quasi-stationary distributions. The discrepancy in the approximation is bounded under the Wasserstein metric using the celebrated method of Stein. Together with tightness estimates, this results in a central limit theorem for certain normalised sequences of models. Finally, inspired by the tau-leaping algorithm from chemical kinetics, Euler and midpoint approximation methods are considered to facilitate rapid simulation of finite spin systems. Strong error bounds and exact error asymptotics are obtained with the additional aid of entropy methods. Using these approximations, predicting and classifying the behaviour of a significant number of practical microscale population models becomes tractable, as demonstrated in a number of noteworthy applications. Some discussion of consequences for the identification of universality classes, and for general practitioners of population models, is also provided. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. Publications included in this thesis No publications included Submitted manuscripts included in this thesis 1. [1] Liam Hodgkinson, Ross McVinish, and Philip K. Pollett, Normal approximations for discrete-time occupancy processes, under revision for Stochastic Processes and their Applica- tions on 28th August 2019. 2. [2] Ross McVinish, Liam Hodgkinson, Fast approximate simulation of finite long-range spin systems, under revision for Annals of Applied Probability on 9th October 2019. Other publications during candidature Submitted manuscripts 1. [3] Liam Hodgkinson, Robert Salomone, and Fred Roosta, Implicit Langevin Algorithms for Sampling From Log-concave Densities, submitted to Journal of Machine Learning Research on 15th April 2019. Articles 1. [4] Liam Hodgkinson, Central limit theorems for dynamic random graph models, Gazette of the Australian Mathematical Society, 46(1):43–45, 2019. Contributions by others to the thesis Ross McVinish made significant contributions to Chapter 5, including conception, initial theoretical derivations for a special case, and components of the proof of Theorem 5.4. Statement of parts of the thesis submitted to qualify for the award of another degree No works submitted towards another degree have been included in this thesis. Research involving human or animal subjects No animal or human subjects were involved in this research. Acknowledgments First and foremost, I express my sincerest gratitude to my advisors, Philip Pollett and Ross McVinish, for their support and supervision throughout this project. I am grateful for their contributions of time and ideas; their academic, writing, and life advice which proved instrumental throughout this project; and their positivity and encouragement which kept my Ph.D. experience enjoyable and kept me going. Their infectious enthusiasm, and their support of my exploration of other topics, kindled my passion for mathematics and served to inform my research in significant ways. I also thank Phil and the Australian Centre of Excellence in Mathematical and Statistical Frontiers (ACEMS) for their financial support during my candidature, and facilitating my interactions with probabilists and statisticians across Australia. During my candidature, I was fortunate to spend cumulatively two months interacting with staff and students at the University of Melbourne; I would like to thank Andrew Barbour, Nathan Ross, Aihua Xia, Jan de Gier, and James McCaw for their discussions and advice during my stay. My discussions with Andrew and Nathan in particular proved to be indispensable for the completion of this project. I would also like to thank Jan and Michael Wheeler for inviting me to the MATRIX Program on Non-Equilibrium Systems and Special Functions in 2018, where I was able to interact with many other researchers at the forefront of the (mathematical) physical approach to interacting particle systems. I would like to thank the other academic staff at the School of Mathematics and Physics at the University of Queensland, including Fred Roosta, Thomas Taimre, Cecilia Gonzalez-Tokman, and Dirk Kroese, with whom I have had many valuable conversations that supported and inspired some of this work. I feel fortunate to have been part of many of the UQ Maths reading groups, which were a great source of enthusiasm and exposure to other mathematical techniques, some of which played a significant role in this project. I am also grateful to Murray Kane for providing invaluable support in administrative matters. This dissertation would not have come to fruition without the ongoing support of friends and family. In particular, I want to thank Timothy Buttsworth, who has been a wonderful friend and companion throughout my academic journey; and Robert Salomone, a great mentor and friend, and a source of support throughout this project. I always greatly enjoyed, and still treasure, my discussions with them both. I thank Robin Pearce for his advice on putting this thesis together, and Sarat Moka, Morgan Grant, Marielle Ong, Seamus Albion, Lachlan Burton, Joe Feng, Leslie Cheung, and Chris van der Heide for their camaraderie, and for making these last four years fun. Most of all, I thank my loving parents, Kim and Brent, for their immense support over all these years, in every facet of my life. They supported me at every step, even during the most challenging periods of my candidature, and I am grateful for their patience in doing so. Their advice and encouragement has been an unwavering source of comfort and stability in an ocean of uncertainty,