Order and Disorder in Interacting Spin Systems and Random Networks

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This research project aims to develop new tools in probability theory that will enhance understanding of interacting particle systems and random networks. The motivating problems touch fundamental questions such as the effect of initial conditions (e.g., ordered or disordered, random or fixed) in Markov chains on their time of convergence to equilibrium, and structure formation (surface shape, subgraphs in networks, etc.) versus disorder under various conditions. Many of the questions under study remain open, despite extensive studies offering detailed heuristics and corroborating experiments; the goal of this project is to make progress on these via new methods of analysis, which are expected to find further applications in probability theory as well as in other areas. The first research project focuses on the interplay between a spin system, such as the Ising or Potts model, and a natural Markov chain modelling its evolution, for example Metropolis or heat-bath Glauber dynamics. The phase transition that both the dynamical and static models undergo has received much attention, yet various basic problems have so far been out of reach of rigorous analysis in all three (high, low, and critical) temperature regimes. This project studies several such problems, as well as newer ones suggested by recent advances, including understanding the effect of the initial configurations (random or deterministic, balanced or alternating, etc.) on the mixing time at high temperature and establishing a power-law for mixing at criticality in dimension 3 and higher. A second research direction focuses on random surface models, as well as their evolution, with ramifications for the Ising model and the roughening transition in crystals. The final research topic addresses random walks on the Erdos-Renyi random graph: the project aims to study the effect of the initial state on the mixing time of random walks, and in different regimes to establish typical structural properties, such as robustness of its features to noise, and atypical ones, such as whether the system organizes to asymmetric structures when conditioned on an atypical event (large deviations).