RUI: Statistical Mechanics Models on Non-Amenable Graphs

The aim of this proposal is to study several random spatial models on various infinite graphs. Several related topics concerning self-avoiding walks, percolation, Ising and random cluster models, and stochastic dynamics of Ising models are proposed. For the self-avoiding walk, the proposer hopes to answer some fundamental questions of the model on non-amenable graphs. Such questions include: (1) What is the average end-to-end distance of an N-step self-avoiding walk? (2) What is the asymptotic behavior of the number of N-step self-avoiding walks starting from a given point? The topic in percolation concerns the critical behavior and critical exponents of the model on d-dimensional hyperbolic graphs, which are regular tessellations of the hyperbolic d-dimensional space. The main project of Ising models on non-amenable graphs concerns problems which are of fundamental importance in physics literature. They include: (1) How many extremal Gibbs states are there in different temperature region? What are they? (2) How many translation-invariant Gibbs states are there and what are they? Problems concerning stochastic dynamics of Ising models on various infinite graphs where spins evolve according to the usual Glauber dynamics are also proposed.

The proposed topics originate from and have a wide range of application in materials science, statistical physics, chemistry, biology and other fields of sciences. The study of the self-avoiding walk arose in chemical physics as a model for long polymer chains. It is the simplest mathematical model which characterizes the self-repulsion between monomers in a polymer chain, yet exhibits critical phenomena. The interest in this model is, however, much broader since it is closely related to other stochastic models such as the Brownian motion and the newly discovered stochastic Loewner evolution. Percolation is a probabilistic model of studying flow through a disordered system, such as particles flowing through the filter of a gas mask, or fluid seeping through the interstices of a random porous medium. Ising models capture the basic magnetic properties of materials. An important goal of this project is to understand how the geometry of the space affects the nature of macroscopic behavior of these models. The proposer will continue to make an effort to involve undergraduate students to undertake small pieces of the problems proposed here. The proposal, if funded, will also support the students to present their findings in undergraduate research/education conferences.