Limit Shapes in Probability and Combinatorics

Brown University, Providence RI

Investigators

Abstract

The main goal of statistical mechanics, a field at the interface of mathematical probability and theoretical physics, is to understand the collective behavior of systems consisting of many interacting identical particles. One of the most interesting and important types of behavior is when an external force, such as an imposed boundary condition or other constraint, results in a non-homogeneity in the resulting system of particles. This non-homogeneity can result in abrupt changes in structure, called spatial phase transitions, where the system breaks into pieces with very different local behaviors. For example under a pressure gradient, water can have both solid and liquid phases in the same container. It is a challenging mathematical problem to understand these phases and their common boundaries. The project will investigate some of the mathematics involved in describing these kinds of phenomena at a generic level, with a goal to characterize the macroscopic shapes of the systems and their internal interfaces based on an understanding of their microscopic interactions. One of the main outcomes of this project, beyond the discovery of mathematical laws describing the complex behavior of condensed matter, will be the training of Ph.D. students in the mathematical sciences who will profit from working on cutting-edge topics in probability theory and combinatorics. The research project will study mathematical models of such limiting-shape behaviors, in several different settings, notably in planar configurational models such as "square ice" and related integrable statistical mechanical models. Bethe Ansatz techniques, although notoriously difficult in general, can be effectively applied in certain limiting situations to obtain mathematically rigorous limit shape theorems, as this project plans to show.