Path Coupling and Aggregate Path Coupling

TL;DR

This paper introduces aggregate path coupling, an extension of the classical path coupling method, to analyze rapid mixing in statistical mechanical models, including those with discontinuous phase transitions, using large deviations.

Contribution

It presents a novel extension called aggregate path coupling and applies it to prove rapid mixing in models with complex phase transition behaviors.

Findings

Proves rapid mixing for the generalized Curie-Weiss-Potts model.

Extends analysis to models with discontinuous phase transitions.

Provides a new tool for analyzing statistical mechanical models.

Abstract

In this survey paper, we describe and characterize an extension to the classical path coupling method applied statistical mechanical models, referred to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. The parameter region for rapid mixing for the generalized Curie-Weiss-Potts model is derived as a new application of the aggregate path coupling method.