Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

TL;DR

This paper analyzes how the mixing times of Glauber dynamics in the mean-field Blume-Capel model relate to its phase transition structure, using an extended path coupling method to identify rapid mixing regions.

Contribution

It introduces a novel extension of the path coupling method to prove rapid mixing without contraction, applied to the complex phase transitions of the Blume-Capel model.

Findings

Identifies the interface between slow and rapid mixing regions.

Proves rapid mixing in the absence of contraction between neighboring states.

Connects phase transition types to mixing time behavior.

Abstract

In this paper we investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.