Mixing times for the Swapping Algorithm on the Blume-Emery-Griffiths Model

TL;DR

This paper studies the mixing times of the Swapping Algorithm, a parallel MCMC method, on the Blume-Emery-Griffiths model, revealing rapid mixing during second order phase transitions and slow mixing during first order transitions.

Contribution

It provides the first rigorous analysis linking phase transition types to the efficiency of the Swapping Algorithm in the mean-field BEG model.

Findings

Rapid mixing during second order phase transitions.

Slow mixing during first order phase transitions.

Supports conjecture by Bhatnagar and Randall.

Abstract

We analyze the so called Swapping Algorithm, a parallel version of the well-known Metropolis-Hastings algorithm, on the mean-field version of the Blume-Emery-Griffiths model in statistical mechanics. This model has two parameters and depending on their choice, the model exhibits either a first, or a second order phase transition. In agreement with a conjecture by Bhatnagar and Randall we find that the Swapping Algorithm mixes rapidly in presence of a second order phase transition, while becoming slow when the phase transition is first order.