Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs

TL;DR

This paper investigates sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs, revealing slow mixing of Glauber dynamics and proposing an efficient alternative algorithm with an FPTAS for the partition function.

Contribution

It demonstrates the limitations of Glauber dynamics on these graphs and introduces a novel efficient sampling algorithm using polymer models and cluster expansion.

Findings

Glauber dynamics mixes exponentially slowly in certain parameters.

An efficient alternative sampling algorithm is proposed.

An FPTAS for the partition function is developed.

Abstract

The antiferromagnetic Ising model samples subsets of vertices of a graph with weight decaying exponentially in the number of edges induced. We study the problem of sampling from this model on the class of bipartite, regular graphs with good vertex expansion. We show that a natural sampler, namely the Glauber dynamics, mixes exponentially slowly in a wide range of parameters. On the other hand, we give an efficient alternative algorithm for sampling from the Ising model and an FPTAS for its partition function, using polymer models and the cluster expansion method.