Sampling from the low temperature Potts model through a Markov chain on flows

TL;DR

This paper introduces a Markov chain approach on flows to efficiently sample from the low temperature Potts model, providing a rapid mixing algorithm with polynomial runtime bounds.

Contribution

It develops a novel Markov chain on flows for the Potts model, enabling efficient sampling at low temperatures, which was previously challenging.

Findings

Markov chain on flows is rapidly mixing at low temperatures

Provides a $oldsymbol{ ext{O}(m^2 ext{log}(moldsymbol{ ext{delta}}^{-1}))}$ sampling algorithm

Enables approximate sampling and partition function estimation for the Potts model

Abstract

In this paper we consider the algorithmic problem of sampling from the Potts model and computing its partition function at low temperatures. Instead of directly working with spin configurations, we consider the equivalent problem of sampling flows. We show, using path coupling, that a simple and natural Markov chain on the set of flows is rapidly mixing. As a result we find a $\delta$-approximate sampling algorithm for the Potts model at low enough temperatures, whose running time is bounded by $O(m^2\log(m\delta^{-1}))$ for graphs $G$ with $m$ edges.