Random cluster dynamics for the Ising model is rapidly mixing

TL;DR

This paper proves that the Glauber dynamics for the random cluster model at q=2, including the Swendsen-Wang algorithm for the Ising model, mixes rapidly with polynomial time bounds.

Contribution

It establishes polynomial mixing time bounds for Glauber dynamics and the Swendsen-Wang algorithm at q=2, advancing understanding of their efficiency.

Findings

Glauber dynamics for the random cluster model at q=2 is rapidly mixing.

Swendsen-Wang algorithm for the ferromagnetic Ising model has polynomial mixing time.

Mixing time bounds are polynomial in the size of the underlying graph.

Abstract

We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at $q=2$ is bounded by a polynomial in the size of the underlying graph. As a consequence, the Swendsen-Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound.