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.

Highlights

  • The Ising model is perhaps the best known model in statistical physics, and it has been widely studied from an algorithmic perspective

  • A direct approach using Markov chain Monte Carlo (MCMC) on the spin configurations described above fails, as the spin model exhibits a phase transition for sufficiently large β

  • There is an equivalent formulation of the Ising model in terms of “even subgraphs” which does form the basis for a successful application of MCMC, as was shown by Jerrum and Sinclair [19]. (See Sections 2 and 3 for details of the various models referred to in this introduction.)

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Summary

Introduction

The Ising model is perhaps the best known model in statistical physics, and it has been widely studied from an algorithmic perspective. We already have a polynomial-time algorithm for estimating the partition function of the Ising model, it is natural to wonder about the mixing time of the Gibbs sampler for random cluster configurations, which makes single edge-flip moves with Metropolis rejection probabilities. For one thing, this dynamics may potentially mix faster than the standard dynamics for the even subgraphs model, and the same is true with even greater force for the closely related Swendsen-Wang algorithm. In the range 0 ≤ q ≤ 2 there is no known barrier to rapid mixing, and there is cause to be optimistic, in the range 1 < q < 2, in which the random cluster model is monotonic

Ising and Random Cluster model
Random Even Subgraphs
Lifting Canonical Paths
A Equivalence of the three views
B Congestion of the worm process

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