Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields

TL;DR

This paper establishes polynomial and near-linear mixing time bounds for Swendsen-Wang and Glauber dynamics in the ferromagnetic Ising model with external fields, extending previous results and simplifying proofs.

Contribution

It introduces a new unified model linking subgraph and random cluster models, enabling improved mixing time bounds for dynamics with external fields.

Findings

Polynomial mixing time bounds for Swendsen-Wang and Glauber dynamics.

Near linear mixing time for bounded degree graphs with bounded external fields.

Simplified proofs extending previous no-field results.

Abstract

We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show: (1) polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018); (2) near linear mixing time for the two dynamics above if the maximum degree is bounded and all fields are (consistent and) bounded away from $1$.