A Near-Linear Time Sampler for the Ising Model with External Field

TL;DR

This paper introduces a near-linear time algorithm for sampling from the Gibbs distribution of ferromagnetic Ising models with external fields on general graphs, leveraging spectral independence and Glauber dynamics.

Contribution

It presents a novel near-linear time sampler for the Ising model with external fields, extending previous methods to more general graph settings.

Findings

Algorithm runs in near-linear time.

Proves spectral independence in low-temperature regime.

Establishes rapid mixing of Glauber dynamics.

Abstract

We give a near-linear time sampler for the Gibbs distribution of the ferromagnetic Ising models with edge activities $\boldsymbol{\beta} > 1$ and external fields $\boldsymbol{\lambda}<1$ (or symmetrically, $\boldsymbol{\lambda}>1$) on general graphs with bounded or unbounded maximum degree. Our algorithm is based on the field dynamics given in [CFYZ21]. We prove the correctness and efficiency of our algorithm by establishing spectral independence of distribution of the random cluster model and the rapid mixing of Glauber dynamics on the random cluster model in a low-temperature regime, which may be of independent interest.