Approximation algorithms for the random-field Ising model

TL;DR

This paper develops polynomial-time approximation schemes for the random field Ising model on bounded-degree graphs with Gaussian external fields, overcoming worst-case hardness by focusing on average-case scenarios.

Contribution

It introduces algorithms that work with high probability for the random field Ising model when external fields are Gaussian with sufficiently large variance, addressing average-case complexity.

Findings

Existence of fully polynomial-time approximation schemes for the model.

Algorithms succeed with high probability over random fields.

Overcomes barriers posed by small external fields in certain regions.

Abstract

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $\Delta$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $\Delta$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $\Theta(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.