Approximating Pairwise Correlations in the Ising Model

TL;DR

This paper develops a polynomial-time randomized algorithm to accurately estimate the covariance between spins in the ferromagnetic Ising model, addressing a key challenge in statistical physics and computational complexity.

Contribution

It introduces a fully polynomial randomized approximation scheme for covariance estimation in ferromagnetic Ising models, and proves hardness results for the antiferromagnetic case.

Findings

FPRAS for covariance in ferromagnetic Ising models

No FPRAS exists for antiferromagnetic models unless RP = #P

Determining the sign of covariance in antiferromagnetic models is #P-hard

Abstract

In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approximation to this covariance by repeatedly sampling from the relevant Gibbs distribution. However, we desire a multiplicative approximation, and it is not clear how to achieve this by sampling, given that the covariance can be exponentially small. Our main contribution is a fully polynomial time randomised approximation scheme (FPRAS) for the covariance. We also show that that the restriction to the ferromagnetic case is essential --- there is no FPRAS for multiplicatively estimating the covariance of an antiferromagnetic Ising model unless RP = #P. In fact, we show that even determining the sign of the covariance is #P-hard in the antiferromagnetic case.