An exactly solvable ansatz for statistical mechanics models

TL;DR

This paper introduces a family of exactly solvable probability distributions for 2D statistical mechanics models, enabling efficient computation of free energies beyond mean-field approximations.

Contribution

It presents a novel approach to approximate partition functions using a nontrivial solution to the marginal problem with linear-time free energy computation.

Findings

Free energies can be computed in linear time.

The distributions are outside the mean-field framework.

The method ensures a consistent global probability distribution.

Abstract

We propose a family of "exactly solvable" probability distributions to approximate partition functions of two-dimensional statistical mechanics models. While these distributions lie strictly outside the mean-field framework, their free energies can be computed in a time that scales linearly with the system size. This construction is based on a simple but nontrivial solution to the marginal problem. We formulate two non-linear constraints on the set of locally consistent marginal probabilities that simultaneously (i) ensure the existence of a consistent global probability distribution and (ii) lead to an exact expression for the maximum global entropy.