Variational principle for weakly dependent random fields

TL;DR

This paper introduces a new simplified framework based on max-entropy to analyze the minimizers of free energy in weakly dependent random fields, extending to various models like the loop O(n) and Ising models.

Contribution

It presents a novel variational principle framework for weakly dependent random fields using max-entropy, applicable to multiple statistical physics models.

Findings

Derived the variational principle for the loop O(n) model and Ising model in random environments.

Extended the framework to models with summable interactions and the random-cluster model.

Provided a unified approach to study free energy minimizers in weakly dependent systems.

Abstract

Using an alternative notion of entropy introduced by Datta, the max-entropy, we present a new simplified framework to study the minimizers of the specific free energy for random fields which are weakly dependent in the sense of Lewis, Pfister, and Sullivan. The framework is then applied to derive the variational principle for the loop $O(n)$ model and the Ising model in a random percolation environment in the nonmagnetic phase, and we explain how to extend the variational principle to similar models. To demonstrate the generality of the framework, we indicate how to naturally fit into it the variational principle for models with an absolutely summable interaction potential, and for the random-cluster model.