Random ordering formula for sofic and Rokhlin entropy of Gibbs measures

TL;DR

This paper derives an explicit formula for sofic and Rokhlin entropy of Gibbs measure actions, providing new examples and criteria for measure uniqueness, notably for the supercritical Ising model.

Contribution

It introduces a novel explicit formula for sofic and Rokhlin entropy applicable to certain Gibbs measures, expanding understanding beyond Bernoulli actions.

Findings

Formula applies to supercritical Ising model

Sofic entropy can be independent of sofic approximations

Criterion for Gibbs measure uniqueness established

Abstract

We prove the explicit formula for sofic and Rokhlin entropy of actions arising from some class of Gibbs measures. It provides a new set of examples with sofic entropy independent of sofic approximations. It is particularilly interresting, since in non-amenable case Rokhlin entropy was computed only in case of Bernoulli actions and for some examples with zero Rokhlin entropy. As an example we show that our formula holds for the supercritical Ising model. We also establish a criterion for uniqueness of Gibbs measure by means of f-invariant.