Kieffer-Pinsker type formulas for Gibbs measures on sofic groups

TL;DR

This paper establishes formulas for sofic entropy and pressure of Gibbs measures on sofic groups, under certain conditions, extending previous results and providing new insights into their independence and locality properties.

Contribution

It introduces sufficient conditions for expressing sofic entropy and pressure of Gibbs measures as limits and integrals, unifying and extending prior work on these topics.

Findings

Sofic entropy can be expressed as a limit of Shannon entropies under certain conditions.

Strong spatial mixing implies a formula for sofic pressure involving a random information function.

Conditions are provided for the independence of sofic pressure and entropy from the choice of sofic approximation.

Abstract

Given a countable sofic group $\Gamma$, a finite alphabet $A$, a subshift $X \subseteq A^\Gamma$, and a potential $\phi: X \to \mathbb{R}$, we give sufficient conditions on $X$ and $\phi$ for expressing, in the uniqueness regime, the sofic entropy of the associated Gibbs measure $\mu$ as the limit of the Shannon entropies of some suitable finite systems approximating $\Gamma \curvearrowright (X,\mu)$. Next, we prove that if $\mu$ satisfies strong spatial mixing, then the sofic pressure admits a formula in terms of the integral of a random information function with respect to any $\Gamma$-invariant Borel probability measure with nonnegative sofic entropy. As a consequence of our results, we provide sufficient conditions on $X$ and $\phi$ for having independence of the sofic approximation for sofic pressure and sofic entropy, and for having locality of pressure in some relevant families of systems, among other applications. These results complement and unify those of Marcus and Pavlov (2015), Alpeev (2017), and Austin and Podder (2018).