Kieffer-Pinsker type formulas for Gibbs measures on sofic groups

Abstract

Given a countable sofic group $\Gamma$, a finite alphabet $A$, a subshift $X\subseteq A^{\Gamma }$, and a potential $\varphi :X\to \mathbb{R}$, we give sufficient conditions on $X$ and $\varphi$ 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 $\varphi$ 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).

arXiv comment36 pages

References in corpus (5)

• Entropy and the variational principle for actions of sofic groups
• Examples in the entropy theory of countable group actions
• Weak containment and Rokhlin entropy
• Random ordering formula for sofic and Rokhlin entropy of Gibbs measures
• Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees