# Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees

## TL;DR

This paper investigates the behavior of Gibbs measures on sequences of finite graphs converging to an infinite regular tree, establishing bounds and convergence results for their entropies related to percolative entropy, especially under strong spatial mixing.

## Contribution

It introduces the concept of percolative entropy for Gibbs measures on infinite trees and links entropy convergence to spatial mixing properties, providing new insights into entropy limits in such models.

## Key findings

- The upper bound of normalized entropy is given by percolative entropy.
- Normalized entropy converges to percolative entropy under strong spatial mixing.
- Examples include high-temperature regimes where results hold.

## Abstract

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.