# Entropy inequalities and exponential decay of correlations for unique Gibbs measures on trees

## TL;DR

This paper demonstrates that entropy inequalities and exponential decay of correlations, previously established for IID processes on trees, also hold for Gibbs measures, confirming exponential decay of correlations in this broader setting.

## Contribution

It extends existing entropy inequalities and decay results from IID processes to Gibbs measures on trees, providing a unified framework for correlation decay analysis.

## Key findings

- Mutual information decays exponentially in Gibbs measures on trees
- Entropy inequalities translate from IID to Gibbs processes
- Correlations decay exponentially for unique Gibbs measures

## Abstract

In a recent paper by A. Backhausz, B. Gerencs\'er and V. Harangi, it was shown that factors of independent identically distributed random processes (IID) on trees obey certain geometry-driven inequalities. In particular, the mutual information shared between two vertices decays exponentially, and there is an explicit bound for this decay. In this note we show that all of these inequalities could be verbatim translated to the setting of factors of processes driven by unique Gibbs measures. As a consequence, we show that correlations decay exponentially for unique Gibbs measures on trees.