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

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.