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

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