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

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