LAUSR

Consider a statistical physical model on the d-regular infinite tree $$T_{d}$$Td described by a set of interactions $$\Phi $$Φ. Let $$\{G_{n}\}$${Gn} be a sequence of finite graphs with vertex sets $$V_n$$Vn that locally converge to $$T_{d}$$Td. From $$\Phi $$Φ one can construct a sequence of corresponding models on the graphs $$G_n$$Gn. Let $$\{\mu _n\}$${μn} be the resulting Gibbs measures. Here we assume that $$\{\mu _{n}\}$${μn} converges to some limiting Gibbs measure $$\mu $$μ on $$T_{d}$$Td in the local weak$$^*$$∗ sense, and study the consequences of this convergence for the specific entropies $$|V_n|^{-1}H(\mu _n)$$|Vn|-1H(μn). We show that the limit supremum of $$|V_n|^{-1}H(\mu _n)$$|Vn|-1H(μn) is bounded above by the percolative entropy$$H_{\textit{perc}}(\mu )$$Hperc(μ), a function of $$\mu $$μ itself, and that $$|V_n|^{-1}H(\mu _n)$$|Vn|-1H(μn) actually converges to $$H_{\textit{perc}}(\mu )$$Hperc(μ) in case $$\Phi $$Φ exhibits strong spatial mixing on $$T_d$$Td. When it is known to exist, the limit of $$|V_n|^{-1}H(\mu _n)$$|Vn|-1H(μn) is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.