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- W3100015600 abstract "Consider a statistical physical model on the d-regular infinite tree $$T_{d}$$ described by a set of interactions $$Phi $$ . Let $${G_{n}}$$ be a sequence of finite graphs with vertex sets $$V_n$$ that locally converge to $$T_{d}$$ . From $$Phi $$ one can construct a sequence of corresponding models on the graphs $$G_n$$ . Let $${mu _n}$$ be the resulting Gibbs measures. Here we assume that $${mu _{n}}$$ converges to some limiting Gibbs measure $$mu $$ on $$T_{d}$$ in the local weak $$^*$$ sense, and study the consequences of this convergence for the specific entropies $$|V_n|^{-1}H(mu _n)$$ . We show that the limit supremum of $$|V_n|^{-1}H(mu _n)$$ is bounded above by the percolative entropy $$H_{textit{perc}}(mu )$$ , a function of $$mu $$ itself, and that $$|V_n|^{-1}H(mu _n)$$ actually converges to $$H_{textit{perc}}(mu )$$ in case $$Phi $$ exhibits strong spatial mixing on $$T_d$$ . When it is known to exist, the limit of $$|V_n|^{-1}H(mu _n)$$ 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." @default.
- W3100015600 created "2020-11-23" @default.
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- W3100015600 date "2018-01-25" @default.
- W3100015600 modified "2023-09-27" @default.
- W3100015600 title "Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees" @default.
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- W3100015600 doi "https://doi.org/10.1007/s10955-018-1959-3" @default.
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