Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

摘要

Consider a statistical physical model on the d-regular infinite tree described by a set of interactions . Let be a sequence of finite graphs with vertex sets that locally converge to . From one can construct a sequence of corresponding models on the graphs . Let be the resulting Gibbs measures. Here we assume that converges to some limiting Gibbs measure on in the local weak sense, and study the consequences of this convergence for the specific entropies . We show that the limit supremum of is bounded above by the percolative entropy , a function of itself, and that actually converges to in case exhibits strong spatial mixing on . When it is known to exist, the limit of 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.