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

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees Consider a statistical physical model on the d-regular infinite tree $$T_{d}$$ T d described by a set of interactions $$\Phi$$ Φ . Let $$\{G_{n}\}$$ { G n } be a sequence of finite graphs with vertex sets $$V_n$$ V n that locally converge to $$T_{d}$$ T d . From $$\Phi$$ Φ one can construct a sequence of corresponding models on the graphs $$G_n$$ G n . Let $$\{\mu _n\}$$ { μ n } be the resulting Gibbs measures. Here we assume that $$\{\mu _{n}\}$$ { μ n } converges to some limiting Gibbs measure $$\mu$$ μ on $$T_{d}$$ T d in the local weak $$^*$$ ∗ sense, and study the consequences of this convergence for the specific entropies $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) . We show that the limit supremum of $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) is bounded above by the percolative entropy $$H_{\textit{perc}}(\mu )$$ H perc ( μ ) , a function of $$\mu$$ μ itself, and that $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) actually converges to $$H_{\textit{perc}}(\mu )$$ H perc ( μ ) in case $$\Phi$$ Φ exhibits strong spatial mixing on $$T_d$$ T d . When it is known to exist, the limit of $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Statistical Physics Springer Journals

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

, Volume 170 (5) – Jan 25, 2018
20 pages

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Physics; Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics
ISSN
0022-4715
eISSN
1572-9613
D.O.I.
10.1007/s10955-018-1959-3
Publisher site
See Article on Publisher Site

### Abstract

Consider a statistical physical model on the d-regular infinite tree $$T_{d}$$ T d described by a set of interactions $$\Phi$$ Φ . Let $$\{G_{n}\}$$ { G n } be a sequence of finite graphs with vertex sets $$V_n$$ V n that locally converge to $$T_{d}$$ T d . From $$\Phi$$ Φ one can construct a sequence of corresponding models on the graphs $$G_n$$ G n . Let $$\{\mu _n\}$$ { μ n } be the resulting Gibbs measures. Here we assume that $$\{\mu _{n}\}$$ { μ n } converges to some limiting Gibbs measure $$\mu$$ μ on $$T_{d}$$ T d in the local weak $$^*$$ ∗ sense, and study the consequences of this convergence for the specific entropies $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) . We show that the limit supremum of $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) is bounded above by the percolative entropy $$H_{\textit{perc}}(\mu )$$ H perc ( μ ) , a function of $$\mu$$ μ itself, and that $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ n ) actually converges to $$H_{\textit{perc}}(\mu )$$ H perc ( μ ) in case $$\Phi$$ Φ exhibits strong spatial mixing on $$T_d$$ T d . When it is known to exist, the limit of $$|V_n|^{-1}H(\mu _n)$$ | V n | - 1 H ( μ 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.

### Journal

Journal of Statistical PhysicsSpringer Journals

Published: Jan 25, 2018

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