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

Loading next page...
 
/lp/springer_journal/gibbs-measures-over-locally-tree-like-graphs-and-percolative-entropy-YFk9O0rDzl
Publisher
Springer US
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off