On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems

J.R. Chazottes, P. Collet, F. Redig

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

We consider Gibbs measures on the configuration space (Formula presented.), where mostly (Formula presented.) and S is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon–McMillan–Breiman theorem, fluctuation bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems.

Original languageEnglish
Pages (from-to)504-546
Number of pages43
JournalJournal of Statistical Physics
Volume169
Issue number3
DOIs
Publication statusPublished - 2017

Keywords

  • d¯-Distance
  • Almost-sure central limit theorem
  • Dobrushin uniqueness
  • Empirical measure
  • Gaussian concentration bound
  • Kantorovich distance
  • Low-temperature Ising model
  • Moment concentration bound
  • Relative entropy

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