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

Chazottes, J.R.; Collet, P.; Redig, Frank

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

Journal article (2017)

Authors

J.R. Chazottes Centre de Physique Théorique

P. Collet Centre de Physique Théorique

Frank Redig Applied Probability -

Research Group

Applied Probability () (TU Delft)

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

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Published Date

2017

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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.

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