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

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Abstract

We consider Gibbs measures on the configuration space \(S^{{\mathbb {Z}}^d}\), where mostly \(d\ge 2\) 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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Authors and Affiliations

  1. Centre de Physique Théorique, CNRS UMR 7644, 91128, Palaiseau Cedex, France

    J.-R. Chazottes & P. Collet

  2. Delft Institute of Applied Mathematics, Technische Universiteit Delft, Delft, The Netherlands

    F. Redig

Authors
  1. J.-R. Chazottes
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  2. P. Collet
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  3. F. Redig
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Correspondence to J.-R. Chazottes.

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Chazottes, JR., Collet, P. & Redig, F. On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems. J Stat Phys 169, 504–546 (2017). https://doi.org/10.1007/s10955-017-1884-x

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  • DOI: https://doi.org/10.1007/s10955-017-1884-x

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