Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems
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Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems J.-R. Chazottes1
· J. Moles1,3 · F. Redig2 · E. Ugalde3
Received: 9 June 2020 / Accepted: 12 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration d space S Z where d ≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound. Keywords Concentration inequalities · Relative entropy · Blowing-up property · Equilibrium states · Large deviations · Hamming distance
1 Introduction and Main Result The phenomenon we are interested in, which goes under the name of “concentration inequalities”, is that if a function of many “weakly dependent” random variables does not depend too much on any of them, then it is concentrated around its expected value. A key feature of this phenomenon is that it is non-asymptotic, in contrast with the usual limit theorems where the number of random variables has to tend to infinity. Recall that the three main types of classical limit theorems are the law of large numbers, the central limit theorem, and large deviations. Another key feature of concentration inequalities is that they allow to deal with functions of random variables defined in an arbitrary way, provided they are “smooth enough”, in contrast with classical limit theorems which deal with sums of random variables. Concentration inequalities made a paradigm shift in probability and statistics, but also in discrete mathematics, in geometry and in functional analysis, see e.g. [2,9,17,25]. d In this paper, we consider Gibbs measures on the configuration space = S Z where S is a finite set and d ≥ 1. Postponing precise definitions till next section, a probability measure
Communicated by Eric A. Carlen.
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J.-R. Chazottes [email protected]
1
Centre de Physique Théorique, CNRS, Institut Polytechnique de Paris, Palaiseau, France
2
Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands
3
Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico
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μ satisfies a Gaussian concentration bound if there exists a constant D > 0 such that, for any local function F : → R, 2 (1) e F− F dμ dμ ≤ e D x∈ F δx (F) where F is the (finite) set of sites x ∈ Zd such that δx (F) = 0, and δx (F) is the largest value of |F(ω) − F(ω )| taken over the configurations ω and ω differing only at site x. Note that D does not depend on F, and in particular not on F . By a standard argument (recalled later on), (1) implies a control on the fluctuations of F around F dμ: for all u > 0, we have u2 μ ω ∈ : F(ω) ≥ F dμ + u ≤ exp − . (2) 4D x∈ F δx (F)2 For instance, if S = {−1, +1}, F can be x∈
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