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November 2019 The structure of low-complexity Gibbs measures on product spaces
Tim Austin
Ann. Probab. 47(6): 4002-4023 (November 2019). DOI: 10.1214/19-AOP1352

Abstract

Let $K_{1},\ldots,K_{n}$ be bounded, complete, separable metric spaces. Let $\lambda_{i}$ be a Borel probability measure on $K_{i}$ for each $i$. Let $f:\prod_{i}K_{i}\longrightarrow \mathbb{R}$ be a bounded and continuous potential function, and let \begin{equation*}\mu (\mathrm{d}\boldsymbol{x})\ \propto \ \mathrm{e}^{f(\boldsymbol{x})}\lambda_{1}(\mathrm{d}x_{1})\cdots \lambda_{n}(\mathrm{d}x_{n})\end{equation*} be the associated Gibbs distribution.

At each point $\boldsymbol{{x}\in \prod_{i}K_{i}}$, one can define a ‘discrete gradient’ $\nabla f(\boldsymbol{x},\cdot )$ by comparing the values of $f$ at all points which differ from $\boldsymbol{{x}}$ in at most one coordinate. In case $\prod_{i}K_{i}=\{-1,1\}^{n}\subset \mathbb{R}^{n}$, the discrete gradient $\nabla f(\boldsymbol{x},\cdot )$ is naturally identified with a vector in $\mathbb{R}^{n}$.

This paper shows that a ‘low-complexity’ assumption on $\nabla f$ implies that $\mu $ can be approximated by a mixture of other measures, relatively few in number, and most of them close to product measures in the sense of optimal transport. This implies also an approximation to the partition function of $f$ in terms of product measures, along the lines of Chatterjee and Dembo’s theory of ‘nonlinear large deviations’.

An important precedent for this work is a result of Eldan in the case $\prod_{i}K_{i}=\{-1,1\}^{n}$. Eldan’s assumption is that the discrete gradients $\nabla f(\boldsymbol{x},\cdot )$ all lie in a subset of $\mathbb{R}^{n}$ that has small Gaussian width. His proof is based on the careful construction of a diffusion in $\mathbb{R}^{n}$ which starts at the origin and ends with the desired distribution on the subset $\{-1,1\}^{n}$. Here our assumption is a more naive covering-number bound on the set of gradients $\{\nabla f(\boldsymbol{x},\cdot ):\boldsymbol{x}\in \prod_{i}K_{i}\}$, and our proof relies only on basic inequalities of information theory. As a result, it is shorter, and applies to Gibbs measures on arbitrary product spaces.

Citation

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Tim Austin. "The structure of low-complexity Gibbs measures on product spaces." Ann. Probab. 47 (6) 4002 - 4023, November 2019. https://doi.org/10.1214/19-AOP1352

Information

Received: 1 December 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212176
MathSciNet: MR4038047
Digital Object Identifier: 10.1214/19-AOP1352

Subjects:
Primary: 60B99
Secondary: 60G99, 82B20, 94A17

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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