## Annals of Probability

### The structure of low-complexity Gibbs measures on product spaces

Tim Austin

#### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 4002-4023.

Dates
Revised: January 2019
First available in Project Euclid: 2 December 2019

https://projecteuclid.org/euclid.aop/1575277346

Digital Object Identifier
doi:10.1214/19-AOP1352

Mathematical Reviews number (MathSciNet)
MR4038047

Zentralblatt MATH identifier
07212176

#### Citation

Austin, Tim. The structure of low-complexity Gibbs measures on product spaces. Ann. Probab. 47 (2019), no. 6, 4002--4023. doi:10.1214/19-AOP1352. https://projecteuclid.org/euclid.aop/1575277346

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