THE STRUCTURE OF LOW-COMPLEXITY GIBBS MEASURES ON PRODUCT SPACES

• 成果类型: Article
• 署名作者: Austin, Tim
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1352
• 发表日期: 2019
• 页码: 4002-4023
• 关键词: INEQUALITIES; divergence

摘要

Let K-1, ..., K-n, be bounded, complete, separable metric spaces. Let lambda(i) be a Borel probability measure on K-i for each i. Let f : Pi(i) K-i -> R be a bounded and continuous potential function, and let mu(dx) proportional to e(f(x))lambda(1)(dx(1)) ... lambda(n) (dx(n)) be the associated Gibbs distribution. At each point x is an element of Pi(i) K-i, one can define a 'discrete gradient' del f (x, .) by comparing the values of f at all points which differ from x in at most one coordinate. In case Pi(i) K-i = {-1, 1} subset of R-n, the discrete gradient del f (x, .) is naturally identified with a vector in R-n. This paper shows that a low-complexity' assumption on del 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 Pi K-i(i) = {1-1}(n), Eldan's assumption is that the discrete gradients del f (x, .) all lie in a subset of R-n that has small Gaussian width. His proof is based on the careful construction of a diffusion in R-n which starts at the origin and ends with the desired distribution on the subset {-1,1}(n). Here our as- sumption is a more naive covering-number bound on the set of gradients {del f (x, .) : x is an element of Pi K-i(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.