The structure of low-complexity Gibbs measures on product spaces
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Let K_1, ..., K_n be bounded, complete, separable metric spaces. Let f:∏_i K_i →R be a bounded and continuous potential function, and let μ ∝ e^f be the associated Gibbs distribution. At each point x∈∏_i K_i one can define a `discrete gradient' ∇_xf by comparing the values of f at all points which differ from x in at most one coordinate. In case ∏_i K_i = {-1,1}^n ⊂R^n, the discrete gradient ∇_xf is naturally identified with a vector in R^n. This paper shows that a `low-complexity' assumption on ∇ f implies that μ can be approximated by a mixture of other measures, relatively few in number, and most of them close in a natural transportation distance to product measures. 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 ∏_i K_i = {-1,1}^n. Eldan's assumption is that the discrete gradients ∇_x f 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 assumption is a more naive covering-number bound on the set of gradients {∇_xf:x∈∏_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 abitrary product spaces.
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