The structure of low-complexity Gibbs measures on product spaces

Let K₁,…,Kₙ be bounded, complete, separable metric spaces. Let λᵢ be a Borel probability measure on Kᵢ for each i. Let f:∏ᵢKᵢ R be a bounded and continuous potential function, and let {equation*}μ (dx)\ ∝ \ eᶠ⁽ˣ⁾λ₁(dx₁)⋯ λₙ(dxₙ){equation*} be the associated Gibbs distribution. At each point x∈ ∏ᵢKᵢ, one can define a ‘discrete gradient’ ∇ f(x,· ) by comparing the values of f at all points which differ from x in at most one coordinate. In case ∏ᵢKᵢ=\-1,1\ⁿ⊂ Rⁿ, the discrete gradient ∇ f(x,· ) is naturally identified with a vector in Rⁿ. 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 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 ∏ᵢKᵢ=\-1,1\ⁿ. Eldan’s assumption is that the discrete gradients ∇ f(x,· ) all lie in a subset of Rⁿ that has small Gaussian width. His proof is based on the careful construction of a diffusion in Rⁿ 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 \∇ f(x,· ):x∈ ∏ᵢKᵢ\, 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.