Taming correlations through entropy-efficient measure decompositions with applications to mean-field approximation

The analysis of various models in statistical physics relies on the existence of decompositions of measures into mixtures of product-like components, where the goal is to attain a decomposition into measures whose entropy is close to that of the original measure, yet with small correlations between coordinates. We prove a related general result: For every measure μ on R n and every ε > 0 , there exists a decomposition μ = ∫ μ θ d m ( θ ) such that H ( μ ) - E θ ∼ m H ( μ θ ) ≤ Tr ( Cov ( μ ) ) ε and E θ ∼ m Cov ( μ θ ) ⪯ Id / ε . As an application, we derive a general bound for the mean-field approximation of Ising and Potts models, which is in a sense dimension free, in both continuous and discrete settings. In particular, for an Ising model on { ± 1 } n or on [ - 1 , 1 ] n , we show that the deficit between the mean-field approximation and the free energy is at most C 1 + p p n ‖ J ‖ S p p 1 + p for all p > 0 , where ‖ J ‖ S p denotes the Schatten- p norm of the interaction matrix. For the case p = 2 , this recovers the result of Jain et al. (Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective. arXiv:1808.07226 , 2018 ), but for an optimal choice of p it often allows to get almost dimension-free bounds.