2018/11/28 by Eldan, Ronen · 4 citations
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1811.11530
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 isotropic measure μ on ℝn and every ε> 0, there exists a decomposition μ= ∫ μθd m(θ) such that H(μ) - 𝔼θ∼ m H(μθ) ≤ n ε and 𝔼θ∼ m Cov(μθ) \preceq Id/ε. As an application, we prove 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‖Sp )(p)/(1+p) for all p>0, where ‖J‖Sp denotes the Schatten-p norm of the interaction matrix. For the case p=2, this recovers the result of [Jain et al., 2018], but for an optimal choice of p it often allows to get almost dimension-free bounds.