Stochastic localization + Stieltjes barrier = tight bound for log-Sobolev

Abstract

Logarithmic Sobolev inequalities are a powerful way to estimate the rate of convergence of Markov chains and to derive concentration inequalities on distributions. We prove that the log-Sobolev constant of any isotropic logconcave density in Rn with support of diameter D is Ω(1/D), resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of Ω(1/D2) by Kannan-Lovász-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size δ=Θ(1/√n) mixes in O*(n2D) proper steps from any starting point. This improves on the previous best bound of O*(n2D2) and is also asymptotically tight. The new bound leads to the following refined large deviation inequality for an L-Lipschitz function g over an isotropic logconcave density p: for any t>0, [complex formula not displayed] where ḡ is the median or mean of g for x∼ p; this improves on previous bounds by Paouris and by Guedon-Milman. Our main proof is based on stochastic localization together with a Stieltjes-type barrier function.