Stochastic Localization + Stieltjes Barrier = Tight Bound for Log-Sobolev

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 $R^n$ with support of diameter $D$ is $1/D$, resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of $1/D^2$ by Kannan-Lov\'asz-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size $\delta = (1/\sqrt{n})$ mixes in $O(n^2D)$ proper steps from any starting point. This improves on the previous best bound of $O(n^2D^2)$ and is also asymptotically tight. The new bound leads to the following refined large deviation inequality for any L-Lipschitz function g over an isotropic logconcave density p: for any t > 0, $$P(|g(x)- \bar{g}(x)| \ge c . L. t) \le \exp(-\frac{t^2}{t+\sqrt{n}})$$ where $\bar{g}$ is the median or mean of $g$ for $x \sim p$; this generalizes/improves on previous bounds by Paouris and by Guedon-Milman. The technique also bounds the "small ball" probability in terms of the Cheeger constant, and recovers the current best bound. Our main proof is based on stochastic localization together with a Stieltjes-type barrier function.