Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics

We analyze the Poincaré and Log-Sobolev constants of logconcave densities in $\mathbb{R}^{n}$. For the Poincaré constant, we give an improved estimate of $O(\sqrt{n})$ for any isotropic logconcave density. For the Log-Sobolev constant, we prove a bound of $\Omega (1/D)$, where $D$ is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in $\mathbb{R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov\’aasz and Simonovits mixes in $O^{*}(n^{2}D)$ proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any $L$-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.