Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing.

We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}sqrt{log n}).$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincaru0027{e} constant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic (slicing) constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start. The proof is based on gradually transforming any logconcave density to one that has a significant Gaussian factor via a Martingale process. Extending this proof technique, we prove that the log-Sobolev constant of any isotropic logconcave density in ${bf R}^{n}$ with support of diameter $D$ is $Omega(1/D)$, resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of $Omega(1/D^{2})$ by Kannan-Lovu0027{a}sz-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size $delta=Theta(1/sqrt{n})$ mixes in $Oleft(n^{2}Dright)$ proper steps from emph{any }starting point. This improves on the previous best bound of $O(n^{2}D^{2})$ and is also asymptotically tight. The new bound leads to the following large deviation inequality for an $L$-Lipschitz function $g$ over an isotropic logconcave density $p$: for any $tu003e0$, [ Pr_{xsim p}left(left|g(x)-bar{g}right|geq Lcdot tright)leqexp(-frac{ccdot t^{2}}{t+sqrt{n}}) ] where $bar{g}$ is the median or mean of $g$ for $xsim p$; this generalizes and improves on previous bounds by Paouris and by Guedon-Milman. The technique also bounds the ``small ballu0027u0027 probability in terms of the Cheeger constant, and recovers the current best bound.