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 $O(n^{1\mathrm{/}4})$, improving on the previous best bound of $O(n^{1\mathrm{/}3}\sqrt{\log n})\mathrm{.}$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincar'{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 ${\mathbf{R}}^n$ converges in $O^{\ast }(n^{2.5})$ 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 ${\mathbf{R}}^n$ with support of diameter $D$ is $\mathrm{\Omega }(1\mathrm{/}D)$, resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of $\mathrm{\Omega }(1\mathrm{/}{D}^2)$ by Kannan-Lov'{a}sz-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size $\delta =\mathrm{\Theta }(1\mathrm{/}\sqrt{n})$ mixes in $O\left(n^{2}D\right)$ 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 $t>0$, $$P{r}_{x\sim p}(\mid g(x)-\bar{g}\mid \ge L\cdot t)\le \exp (-\frac{c\cdot t^2}{t+\sqrt{n}})$$ where $\bar{g}$ is the median or mean of $g$ for $x\sim p$; this generalizes and 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.