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(n1/4), improving on the previous best bound of O(n1/3√log n). 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 bf Rn converges in O*(n2.5) 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 Rn 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/D2) by Kannan-Lov'asz-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size δ=Theta(1/√n) mixes in Oleft(n2Dright) 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 large deviation inequality for an L-Lipschitz function g over an isotropic logconcave density p: for any t>0, [ Prx~ pleft(left|g(x)-bargright|≥ L· tright)≤exp(-fracc· t2t+√n) ] where barg is the median or mean of g for x~ 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.