On tail probability of the covariance matrix in Eldan's stochastic localization

TL;DR

This paper provides exponential tail probability estimates for the covariance matrix process in Eldan's stochastic localization, advancing understanding of high-dimensional convex bodies and partially confirming a conjecture by Klartag and Lehec.

Contribution

It offers the first exponential tail bounds for the covariance process in a simplified stochastic localization framework, contributing to convex geometry and probability theory.

Findings

Exponential tail bounds for covariance matrix process.

Implication of weaker form of Klartag-Lehec p-moment conjecture.

Enhanced understanding of high-dimensional convex bodies.

Abstract

The Eldan's stochastic localization is a new kind of stochastic evolution in the space of probability measures which provides a novel way to study high dimensional convex body. A central object in the study of the stochastic localization is the stochastic process of its covariance matrix. The main result of this paper is some exponential-type tail probability estimate of the covariance process for the general time. This estimate implies a weaker version of a $p$-moment conjecture by Klartag and Lehec. The stochastic localization considered here is a simplified version by Lee and Vemplala.