Thin-shell concentration for random vectors inOrlicz balls via moderate deviations and Gibbs measures

In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector X-n being in a thin shell of radius root n ntimes the asymptotic value of n(-1/2)(E[parallel to X-n parallel to 2/2])(1/2)(as n -> infinity), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter t = t(n) goes down to zero as the dimension nof the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno (2021) [32]. (C) 2021 Elsevier Inc. All rights reserved.