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

TL;DR

This paper investigates the concentration of random vectors in Orlicz balls, providing sharp bounds and asymptotic constants using moderate deviations and Gibbs measure techniques, advancing understanding of high-dimensional geometric probability.

Contribution

It introduces new asymptotic bounds for thin-shell concentration in Orlicz balls and links uniform distributions to Gibbs measures at critical temperatures.

Findings

Improved bounds on thin-shell probability decay rates.

Determined the asymptotic isotropic constant for Orlicz balls.

Enhanced understanding of high-dimensional geometry in Orlicz spaces.

Abstract

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 $\sqrt{n}$ times the asymptotic value of $n^{-1/2}\left(\mathbb E\left[\| X_n\|_2^2\right]\right)^{1/2}$ (as $n\to\infty$), 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 $n$ of 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 in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {\bf 495}(1) 2021, 1--19].