Thin-shell theory for rotationally invariant random simplices

TL;DR

This paper establishes a Gaussian thin-shell phenomenon for rotationally invariant high-dimensional distributions, applies it to the volume of random simplices, and refines the Gaussian approximation for the log-determinant of Gaussian matrices.

Contribution

It introduces a universal Gaussian concentration result for norms of rotationally invariant measures and applies it to analyze the volume of random simplices and the determinant of Gaussian matrices.

Findings

Norms of random vectors concentrate around a specific value with Gaussian fluctuations.

Logarithmic volume of random simplices exhibits Gaussian behavior in high dimensions.

Improves the rate of convergence in the Gaussian approximation of the log-determinant of Gaussian matrices.

Abstract

For fixed functions $G,H:[0,\infty)\to[0,\infty)$, consider the rotationally invariant probability density on $\mathbb{R}^n$ of the form \[ \mu^n(ds) = \frac{1}{Z_n} G(\|s\|_2)\, e^{ - n H( \|s\|_2)} ds. \] We show that when $n$ is large, the Euclidean norm $\|Y^n\|_2$ of a random vector $Y^n$ distributed according to $\mu^n$ satisfies a Gaussian thin-shell property: the distribution of $\|Y^n\|_2$ concentrates around a certain value $s_0$, and the fluctuations of $\|Y^n\|_2$ are approximately Gaussian with the order $1/\sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,\ldots,Y_p^n$ distributed according to $\mu^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Th\"ale [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n \times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1\in(0,\infty)$ and an absolute constant $C\in(0,\infty)$ such that \[\sup_{ s \in \mathbb{R}} \left| \mathbb{P} \left[ \frac{ \log \mathrm{det}(A^n) - \log(n-1)! - c_0 }{ \sqrt{ \frac{1}{2} \log n + c_1 }} < s \right] - \int_{-\infty}^s \frac{e^{ - u^2/2} du}{ \sqrt{ 2 \pi }} \right| < \frac{C}{\log^{3/2}n}, \] sharpening the $1/\log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].