Thin-shell theory for rotationally invariant random simplices

Abstract

For fixed functions $G,H:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, consider the rotationally invariant probability density on ${\mathbb{R}}^n$ of the form

$${\mathrm{\mu }}^n(\mathrm{d}s)=\frac{1}{Z_n}G(\Vert s{\Vert }_2)e^{-nH(\Vert s{\Vert }_2)}\mathrm{d}s.$$

We show that when n is large, the Euclidean norm $\Vert Y^n{\Vert }_2$ of a random vector $Y^n$ distributed according to ${\mathrm{\mu }}^n$ satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value $s_0$ minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from $s_0$ have the order $1∕\sqrt{n}$ and are approximately Gaussian when n is large.

We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors $Y_1^n,\dots ,Y_p^n$ distributed according to ${\mathrm{\mu }}^n$, ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on ${\mathbb{R}}^n$, provided a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA 16, 141–177 (2019)].

Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to 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 $c_0,c_1\in (0,\mathrm{\infty })$ and an absolute constant $C\in (0,\mathrm{\infty })$ such that

$$\underset{s\in \mathbb{R}}{sup}\left|\mathbb{P}\left[\frac{log\det (A^n)-log(n-1)!-c_0}{\sqrt{\frac{1}{2}logn+c_1}}<s\right]-{\int }_{-\mathrm{\infty }}^s\frac{e^{-u^2∕2}\mathrm{d}u}{\sqrt{2\mathrm{\pi }}}\right|<\frac{C}{{log}^{3∕2}n},$$

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].