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-Thale [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 $c_0,c_1in(0,infty)$ and an absolute constant $Cin(0,infty)$ 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].
Download