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2022 Thin-shell theory for rotationally invariant random simplices
Johannes Heiny, Samuel Johnston, Joscha Prochno
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Electron. J. Probab. 27: 1-41 (2022). DOI: 10.1214/21-EJP734

Abstract

For fixed functions G,H:[0,)[0,), consider the rotationally invariant probability density on Rn of the form

μn(ds)=1ZnG(s2)enH(s2)ds.

We show that when n is large, the Euclidean norm Yn2 of a random vector Yn distributed according to μn satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value s0 minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from s0 have the order 1n 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 Y1n,,Ypn distributed according to μn, 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 Rn, 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 An is an n×n random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants c0,c1(0,) and an absolute constant C(0,) such that

supsRPlogdet(An)log(n1)!c012logn+c1<sseu22du2π<Clog32n,

sharpening the 1log13+o(1)n bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146–167].

Funding Statement

JH was supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity. SJ and JP have been supported by the Austrian Science Fund (FWF) Project P32405 “Asymptotic Geometric Analysis and Applications” of which JP is principal investigator. JP has also been supported by the FWF Project F5508-N26, which is part of the Special Research Program ‘Quasi-Monte Carlo Methods: Theory and Applications’.

Citation

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Johannes Heiny. Samuel Johnston. Joscha Prochno. "Thin-shell theory for rotationally invariant random simplices." Electron. J. Probab. 27 1 - 41, 2022. https://doi.org/10.1214/21-EJP734

Information

Received: 8 April 2021; Accepted: 14 December 2021; Published: 2022
First available in Project Euclid: 11 January 2022

Digital Object Identifier: 10.1214/21-EJP734

Subjects:
Primary: 52A23 , 60F05
Secondary: 60B20 , 60D05

Keywords: central limit theorem , high dimension , logarithmic volume , Random matrix , random simplex , Stochastic geometry

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