Phase transition for the volume of high-dimensional random polytopes

TL;DR

This paper investigates the phase transition in the expected normalized volume of high-dimensional beta polytopes, revealing a sharp change in behavior as parameters vary, with implications for understanding their geometric properties.

Contribution

It establishes the existence and shape of a phase transition in the volume of beta polytopes and extends results to intrinsic volumes and vertex counts for specific cases.

Findings

Expected normalized volume exhibits a phase transition.

Derived analogous results for intrinsic volumes.

Analyzed vertex counts for the uniform case.

Abstract

The beta polytope $P_{n,d}^\beta$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^{\beta}$ if $\beta>-1$ (in particular, $\beta=0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $\beta=-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when $\beta=0$, their number of vertices.