Phase transition for the volume of high‐dimensional random polytopes. Issue 4 (28th December 2020)

• Record Type: Journal Article
• Authors: Bonnet, Gilles; Kabluchko, Zakhar; Turchi, Nicola
• Is Part Of: Random structures & algorithms. Volume 58:Issue 4(2021)
• Page Start: 648
• Page End: 663
• Publication Date: 2020-12-28
• Subjects: Beta distribution -- convex hull -- expected volume -- phase transition -- random polytopes; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20986
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: The beta polytope P n, d β is the convex hull of n i.i.d. random points distributed in the unit ball of ℝ d according to a density proportional to ( 1 − ‖ x ‖ 2 ) β if β > − 1 (in particular, β = 0 corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if β = − 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 β = 0, their number of vertices.