Haagerup's phase transition at polydisc slicing

TL;DR

This paper investigates a phase transition phenomenon related to the behavior of sums of random vectors on spheres, extending known inequalities and revealing a sharp transition point in the context of polydisc slicing.

Contribution

It establishes a new comparison inequality for negative and second moments of sums of spherical random vectors, extending the polydisc slicing results to a probabilistic setting.

Findings

Identifies the exact point of phase transition where p-norm recovers volume.

Provides a probabilistic extension of the Oleszkiewicz-Pelczyński result.

Obtains partial results in higher dimensions.

Abstract

We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz-Pelczy\'nski polydisc slicing result. The Haagerup-type phase transition occurs exactly when the p-norm recovers volume, in contrast to the real case. We also obtain partial results in higher dimensions.