Principal angles between random subspaces and polynomials in two free projections

TL;DR

This paper explores the geometric concept of principal angles between subspaces to analyze free projections, deriving formulas for noncommutative distributions and revealing asymptotic behaviors of random subspaces.

Contribution

It introduces a novel geometric approach to compute distributions involving free projections and simplifies existing formulas in free probability theory.

Findings

Explicit formula for free additive convolution of Bernoulli distributions

Asymptotic uniform distribution of principal angles between random subspaces

Simplification of the free Bernoulli anticommutator formula

Abstract

We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure.