Freely Independent Random Variables with Non-Atomic Distributions

TL;DR

This paper studies the distributions of non-commutative polynomials of freely independent variables, proving an analogue of the Strong Atiyah Conjecture and analyzing the algebraic nature of their Cauchy transforms.

Contribution

It establishes a free probability analogue of the Strong Atiyah Conjecture and characterizes the distributional properties of matricial polynomials of free semicircular variables.

Findings

Atoms of matricial polynomials have measures as integer multiples of 1/n

Cauchy transforms are algebraic functions

Distributions are real-analytic except at finitely many points

Abstract

We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups thus proving that the measure of each atom of any $n \times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $n^{-1}$. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real-analytic except at a finite number of points.