Free Convolution Operators and Free Hall Transform

TL;DR

This paper extends polynomial calculus in free probability to define transition operators for free convolutions, characterizes free transforms, and proves asymptotic convergence results for random matrices and free transforms.

Contribution

It introduces an algebraic extension of polynomial calculus to better understand free convolutions and transforms, linking classical and free probability frameworks.

Findings

Convergence of Brownian motion on GL_N(C) to free multiplicative circular Brownian motion.

Convergence of classical Hall transform on U(N) to free Hall transform.

New algebraic framework for free convolution operators and transforms.

Abstract

We define an extension of the polynomial calculus on a W*-probability space by introducing an abstract algebra which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal-Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to infinity, of the *-distribution of the Brownian motion on the linear group GL_N(C) to the *-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N) to the free Hall transform.