Regularization by free additive convolution, square and rectangular cases

TL;DR

This paper investigates the regularization effects of free convolution operations on probability measures, showing conditions under which these convolutions become absolutely continuous with positive analytic densities, with distinctions between square and rectangular cases.

Contribution

It provides new insights into the regularization properties of free and rectangular free convolutions, including conditions for absolute continuity and analyticity of resulting measures.

Findings

In the square case, no measure in the semigroup can have a finite second moment.

In the rectangular case, the convolution often has an atom at zero or no mass nearby, limiting regularization.

Sufficient conditions for the analyticity and existence of densities in rectangular convolutions are established.

Abstract

The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups $(\mu_t)$ of probability measures such that $\mu_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $\nu$, the free convolution of $\mu_t$ with $\nu$ (or, in the rectangular context, the rectangular free convolution of $\mu_t$ with $\nu$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for $\mu$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $\mu$ with $\nu$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $\mu$ with $\nu$ except on a negligible set of points, as well as existence and continuity of a density everywhere.