The Hadamard product and the free convolutions

TL;DR

This paper explores the properties of free multiplicative convolutions involving measures supported away from zero, demonstrating absolute continuity, and introduces new results on the spectral distribution of Hadamard products of matrices.

Contribution

It establishes absolute continuity of free multiplicative convolutions with measures supported on positive sets and links Hadamard products to stationary Gaussian processes.

Findings

Free multiplicative convolution with measures supported on (0,∞) is absolutely continuous.

Spectral distribution of Hadamard product matches that of certain Gaussian matrices.

Hadamard product results connect deterministic matrices with random matrix theory.

Abstract

It is shown that if a probability measure $\nu$ is supported on a closed subset of $(0,\infty)$, that is, its support is bounded away from zero, then the free multiplicative convolution of $\nu$ and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.