Rank one HCIZ at high temperature: interpolating between classical and free convolutions

TL;DR

This paper introduces a new one-parameter family of convolutions, called the c-convolution, interpolating between classical and free convolutions using the high temperature limit of the rank one HCIZ integral, with applications to distribution transformations.

Contribution

It constructs the c-convolution via the high temperature limit of the rank one HCIZ integral and relates it to the Markov-Krein transform, providing new analytical tools.

Findings

Derived cumulants-moments relations for c-convolution.

Established a central limit theorem for c-convolution.

Presented numerical examples demonstrating the properties of c-convoluted distributions.

Abstract

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where $\frac{N \beta}{2} \to c $, called the high temperature regime and show that it can be used to construct a promising one-parameter interpolation, with parameter $c$ between the classical and the free convolution. This $c$-convolution has a simple interpretation in terms of another associated family of distribution indexed by $c$, called the Markov-Krein transform: the $c$-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulants-moments relations, a central limit theorem, a Poisson limit theorem and shows several numerical examples of $c$-convoluted distributions.