Matrix addition and the Dunkl transform at high temperature

TL;DR

This paper establishes a Law of Large Numbers for eigenvalues of random matrices at high temperature using Dunkl transform analysis, revealing a new family of interpolating convolutions and deformed cumulants.

Contribution

It introduces a novel framework connecting Dunkl transform analysis with high-temperature limits in random matrix theory, leading to new interpolating convolutions and cumulants.

Findings

LLN for eigenvalues as matrix size and inverse temperature grow

A new one-parameter family of binary operations interpolating classical and free convolutions

Introduction of deformed cumulants linearizing the new operation

Abstract

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.