# Matrix harmonic analysis at high temperature via the Dirichlet process

## TL;DR

This paper studies the behavior of large random matrices at high temperature, showing convergence of spectral functions to transforms related by the Markov-Krein correspondence, with analysis rooted in the Dirichlet process.

## Contribution

It introduces a novel connection between high-temperature harmonic analysis of matrices and the Dirichlet process via the Markov-Krein correspondence.

## Key findings

- Convergence of multivariate Bessel functions to Fourier/Mellin transforms
- Establishment of the Markov-Krein correspondence in this regime
- Application of Dirichlet process theory to spectral analysis

## Abstract

We investigate harmonic analysis of random matrices of large size with their Dyson indices going simultaneous to zero, that is in the high temperature limit. In this regime, we show that the multivariate Bessel function/Heckman-Opdam hypergeometric function of the empirical spectral distribution converges to the Fourier/Mellin transform of a measure, which and the limiting empirical distribution are intimately related by the Markov-Krein correspondence. The uniqueness, existence and other properties of the Markov-Krein correspondence can be studied using the theory of the Dirichlet process.