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

### Submission summary

 Authors (as Contributors): Pierre Mergny
Submission information
Date accepted: 2021-11-18
Date submitted: 2021-08-18 12:03
Submitted by: Mergny, Pierre
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Mathematical Physics
Approach: Theoretical

### 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.

Published as SciPost Phys. 12, 022 (2022)

### Submission & Refereeing History

Resubmission scipost_202108_00047v1 on 18 August 2021
Submission 2101.01810v2 on 25 January 2021

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### Report

The paper looks now OK to me and may be published in that form.

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