SciPost Submission Page

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

by Pierre Mergny, Marc Potters

Submission summary

As Contributors:	Pierre Mergny
Preprint link:	scipost_202108_00047v1
Date accepted:	2021-11-18
Date submitted:	2021-08-18 12:03
Submitted by:	Mergny, Pierre
Submitted to:	SciPost Physics
Academic field:	Physics
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)

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