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Rank one HCIZ at high temperature: interpolating between classical and free convolutions
by Pierre Mergny, Marc Potters
This Submission thread is now published as
Submission summary
| Submission information |
| Preprint Link: |
scipost_202108_00047v1
(pdf)
|
| Date accepted: |
2021-11-18 |
| Date submitted: |
2021-08-18 12:03 |
| Submitted by: |
Mergny, Pierre |
| Submitted to: |
SciPost Physics |
| Ontological classification |
| Academic field: |
Physics |
| Specialties: |
|
| 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.
