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Rank one HCIZ at high temperature: interpolating between classical and free convolutions
by Pierre Mergny, Marc Potters
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Pierre Mergny |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2101.01810v2 (pdf) |
| Date submitted: | Jan. 25, 2021, 5:41 p.m. |
| Submitted by: | Pierre Mergny |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| 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.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 2) on 2021-6-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2101.01810v2, delivered 2021-06-08, doi: 10.21468/SciPost.Report.3040
Strengths
1) Novel and interesting construction of a new convolution operation.
2) Creates new directions for future research.
Weaknesses
1) Poor presentation makes the paper unusually difficult to read.