Abstract
We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, 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 cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.
Highlights
The HCIZ integral has applications in problems directly linked to random matrix theory (RMT) such as the study of the sum of invariant ensembles [22,23,24], the development of large deviation principles [25], the study of the so-called orbital beta processes [26]
Let μ be a measure with zero mean and unit variance, we look at the following c-Central Limit Theorem (c-CLT): μ[Gc](.)
Integral in the high temperature regime c, which is multiplicative for this convolution
Summary
For a self-adjoint random matrix A, of size N with real, complex, or quaternionic entries, under mild assumptions and up to a rescaling of the entries, we know from Random Matrix Theory (RMT) that the (random) spectral measure of A tends to a deterministic limiting measure μA in the limit N → ∞ (see for example [1]). Georges and Lévy in [11], which interpolates between the classical convolution (t = 0) and the free convolution (t → ∞) but for which it is not possible to construct a transform that linearizes the convolution and from which one can define cumulants at any order.
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