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arXiv · 1504.03526

A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices

Fabio Deelan Cunden·Francesco Mezzadri·Pierpaolo Vivo
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Abstract

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.

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Fabio Deelan Cunden, Francesco Mezzadri, Pierpaolo Vivo. 2015-04-14. A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices. https://doi.org/10.1088/1751-8113/48/31/315204

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