Rotationally invariant family of Lévy like random matrix ensembles

Abstract

We introduce a family of rotationally invariant random matrix ensembles characterized by a parameter $\lambda$. While $\lambda =1$ corresponds to well-known critical ensembles, we show that $\lambda \ne 1$ describes "Lévy like" ensembles, characterized by power law eigenvalue densities. For $\lambda >1$ the density is bounded, as in Gaussian ensembles, but $\lambda <1$ describes ensembles characterized by densities with long tails. In particular, the model allows us to evaluate, in terms of a novel family of orthogonal polynomials, the eigenvalue correlations for Lévy like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles.

arXiv comment9 pages, 5 figures

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