Rotationally invariant family of Lévy like random matrix ensembles

Details Rotationally invariant family of Lévy like random matrix ensembles Rotationally invariant family of Lévy like random matrix ensembles

2009-03-30

arxiv.org/abs/0903.5266v1

J. Phys. A: Math. Theor. 42 (2009) 152001

Physics

Condensed Matter

Statistical Mechanics

9 pages, 5 figures

Scientific paper

10.1088/1751-8113/42/15/152001

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 "L\'evy 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 L\'evy like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles.

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