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Free Random Levy Matrices
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Free Random Levy Matrices
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Using the theory of free random variables (FRV) and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable L\'{e}vy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit L\'{e}vy tails. For the analytically known L\'{e}vy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the L\'{e}vy tail-distributions are characterized by a novel form of microscopic universality.
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Random matrix theory of integrability-to-chaos transition
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