Free Random Levy Matrices

Details Free Random Levy Matrices Free Random Levy Matrices

2000-11-27

arxiv.org/abs/cond-mat/0011451v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

5 pages

Scientific paper

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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