Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method

Abstract

We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the hermitian reduction method, as considered in the physical literature, combined with subordination ideas and the linearization trick.

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References in corpus (3)

• Products of independent elliptic random matrices
• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• Polynomials in Asymptotically Free Random Matrices

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• Complete diagrammatics of the single ring theorem
• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
• The Brown measure of the sum of a self-adjoint element and an elliptic element
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• Summing free unitary Brownian motions with applications to quantum information