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arXiv · 1506.02017

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

Serban Belinschi·Piotr Sniady·Roland Speicher
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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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Serban Belinschi, Piotr Sniady, Roland Speicher. 2017-09-08. Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method. https://doi.org/10.1016/j.laa.2017.09.024

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