Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method
By Serban Teodor Belinschi, Piotr Śniady, Roland Speicher. 2015
49 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/0c5becf1493fb10c25af3ecb31e44aafa4d77e3b
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Citations in the corpus
- Free Probability and Random Matrices (2014, background): “In [24] it was shown that operatorvalued free probability is the right frame to deal with this rigorously.”