Multiplication law and S transform for non-Hermitian random matrices.
We derive a multiplication law for free non-Hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-Hermitian S transform being a natural generalization of the Voiculescu S transform. In addition, we extend the classical Hermitian S transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.
Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, Reymonta 4, PL-30-059 Kraków, Poland. zdzislaw.burda@uj.edu.pl
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Janik
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Nowak