Theory Seminar

KEK 素粒子原子核研究所 理論セミナー

TITLE Spectrum of the Product of Independent Random Gaussian Matrices
(in English)
SPEAKER Prof. Zdzislaw Burda
(Jagellonian University)
DATE: September 20 (Tue.) 11:00 − 12:00
PLACE: Kenkyu Honkan 1F, Meeting Room 1


We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the limit N\rightarrow \infty is rotationally symmetric in the complex plane and is given by a simple expression \rho(z,\bar{z}) = \frac{1}{M\pi} \sigma^{-\frac{2}{M}} |z|^{-2+\frac{2}{M}} for |z|\le \sigma, and is zero for |z|> \sigma. The parameter \sigma corresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not change even if the matrices in the product are taken from different Gaussian ensembles. We present a self-contained derivation of this result using a planar diagrammatic technique. Additionally, we conjecture that this distribution also holds for any matrices whose elements are independent, centered random variables with a finite variance or even more generally for matrices which fulfill Pastur-Lindeberg's condition. We provide a numerical evidence supporting this conjecture.