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KTH/Nordita/SU seminar in Theoretical Physics

Eigenvalues of the product of independent random Gaussian matrices

by Zdzislaw Burda (Jagellonian University)

FA31 ()


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 large-$N$ limit is rotationally symmetric in the complex plane and is given by a simple expression $\rho(z,\bar{z}) = 1/(M\pi\sigma^2} |z|^{-2+2/M}$ for $|z|<\sigma$, 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 for Gaussian matrices. We also give a numerical evidence suggesting that this result applies also to matrices whose elements are independent, centered random variables with a finite v ariance. We generalize the result to rectangular matrices.
Based on papers: Z. Burda, R.A. Janik and B. Waclaw, Phys. Rev. E 81, 041132 (2010);
Z. Burda, A. Jarosz, G. Livan, M. A. Nowak, A. Swiech, arXiv:1007.3594, to appear in PRE;