Spectrum of the Product of Independent Random Gaussian Matrices

TL;DR

This paper derives a universal eigenvalue density for the product of independent Gaussian random matrices, showing it is rotationally symmetric and applies broadly across different matrix types and ensembles.

Contribution

It provides a simple, universal formula for the eigenvalue density of matrix products, valid for various Gaussian ensembles and potentially for other independent, finite-variance variables.

Findings

Eigenvalue density is rotationally symmetric in the complex plane.

The density follows a universal power-law form within a circular support.

Numerical evidence suggests applicability to broader classes of random matrices.

Abstract

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 variance.