Spectrum of the product of independent random Gaussian matrices

We show that the eigenvalue density of a product X=X₁X₂⋯XM of M independent N×N Gaussian random matrices in the limit N→∞ is rotationally symmetric in the complex plane and is given by a simple expression ρ(z,z̄)=1Mπσ^-2/M|z|^-2+(2/M) for |z|≤σ, and is zero for |z|>σ. The parameter σ 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, and 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.