Hole probabilities and overcrowding estimates for products of complex Gaussian matrices

TL;DR

This paper analyzes eigenvalue distributions of products of complex Gaussian matrices, deriving asymptotics for hole and overcrowding probabilities in the eigenvalue point process as the matrix size grows large.

Contribution

It provides a novel characterization of the eigenvalue distribution for products of Gaussian matrices and derives asymptotic estimates for hole and overcrowding probabilities.

Findings

Distribution of eigenvalue moduli matches product of Gamma variables

Asymptotic formulas for hole probabilities as radius increases

Asymptotic estimates for overcrowding probabilities as number of points grows

Abstract

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda J. Phys A: Math. Theor. 45 (2012) 465201), which can be understood as a generalization of the finite Ginibre ensemble. As N\rightarrow\infty, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as {R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2 is distributed as the product of n independent Gamma variables Gamma(k,1). This enables us to find the asymptotics for the hole probabilities, i.e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r\rightarrow\infty. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m\rightarrow\infty.