Universal microscopic correlation functions for products of independent Ginibre matrices

TL;DR

This paper derives exact microscopic eigenvalue correlation functions for products of independent Ginibre matrices, revealing universal bulk and edge behavior, and novel correlations at the origin described by a hypergeometric kernel.

Contribution

It provides explicit formulas for eigenvalue correlations of matrix products, extending known results to arbitrary n and uncovering new universal and non-universal features.

Findings

Eigenvalue correlations are given by explicit determinantal kernels.

Bulk and edge correlations are universal and match the Ginibre ensemble.

Origin correlations are new and described by a hypergeometric kernel.

Abstract

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.