Universal microscopic correlation functions for products of independent Ginibre matrices
Abstract
We consider the product of n complex nonHermitian, independent random matrices, each of size N × N 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 Gfunction 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 largeN 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 generalize the known Bessel law in the complex plane for n = 2 to a new hypergeometric kernel _{0}F_{n  1}.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 November 2012
 DOI:
 10.1088/17518113/45/46/465201
 arXiv:
 arXiv:1208.0187
 Bibcode:
 2012JPhA...45T5201A
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics
 EPrint:
 20 pages, v2 published version: typos corrected and references added