44 research outputs found
Universal microscopic correlation functions for products of independent Ginibre matrices
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.Comment: 20 pages, v2 published version: typos corrected and references adde