Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Abstract

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.

Supersymmetry for Products of Random Matrices

“…The polynomial (42) is a hypergeometric function and, thus, a Meijer G-function [14]. It agrees for certain values of the parameters L WL , L CL , and L J with known results [11,13]. What is completely new are the results for β = 1, 4 and k = 1 which are essentially the same polynomials.…”

“…Indeed we could also have chosen another scaling which still leads to a hard edge scaling limit. Then we would get finite rank deformations of the result (44) which was recently discovered for a product of truncated unitary matrices in [13]. Nevertheless the limiting kernel is still a Meijer G-kernel but with other parameters.…”

“…Since the projection formula works in a unifying way for all three Dyson indices β = 1, 2, 4 this approach is an ideal alternative compared to other methods like orthogonal polynomials and free probability when studying real or quaternion matrices. Note that up to now free probability only applies to global spectral properties and to use orthogonal polynomials we need to know group integrals like the Itzykson-Zuber integral [18] or its polynomial counterpart [19,13]. The projection formula circumvents this problem.…”

“…For example, free probability has proven as an efficient tool for calculating the macroscopic level density [9]. With the help of orthogonal polynomials one could calculate algebraic structures like determinants and Pfaffians, their kernels, and certain universal statistics on the local scale of the spectrum [6,10,11,12,13]. In particular products of random matrices drawn from Meijer G-ensembles (the weight is essentially given by Meijer G-functions, see [14] for a definition of these functions) exhibit a new kind of universal kernel in the hard scaling limit (microscopic limit around the origin).…”

“…The only exception, the macroscopic level density, can be computed for real (β = 1) and quaternion (β = 4) matrices with free probability [9] because they share the level density with β = 2. However the local statistics of the singular values is still highly involved for β = 1, 4 due to unknown group integrals like the Itzykson-Zuber integral [18] and its polynomial counterpart [19,13]. The projection formula recently proposed [20] circumvents such problems.…”