Products of Complex Rectangular and Hermitian Random Matrices

Abstract

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's

“…The general transformation formula of the finite n kernel from the one of H − nx1 1 n to G(H − nx1 1 n )G * is briefly recalled in Subsec. II C. This formula has been very recently derived in [22].…”

“…In Subsec. II B, we introduce the Pólya ensembles [18,22] and their properties. In particular, we state the conditions under which Theorem III.1 holds.…”

“…This influence is born out the fact that the singular value statistics of the single factors play a crucial role and not the eigenvalues. This becomes apparent when considering the harmonic analysis approach [20] for products of matrices exploited for the complex linear group Gl(n, C) in [25,26] and modified to products involving elements of some Lie algebras in [22,24]. Singular values have a natural lower boundary at the origin which is the source of the non-trivial effect on the hard edge statistics.…”

“…By means of this model, we want to investigate the impact of the local and macroscopic spectral statistics of the matrix H on the hard edge statistics of the product. For this aim, we make use of the recently derived statistics of the products of Pólya ensembles on Gl(n, C) and polynomial ensembles [31] on the Hermitian matrices Herm(n) at finite matrix size, see [22]. Such product matrices satisfy a determinantal point process [8], which simplifies the analysis a lot since it reduces the whole statistics to a single kernel.…”

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

“…As we have seen in the introduction we have pointed out, a particular class of invariant ensembles which have a rich analytic and algebraic structure are the Pólya ensembles. There are different kinds of these ensembles, multiplicative ones [53,52,34,51,50,49,54] with additive ones [55,34,48]. We are naturally interested in those which are related to the additive convolution on Herm(N ) that have a joint probability density of the eigenvalues…”