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

Abstract

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.

Products of Complex Rectangular and Hermitian Random Matrices

“…Those formulas are an ideal basis to start a large n-analysis. A work on the hard edge statistics of the products of multiplicative Pólya ensembles with a GUE is currently in preparation [21].…”

“…Let us underline that all calculations are done at finite matrix dimensions, and that it is not the goal of the present work to investigate any limiting statistics. A technical work on a general product involving a GUE is currently in preparation [21].…”

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…”