Hard edge asymptotics of correlation functions between singular values and eigenvalues

TL;DR

This paper derives the large $n$ asymptotics of the joint probability density between one eigenvalue and multiple singular values near the origin for certain random matrix ensembles, revealing universal behaviors at the hard edge.

Contribution

It provides explicit large $n$ asymptotics of the $1,k$-point function at the hard edge for polynomial and Pólya ensembles, including classical Wishart-Laguerre, Jacobi, and Cauchy-Lorentz ensembles.

Findings

Universal asymptotic behavior at the hard edge for multiple ensembles.

Same asymptotics for Wishart-Laguerre, Jacobi, and Cauchy-Lorentz ensembles.

Explicit asymptotics at the soft-hard edge for Jacobi ensembles.

Abstract

Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random complex matrices drawn from a bi-unitarily invariant ensemble, a bijection exists between the underlying singular value ensemble and the corresponding eigenvalue ensemble. This enabled the recent finding of an explicit formula for the joint probability density between $1$ eigenvalue and $k$ singular values, coined $1,k$-point function. We derive here the large $n$ asymptotic of the $1,k$-point function around the origin (hard edge) for a large subclass of bi-unitarily invariant ensembles called polynomial ensembles and its subclass P\'olya ensembles. This latter subclass contains all Meijer-G ensembles and, in particular, Muttalib-Borodin ensembles and the classical Wishart-Laguerre (complex Ginibre), Jacobi (truncated unitary), Cauchy-Lorentz ensembles. We show that the latter three ensembles share the same asymptotic of the $1,k$-point function around the origin. In the case of Jacobi ensembles, there exists another hard edge for the singular values, namely the upper edge of their support, which corresponds to a soft edge for the eigenvalue (soft-hard edge). We give the explicit large $n$ asymptotic of the $1,k$-point function around this soft-hard edge.