Scaling limits of complex and symplectic non-Hermitian Wishart ensembles

TL;DR

This paper studies the eigenvalue behavior of non-Hermitian Wishart matrices in complex and symplectic classes, introducing a new differential equation approach to analyze universal scaling limits in various regimes.

Contribution

It introduces a generalized Christoffel-Darboux formula as a second-order differential equation for analyzing correlation functions in non-Hermitian Wishart ensembles.

Findings

Derived universal bulk and edge scaling limits for eigenvalues.

Established a unified method applicable across all scaling regimes.

Extended understanding of non-Hermitian Wishart matrices in quantum chromodynamics.

Abstract

Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel-Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analyzing correlation functions across all scaling regimes in the model. By employing this method, we derive universal bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.