The Elliptic Ginibre Ensemble: A Unifying Approach to Local and Global Statistics for Higher Dimensions

TL;DR

This paper develops a unifying mathematical framework for analyzing local and global spectral statistics of the elliptic Ginibre ensemble in higher dimensions, connecting non-Hermitian and Hermitian random matrices through saddle point analysis.

Contribution

It introduces a rigorous saddle point method to derive spectral statistics for the elliptic Ginibre ensemble and extends results to higher dimensions, including new local bulk and edge kernels.

Findings

Proved global spectral statistics for the elliptic Ginibre ensemble.

Derived local bulk and edge kernels in higher dimensions.

Connected non-Hermitian matrices with higher-dimensional harmonic oscillators.

Abstract

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows to interpolate between the rotational invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field, at inverse temperature $\beta=2$. Furthermore, it represents a determinantal point process in the complex plane with corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a know correspondence between non-interacting fermions in a trap in $d$ real dimensions $\mathbb{R}^d$ and the $d$-dimensional harmonic oscillator. We present a rigorous proof for the local $d$-dimensional bulk (sine-) and edge (Airy-) kernel first defined by Dean et al., complementing recent results by Deleporte and Lambert. Using the same relation to the $d$-dimensional harmonic oscillator in $d$ complex dimensions $\mathbb{C}^d$, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in $d$ real and $d$ complex dimensions. For $\mathbb{C}^d$ with $d=1$ this corresponds to non-interacting fermions in a rotating trap.