Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble

TL;DR

This paper analyzes the asymptotic behavior of the characteristic polynomial of the Elliptic Ginibre Ensemble, showing convergence to a Gaussian analytic function and advancing understanding of spectral properties in random matrix theory.

Contribution

It establishes the convergence of the normalized characteristic polynomial outside the ellipse for the Elliptic Ginibre Ensemble, linking it to Gaussian analytic functions and addressing an open interpolation problem.

Findings

Convergence in law of the normalized characteristic polynomial outside the ellipse.

Identification of the limit as the exponential of a Gaussian analytic function.

Connection between Hermite polynomials and the spectral properties of the ensemble.

Abstract

We consider the complex Elliptic Ginibre Ensemble, a family of random matrix models introduced by Girko that interpolates between the Ginibre Ensemble and the Gaussian Unitary Ensemble and such that its empirical spectral measure converges to the uniform measure on an ellipse. We show the convergence in law of its normalised characteristic polynomial outside of this ellipse. Our proof contains two main steps. We first show the tightness of the normalised characteristic polynomial using the link between the Elliptic Ginibre Ensemble and Hermite polynomials. This part relies on the uniform control of the Hermite kernel which is derived from the recent work of Akemann, Duits and Molag. In the second step, we identify the limiting object as the exponential of a Gaussian analytic function. The limit expression is derived from the convergence of traces of Chebyshev polynomials of random matrices by the method of moments. These traces of Chebyshev polynomials appear naturally as a kind of centered version, or normal ordering, of the traces of the monomials. This work answers the interpolation problem raised in the work of Bordenave, Chafa{\"i} and the second author of this paper for the integrable case of the Elliptic Ginibre Ensemble and is therefore a fist step towards the conjectured universality of this result.