| Popis: |
We consider the Elliptic Ginibre Ensemble, a family of random matrix models interpolating 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 as a random holomorphic function 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 random matrices, based on an adaptation of techniques that were used to study fluctuations of Wigner and deterministic matrices by Male, Mingo, P{\'e}ch{\'e} and Speicher.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. |
