Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble

TL;DR

This work analyzes the asymptotics of the normalised characteristic polynomial for the Elliptic Ginibre Ensemble outside the limiting ellipse. It proves that, for each fixed $t\in[0,1]$, the normalised polynomial $f_{n,t}$ converges in law to $\exp(-F_t)$ where $F_t(z)=\sum_{k\ge1} X_k\frac{z^k}{\sqrt{k}}$ with independent complex Gaussians $X_k$ satisfying $\mathbb E[X_k^2]=t^k$ and $\mathbb E|X_k|^2=1$, thereby identifying a Gaussian analytic limit. The proof combines tightness from the Hermite kernel (via Akemann–Duits–Molag results) with a coefficient-wise convergence argument based on traces of Chebyshev polynomials, analyzed through a moment-method framework. The paper also derives the average characteristic polynomial limit and discusses universality aspects, including potential extensions to moment-matching settings and to broader elliptic/determinantal-Coulomb gas models. Overall, the results provide a first rigorous step toward universality of the limiting object and outline avenues for generalizing to more general elliptic ensembles.

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{ï} 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.

Figures (4)

• Figure 1: Illustration of Theorem \ref{['theorem:main_result']}. Phase portrait of the normalised characteristic polynomial of an EGE matrix of size $250$ for different values of $t$: 0 (top left), $0.3$ (top right), $0.6$ (bottom left) and $1$ (bottom right). The unit circle is represented in red.
• Figure 2: Graphs for which $n^{-k/2}\mathcal{A}_\pi^{(n)}$ has a non--trivial limit. Top left, top right, bottom left, bottom right correspond respectively to cases 1, 2, 3 and 4 above. Green dots denote the vertices and red stars denote initial points.
• Figure 3: A tree with a quadruple edge with the two possible ways of ungluing it to obtain a planar tree.
• Figure 4: Two elements of $\mathcal{P}_2(\pi_1, \pi_2)$ where $\pi_1, \pi_2$ are both rooted unicyclic graphs (left) or both plane rooted trees (right).

Theorems & Definitions (44)

• Theorem 1.1: Convergence of the normalised characteristic polynomial
• Corollary 1.2: Lack of outliers
• proof
• Theorem 1.3: Average characteristic polynomial
• Lemma 2.1: Tightness and convergence of coefficients imply convergence of functions
• Theorem 2.2: Tightness
• Definition 2.3: Modified Chebyshev polynomials
• Theorem 2.4: Convergence of the traces of Chebyshev polynomials
• proof : Conclusion of the proof of Theorem \ref{['theorem:main_result']}
• Lemma 2.5: Montel's theorem