On the characteristic polynomial of the eigenvalue moduli of random normal matrices

TL;DR

This paper analyzes the characteristic polynomial of eigenvalue moduli for random normal matrices from the Mittag-Leffler ensemble, deriving precise large n asymptotics involving novel associated Hermite polynomials.

Contribution

It provides the first large n asymptotics for a complex determinant with a circular root-type singularity, introducing associated Hermite polynomials as a new asymptotic ingredient.

Findings

Derived large n asymptotics for the moment generating function in the bulk.

Identified the role of associated Hermite polynomials in the asymptotic analysis.

Extended understanding of Fisher-Hartwig singularities to circular root-type cases.

Abstract

We study the characteristic polynomial $p_{n}(x)=\prod_{j=1}^{n}(|z_{j}|-x)$ where the $z_{j}$ are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large $n$ asymptotics for the moment generating function $\mathbb{E}[e^{\frac{u}{\pi} \, \mathrm{Im} \ln p_{n}(r)}e^{a \, \mathrm{Re} \ln p_{n}(r)}]$, in the case where $r$ is in the bulk, $u \in \mathbb{R}$ and $a \in \mathbb{N}$. This expectation involves an $n \times n$ determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities along the circle centered at $0$ of radius $r$. This "circular" root-type singularity differs from earlier works on Fisher-Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.