Large gap asymptotics on annuli in the random normal matrix model

TL;DR

This paper derives detailed large gap asymptotics for a two-parameter generalization of the complex Ginibre point process, explicitly computing constants and revealing a novel oscillatory term involving the Jacobi theta function.

Contribution

It provides the first explicit large gap asymptotics for the generalized process, including all constants and the oscillatory term, extending previous results for the complex Ginibre process.

Findings

Explicit formulas for large gap asymptotics constants C1 to C6.

Identification of the oscillatory term as a Jacobi theta function.

Improved results for special cases like disks and unbounded annuli.

Abstract

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at $0$ satisfies large $n$ asymptotics of the form \begin{align*} \exp \bigg( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + \mathcal{F}_{n} + \mathcal{O}\big( n^{-\frac{1}{12}}\big)\bigg), \end{align*} where $n$ is the number of points of the process. We determine the constants $C_{1},\ldots,C_{6}$ explicitly, as well as the oscillatory term $\mathcal{F}_{n}$ which is of order $1$. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only $C_{1},\ldots,C_{4}$ were previously known, (ii) when the hole region is an unbounded annulus, only $C_{1},C_{2},C_{3}$ were previously known, and (iii) when the hole region is a regular annulus in the bulk, only $C_{1}$ was previously known. For general values of our parameters, even $C_{1}$ is new. A main discovery of this work is that $\mathcal{F}_{n}$ is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.