Exponential moments for disk counting statistics at the hard edge of random normal matrices

TL;DR

This paper analyzes the asymptotic behavior of the exponential moments of disk counting statistics at the hard edge of a random normal matrix ensemble, revealing explicit formulas and central limit theorems.

Contribution

It provides explicit asymptotic formulas for the moment generating function and cumulants at the hard edge, a regime less understood compared to bulk and soft edge.

Findings

Asymptotic formulas for the moment generating function in two regimes

Explicit constants for asymptotic expansions

Central limit theorems for disk counting statistics

Abstract

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let $n$ be the number of points. We focus on two regimes: (a) the ``hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{n}$ from the hard wall, and (b) the ``semi-hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{\sqrt{n}}$ from the hard wall. As $n \to + \infty$, we prove that the moment generating function enjoys asymptotics of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), & & \mbox{for the hard edge}, \\ & \exp \bigg(C_{1}n + C_{2}\sqrt{n} \hspace{0.12cm} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}\bigg(\frac{(\ln n)^{4}}{n}\bigg)\bigg), & & \mbox{for the semi-hard edge}. \end{align*} In both cases, we determine the constants $C_{1},\dots,C_{4}$ explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the ``bulk", ``soft edge" and ``semi-hard edge" regimes, the second and higher order cumulants of the disk counting function in the ``hard edge" regime are proportional to $n$ and not to $\sqrt{n}$.