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

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 $n\to +\mathrm{\infty }$, 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 $\sqrt{n}$.