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 $\sqrt{ n}$.
Categories: math-ph math.MP math.PR