  We consider a two-dimensional point process whose points are separated into
two disjoint components by a hard wall, and study the multivariate moment
generating function of the corresponding disk counting statistics. We
investigate the ``hard edge regime" where all disk boundaries are a distance of
order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points.
We prove that as $n \to + \infty$, the asymptotics of the moment generating
function are of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n +
C_{3} + \mathcal{F}_{n} + \frac{C_{4}}{\sqrt{n}} +
\mathcal{O}(n^{-\frac{3}{5}})\bigg), \end{align*} and we determine the
constants $C_{1},\dots,C_{4}$ explicitly. The oscillatory term
$\mathcal{F}_{n}$ is of order $1$ and is given in terms of the Jacobi theta
function. Our theorems allow us to derive various precise results on the disk
counting function. For example, we prove that the asymptotic fluctuations of
the number of points in one component are of order $1$ and are given by an
oscillatory discrete Gaussian. Furthermore, the variance of this random
variable enjoys asymptotics described by the Weierstrass $\wp$-function.
