The Hyperbolic-type Point Process

In this paper, we introduce a two-parameters determinantal point process in the Poincar\'e disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r$ tends to $1$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.