The hyperbolic-type point process

摘要

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