Limiting behavior of determinantal point processes associated with weighted Bergman kernels

Abstract

Let $\hat{I}\text{c◯}$ be a bounded pseudoconvex domain in ${\mathbb{C}}^n$, and let $\ddot{I}•$ be a strictly plurisubharmonic function on $\hat{I}\text{c◯}$. For each $k\in \mathbb{N}$, we consider determinantal point process $\hat{I}{›}_k$ with kernel $K_{k\ddot{I}•}$, where $K_{k\ddot{I}•}$ is the reproducing kernel of infinite dimensional weighted Bergman space $H(k\ddot{I}•)$ with weight $e^{-k\ddot{I}•}$. We show that the scaled cumulant generating function for $\hat{I}{›}_k$ converges as $k\to \infty$ to a certain limit, which can be explicitly expressed in terms of $\ddot{I}•$ and a test function $u$. Note that we need to restrict the type of test function $u$ to those that are $\ddot{I}•$-admissible.

10 pages, proof of the Proposition 5 is added