Limiting behavior of determinantal point processes associated with weighted Bergman kernels

Abstract

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