Limiting behavior of determinantal point processes associated with weighted Bergman kernels

Abstract

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