Limiting behavior of determinantal point processes associated with weighted Bergman kernels

TL;DR

This paper investigates the asymptotic behavior of determinantal point processes derived from weighted Bergman kernels on pseudoconvex domains, revealing explicit limiting cumulant generating functions as the weight parameter grows.

Contribution

It establishes the limiting behavior of scaled cumulant generating functions for determinantal point processes associated with weighted Bergman kernels, extending understanding of their asymptotics.

Findings

Convergence of scaled cumulant generating functions as weight parameter increases

Explicit expression of the limit in terms of domain weight function and test functions

Restriction to -dmissible test functions for the limit process

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.