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Mathematics > Complex Variables
Title: Limiting behavior of determinantal point processes associated with weighted Bergman kernels
(Submitted on 23 Apr 2024)
Abstract: Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\phi$ be 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 weighted Bergman space $H(k\phi)$ with weight $e^{-k\phi}$. We show that the limit of scaled cumulant generating funcion 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 shape of the test functions $u$ to be $\phi$-admissible.
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