Limiting behavior of determinantal point processes associated with weighted Bergman kernels
Kiyoon Eum
TL;DR
This paper studies determinantal point processes (DPPs) associated with weighted Bergman kernels on bounded pseudoconvex domains. It extends finite-dimensional limit results to an infinite-rank setting by expressing the scaled cumulant generating function of the DPP $\Lambda_k$ with kernel $K_{k\phi}$ and weight $e^{-k\phi}$ as a Fredholm determinant, and then taking the large-$k$ limit. The authors prove that, for $\phi\in SPSH(\Omega)$ and $\phi$-admissible $u$, the limit $\displaystyle \lim_{k\to\infty} \frac{1}{k^{n+1}} \log \mathbb{E}\big[e^{-k\langle u,\Lambda_k\rangle}\big]$ equals the Monge-Ampère energy expression $-\frac{1}{(n+1)!}\sum_{j=0}^{n} \int_{\Omega} u\, (dd^c(\phi+u))^{j} \wedge (dd^c\phi)^{n-j}$, identifying the limit with $-\mathcal{E}(\phi+u)$. The method combines asymptotics of $K_{k\phi}$, infinite-dimensional Jacobi calculus for determinants, and Toeplitz-operator trace formulas to bridge complex-analytic Bergman geometry with probabilistic large deviations for DPPs. This yields a precise, explicit large-$k$ limit for the cumulant generating function in terms of complex-geometry energies, with potential implications for random point configurations in several complex variables.
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.