Limiting behavior of determinantal point processes associated with weighted Bergman kernels
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908343322279936 |
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| author | Eum, Kiyoon |
| author_facet | Eum, Kiyoon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limiting behavior of determinantal point processes associated with weighted Bergman kernels Eum, Kiyoon Complex Variables Probability 32A36, 60G55 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. |
| title | Limiting behavior of determinantal point processes associated with weighted Bergman kernels |
| topic | Complex Variables Probability 32A36, 60G55 |
| url | https://arxiv.org/abs/2404.14793 |