A Law of large numbers for vector-valued linear statistics of Bergman DPP
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916219646377984 |
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| author | Lin, Zhaofeng Qiu, Yanqi Wang, Kai |
| author_facet | Lin, Zhaofeng Qiu, Yanqi Wang, Kai |
| contents | We establish a law of large numbers for a certain class of vector-valued linear statistics for the Bergman determinantal point process on the unit disk. Our result seems to be the first LLN for vector-valued linear statistics in the setting of determinantal point processes. As an application, we prove that, for almost all configurations $X$ with respect to with respect to the Bergman determinantal point process, the weighted Poincaré series (we denote by $d_{h}(\cdot,\cdot)$ the hyperbolic distance on $\mathbb{D}$) \begin{align*} \sum_{k=0}^\infty\sum_{x\in X\atop k\le d_{h}(z,x)<k+1}e^{-sd_{\mathrm{h}}(z,x)}f(x) \end{align*} cannot be simultaneously convergent for all Bergman functions $f\in A^2(\mathbb{D})$ whenever $1<s<3/2$. This confirms a result announced without proof in Bufetov-Qiu's work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14978 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Law of large numbers for vector-valued linear statistics of Bergman DPP Lin, Zhaofeng Qiu, Yanqi Wang, Kai Probability Functional Analysis We establish a law of large numbers for a certain class of vector-valued linear statistics for the Bergman determinantal point process on the unit disk. Our result seems to be the first LLN for vector-valued linear statistics in the setting of determinantal point processes. As an application, we prove that, for almost all configurations $X$ with respect to with respect to the Bergman determinantal point process, the weighted Poincaré series (we denote by $d_{h}(\cdot,\cdot)$ the hyperbolic distance on $\mathbb{D}$) \begin{align*} \sum_{k=0}^\infty\sum_{x\in X\atop k\le d_{h}(z,x)<k+1}e^{-sd_{\mathrm{h}}(z,x)}f(x) \end{align*} cannot be simultaneously convergent for all Bergman functions $f\in A^2(\mathbb{D})$ whenever $1<s<3/2$. This confirms a result announced without proof in Bufetov-Qiu's work. |
| title | A Law of large numbers for vector-valued linear statistics of Bergman DPP |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2404.14978 |