Smallest distances between zeros of Gaussian analytic functions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866909000192229376 |
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| author | Feng, Renjie Yao, Dong |
| author_facet | Feng, Renjie Yao, Dong |
| contents | In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a Poisson point process with a universal rate. Furthermore, the locations where these smallest distances occur tend to follow a uniform measure with respect to the volume form. As a consequence, the limiting density of the $k$-th rescaled smallest distance is proportional to $x^{4k-1}e^{-x^4}$ for any $k\geq 1$. Analogous results hold for the classical Gaussian Entire Functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Smallest distances between zeros of Gaussian analytic functions Feng, Renjie Yao, Dong Probability 53C55, 60D05 In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a Poisson point process with a universal rate. Furthermore, the locations where these smallest distances occur tend to follow a uniform measure with respect to the volume form. As a consequence, the limiting density of the $k$-th rescaled smallest distance is proportional to $x^{4k-1}e^{-x^4}$ for any $k\geq 1$. Analogous results hold for the classical Gaussian Entire Functions. |
| title | Smallest distances between zeros of Gaussian analytic functions |
| topic | Probability 53C55, 60D05 |
| url | https://arxiv.org/abs/2604.26316 |