On the pair correlation function of the Sine$_β$ process
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915502580826112 |
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| author | Qu, Yahui Valkó, Benedek |
| author_facet | Qu, Yahui Valkó, Benedek |
| contents | We study the Sine$_β$ process, the bulk point process scaling limit of beta-ensembles. We provide a representation of its pair correlation function for all $β>0$ via a stochastic differential equation. We show that the pair correlation function is continuous in $β$, and provide estimates for its asymptotic decay. We recover the classical explicit formula for the pair correlation function in the $β=2$ and $4$ cases. For $β=2n$, we derive the power series expansion of the pair correlation function, and express it in terms of a size $n$ linear ordinary differential equation system. We obtain our results by studying the density of the $\operatorname{HP}_{β,δ}$ process, the point process limit of the circular Jacobi beta-ensembles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the pair correlation function of the Sine$_β$ process Qu, Yahui Valkó, Benedek Probability Mathematical Physics We study the Sine$_β$ process, the bulk point process scaling limit of beta-ensembles. We provide a representation of its pair correlation function for all $β>0$ via a stochastic differential equation. We show that the pair correlation function is continuous in $β$, and provide estimates for its asymptotic decay. We recover the classical explicit formula for the pair correlation function in the $β=2$ and $4$ cases. For $β=2n$, we derive the power series expansion of the pair correlation function, and express it in terms of a size $n$ linear ordinary differential equation system. We obtain our results by studying the density of the $\operatorname{HP}_{β,δ}$ process, the point process limit of the circular Jacobi beta-ensembles. |
| title | On the pair correlation function of the Sine$_β$ process |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2509.15446 |