On the pair correlation function of the Sine$_β$ process Article Swipe

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YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2509.15446

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.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2509.15446
PDF: https://arxiv.org/pdf/2509.15446
OA Status: green
OpenAlex ID: https://openalex.org/W4414695966

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DOI: https://doi.org/10.48550/arxiv.2509.15446

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Title: On the pair correlation function of the Sine$_β$ process

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-09-18

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Authors: Yan Qu, Benedek Valkó

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Landing page: https://arxiv.org/abs/2509.15446

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PDF URL: https://arxiv.org/pdf/2509.15446

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