Long-Range Correlation of the Sine$_β$ point Process
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914398077976576 |
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| author | Dumaz, Laure Malvy, Martin |
| author_facet | Dumaz, Laure Malvy, Martin |
| contents | We study the correlations of the celebrated Sine$_β$ point process. This point process arises as the bulk scaling limit of $β$-ensembles and has a geometric description through the Brownian carousel, as shown by Valkó and Virág (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/β$ for large $β$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valkó (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $β>0$ and $k \geq 1$, significantly extending previous results limited to specific values of $β$ or $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-Range Correlation of the Sine$_β$ point Process Dumaz, Laure Malvy, Martin Probability Mathematical Physics We study the correlations of the celebrated Sine$_β$ point process. This point process arises as the bulk scaling limit of $β$-ensembles and has a geometric description through the Brownian carousel, as shown by Valkó and Virág (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/β$ for large $β$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valkó (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $β>0$ and $k \geq 1$, significantly extending previous results limited to specific values of $β$ or $k$. |
| title | Long-Range Correlation of the Sine$_β$ point Process |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2603.15289 |