Bulk asymptotics of the Gaussian $β$-ensemble characteristic polynomial
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915423280168960 |
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| author | Lambert, Gaultier Paquette, Elliot |
| author_facet | Lambert, Gaultier Paquette, Elliot |
| contents | The Gaussian $β$-ensemble (G$β$E) is a fundamental model in random matrix theory. In this paper, we provide a comprehensive asymptotic description of the characteristic polynomial of the G$β$E anywhere in the bulk of the spectrum that simultaneously captures both local-scale fluctuations (governed by the Sine-$β$ point process) and global/mesoscopic log-correlated Gaussian structure, which is accurate down to vanishing errors as $N\to\infty$.
As immediate corollaries, we obtain several important results: (1) convergence of characteristic polynomial ratios to the stochastic zeta function, extending known results from Valko and Virag to the G$β$E; (2) a martingale approximation of the log-characteristic polynomial which immediately recovers the central limit theorem from Bourgade, Mody and Pain; (3) a description of the order one correction to the martingale in terms of the stochastic Airy function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bulk asymptotics of the Gaussian $β$-ensemble characteristic polynomial Lambert, Gaultier Paquette, Elliot Probability Mathematical Physics 60B20 The Gaussian $β$-ensemble (G$β$E) is a fundamental model in random matrix theory. In this paper, we provide a comprehensive asymptotic description of the characteristic polynomial of the G$β$E anywhere in the bulk of the spectrum that simultaneously captures both local-scale fluctuations (governed by the Sine-$β$ point process) and global/mesoscopic log-correlated Gaussian structure, which is accurate down to vanishing errors as $N\to\infty$. As immediate corollaries, we obtain several important results: (1) convergence of characteristic polynomial ratios to the stochastic zeta function, extending known results from Valko and Virag to the G$β$E; (2) a martingale approximation of the log-characteristic polynomial which immediately recovers the central limit theorem from Bourgade, Mody and Pain; (3) a description of the order one correction to the martingale in terms of the stochastic Airy function. |
| title | Bulk asymptotics of the Gaussian $β$-ensemble characteristic polynomial |
| topic | Probability Mathematical Physics 60B20 |
| url | https://arxiv.org/abs/2508.01458 |