Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros
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| Formato: | Preprint |
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2025
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| _version_ | 1866915410156191744 |
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| author | Nishigaki, Shinsuke M. |
| author_facet | Nishigaki, Shinsuke M. |
| contents | We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed $\mathrm{U}(N)$ matrices (the circular unitary ensemble) using our framework for Jánossy densities in random matrix theory, formulated via the Tracy-Widom system of nonlinear PDEs. Our result shows that the leading finite-$N$ correction in the gap-ratio distribution relative to the universal sine-kernel limit is of $\mathcal{O}(N^{-4})$, reflecting a nontrivial cancellation of the $\mathcal{O}(N^{-2})$ part present in the joint distributions of consecutive spacings. This finding suggests the potential to extract subtle finite-size corrections from the energy spectra of quantum-chaotic systems and explains why the deviation of the gap-ratio distribution of the Riemann zeta zeros $\{1/2+iγ_n\}, γ_n\approx T\gg1$ from the sine-kernel prediction scales as $\left(\log(T/2π)\right)^{-3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros Nishigaki, Shinsuke M. Mathematical Physics Disordered Systems and Neural Networks High Energy Physics - Theory Probability We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed $\mathrm{U}(N)$ matrices (the circular unitary ensemble) using our framework for Jánossy densities in random matrix theory, formulated via the Tracy-Widom system of nonlinear PDEs. Our result shows that the leading finite-$N$ correction in the gap-ratio distribution relative to the universal sine-kernel limit is of $\mathcal{O}(N^{-4})$, reflecting a nontrivial cancellation of the $\mathcal{O}(N^{-2})$ part present in the joint distributions of consecutive spacings. This finding suggests the potential to extract subtle finite-size corrections from the energy spectra of quantum-chaotic systems and explains why the deviation of the gap-ratio distribution of the Riemann zeta zeros $\{1/2+iγ_n\}, γ_n\approx T\gg1$ from the sine-kernel prediction scales as $\left(\log(T/2π)\right)^{-3}$. |
| title | Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros |
| topic | Mathematical Physics Disordered Systems and Neural Networks High Energy Physics - Theory Probability |
| url | https://arxiv.org/abs/2507.10193 |