标题： 圆酉系综连续能级间距的分布及其比值：有限尺寸修正和黎曼$ζ$零点 显示英文标题

标题： Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros

摘要： 我们使用我们在随机矩阵理论中关于Jánossy密度的框架，通过Tracy-Widom非线性PDE系统来计算Haar分布的$\mathrm{U}(N)$矩阵（圆酉系综）中两个连续本征相位间距及其比值的联合分布。 我们的结果表明，相对于普遍的正弦核极限，间隙比分布中的主要有限-$N$修正项为$\mathcal{O}(N^{-4})$，这反映了在连续间距联合分布中存在的$\mathcal{O}(N^{-2})$部分的非平凡抵消。 这一发现表明，可以从量子混沌系统的能谱中提取细微的有限尺寸修正，并解释了黎曼zeta零点的间隙比分布$\{1/2+i\gamma_n\}, \gamma_n\approx T\gg1$相对于正弦核预测的偏差为何按$\left(\log(T/2\pi)\right)^{-3}$的量级变化。 显示英文摘要

摘要： 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 J\'{a}nossy 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 $\{1/2+i\gamma_n\}, \gamma_n\approx T\gg1$ from the sine-kernel prediction scales as $\left(\log(T/2\pi)\right)^{-3}$.