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

Abstract

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 $\{1/2+i{\gamma }_n\},{\gamma }_n\approx T\gg 1$ from the sine-kernel prediction scales as ${\left(\log (T/2\pi )\right)}^{-3}$.

15 pages in ptephy_v1.cls, 11 figures