Distributions of Consecutive Level Spacings of Circular Unitary Ensemble and Their Ratio: Finite-Size Corrections and Riemann ζ 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 partial differential equations. 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 points to the possibility of extracting subtle finite-size corrections from the energy spectra of quantum-chaotic systems, and provides a potential explanation of why the deviation of the gap-ratio distribution of the Riemann zeta zeros $\lbrace 1/2+i\gamma _n\rbrace , \gamma _n\approx T\gg 1$ from the sine-kernel prediction scales as $\left(\log (T/2\pi )\right)^{-3}$.