Abstract We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed 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 O (N^-4), reflecting a nontrivial cancellation of the 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 1/2 + iγn, γn ≈ T ≫ 1 from the sine-kernel prediction scales as (log (T/2π) ) −3.
Shinsuke M. Nishigaki (Sat,) studied this question.