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

TL;DR

This paper analyzes the distribution of consecutive eigenphase spacings and their ratios in the circular unitary ensemble, revealing finite-size corrections and connecting these findings to the zeros of the Riemann zeta function.

Contribution

It introduces a framework for computing joint distributions of eigenphase spacings and ratios, highlighting finite-size effects and their implications for quantum chaos and number theory.

Findings

Finite-$N$ correction in gap-ratio distribution is of order $N^{-4}$.

Cancellation of $N^{-2}$ terms in joint spacing distributions.

Deviation of Riemann zeta zeros gap-ratio from universal prediction scales as $( ext{log}(T/2 extpi))^{-3}$.

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 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 $\left(\log(T/2\pi)\right)^{-3}$.