Average weighted ratio of consecutive level spacings for infinite-dimensional orthogonal random matrices

TL;DR

This work defines and computes the unfolding-free, weighted average of the logarithmic ratio of consecutive level spacings, $\langle q \rangle$, for infinite-dimensional orthogonal random matrices to quantify quantum ergodicity under time-reversal symmetry. It computes $\langle q \rangle$ via a Painlevé V equation for the orthogonal class, obtaining $\langle q \rangle \approx 0.8100699350$, and contrasts this with Poisson statistics, which yield $\langle q \rangle = 1$. Complementary Wigner-surmise-inspired analytics provide qualitatively accurate estimates for high-dimensional ensembles across symmetry classes, while finite-$N$ analysis reveals slow, approximately power-law convergence toward the infinite-N value. The results offer a robust diagnostic tool for ergodicity and set the stage for extending the analysis to unitary and symplectic cases, as well as to edge statistics and alternative matrix models.

Abstract

The onset of quantum ergodicity is often quantified by the average ratio of consecutive level spacings. The reference values for ergodic quantum systems have been obtained numerically from the spectra of large but finite-dimensional random matrices. This work introduces a weighted ratio of consecutive level spacings, having the propery that the average can be computed numerically for random matrices of infinite dimension. A Painlevé differential equation is solved numerically in order to determine this average for infinite-dimensional orthogonal random matrices, thereby providing a reference value for ergodic quantum systems obeying time-reversal symmetry (provided that the time-reversal operator squares to the identity matrix). A Wigner surmise-inspired analytical calculation is found to yield a qualitatively accurate picture for the statistics of high-dimensional random matrices from each of the symmetry classes. For Poissonian level statistics, a significantly different average is found, indicating that the average weighted ratio of consecutive level spacings can be used as a probe for quantum ergodicity.

Figures (2)

• Figure 1: The distributions of $q$ for high-dimensional orthogonal ($\beta = 1$), unitary ($\beta = 2$) and symplectic ($\beta =4$) random matrices. The histogram data has been obtained numerically from the spectra of $10000$ matrices sampled from circular ensembles of dimension $10,000$ ($\beta = 1,2$) or $5000$ ($\beta = 4$). The Wigner surmise-inspired probability distributions of Eq. \ref{['eq: Wigner-surmise-inspired-q']}, plotted in black, are qualitatively accurate. The distribution for Poissonian level statistics is shown by a dashed line.
• Figure 2: The deviation of $\langle q \rangle_N$ from $\langle q \rangle_\infty$ versus $N$. The data points show numerically obtained averages over the spectra of $10^9$ matrices sampled from the circular orthogonal ensemble. For all data points, the sample standard deviation is less than $0.025$ times the sample mean. The line shows a fitted power law with coefficient $-1.02$. This coefficient has been obtained from a linear least-squares fit on the logarithms of the deviations versus $\ln(N)$. The first data point ($N = 18$) has been omitted in the fitting procedure.