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

TL;DR

This paper introduces a weighted ratio of consecutive level spacings for infinite-dimensional orthogonal random matrices, providing a new reference for quantum ergodicity and distinguishing between ergodic and Poissonian spectra.

Contribution

It develops a method to compute the average ratio for infinite-dimensional matrices using a Painlevé differential equation, advancing the analysis of quantum ergodicity.

Findings

Numerical solution of Painlevé equation for infinite-dimensional orthogonal matrices.

Analytical approximation inspired by Wigner surmise.

Significant difference in ratios for Poissonian statistics.

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 Painlev\'e 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.