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  • Open Access

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

Wouter Buijsman*

  • *Contact author: buijsman@pks.mpg.de
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Phys. Rev. E 113, 014111 – Published 12 January, 2026

DOI: https://doi.org/10.1103/ny3r-6cmg

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.

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Phys. Rev. E 113, 014111– Published 12 January, 2026
  • Received 21 August 2025
  • Accepted 24 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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