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Statistics of the random matrix spectral form factor

Alex Altland1, Francisco Divi2,3, Tobias Micklitz4, Silvia Pappalardi1, and Maedeh Rezaei1

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Phys. Rev. Research 7, 033138 – Published 11 August, 2025

DOI: https://doi.org/10.1103/n7rj-gwwj

Abstract

The spectral form factor of random matrix theory plays a key role in the description of disordered and chaotic quantum systems. While its moments are known to be approximately Gaussian, corrections subleading in the matrix dimension, D, have recently come to researchers' attention, with conflicting results in the literature. In this work, we investigate these departures from Gaussianity for both circular and Gaussian ensembles. Using two independent approaches—sine-kernel techniques and supersymmetric field theory—we identify the form factor statistics to next leading order in a D−1 expansion. Our sine-kernel analysis highlights inconsistencies with previous studies, while the supersymmetric approach backs these findings and suggests an understanding of the statistics from a complementary perspective. Our findings fully agree with numerics. They are presented in a pedagogical way, highlighting pathways (and pitfalls) in the study of statistical signatures at next leading order, which are increasingly becoming important in applications.

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Phys. Rev. Research 7, 033138– Published 11 August, 2025
  • Received 28 April 2025
  • Accepted 21 July 2025
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