In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.
Communications in Mathematical Physics
We are grateful to the authors of  for sharing with us their insights and preliminary numerical results. We are especially thankful to Stephen Shenker for very valuable advice over several email communications. Helpful comments on the manuscript from Peter Forrester and from the anonymous referees are also acknowledged. Open access funding provided by Institute of Science and Technology (IST Austria).
Cipolloni G, Erdös L, Schröder DJ. On the spectral form factor for random matrices. Communications in Mathematical Physics. 2023. doi:10.1007/s00220-023-04692-y
Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). On the spectral form factor for random matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-023-04692-y
Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “On the Spectral Form Factor for Random Matrices.” Communications in Mathematical Physics. Springer Nature, 2023. https://doi.org/10.1007/s00220-023-04692-y.
G. Cipolloni, L. Erdös, and D. J. Schröder, “On the spectral form factor for random matrices,” Communications in Mathematical Physics. Springer Nature, 2023.
Cipolloni G, Erdös L, Schröder DJ. 2023. On the spectral form factor for random matrices. Communications in Mathematical Physics.
Cipolloni, Giorgio, et al. “On the Spectral Form Factor for Random Matrices.” Communications in Mathematical Physics, Springer Nature, 2023, doi:10.1007/s00220-023-04692-y.