On the Spectral Form Factor for Random Matrices

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.

Automated summary

In the physics literature, the spectral form factor (SFF) is commonly used to test universality for disordered quantum systems. Previous mathematical results were limited to two solvable models. However, using the multi-resolvent local laws, we rigorously prove the physics prediction on SFF for a large class of random matrices, including the monoparametric ensemble. Our findings supplement the recently proven Wigner-Dyson universality and extensive numerics support the accuracy of our formulas in predicting the SFF.