The spectral form factor is believed to exhibit a special type of behavior called “dip-ramp-plateau” in chaotic quantum systems that originates from random matrix theory. This suggests that the shape of the spectral form factor could serve as an indicator of chaos in various quantum systems. It has been shown recently that the dip-ramp-plateau structure appears in the spectral form factor when the normal modes of a massless scalar field theory in the brick-wall model of the Bañados-Teitelboim-Zanelli black hole are treated as eigenvalues of a quantum Hamiltonian. At the same time, the level spacing distribution of these normal modes differs from that associated with random matrix theory ensembles if no averaging over randomly distributed boundary conditions is taken into account. In this paper, we extend the results for BTZ background to the case of a nonzero mass of the field, study the generalized spectral form factor, and consider the same context for another nontrivial background—de Sitter space. We compare the generalized spectral form factor for simple integrable quantum systems and for backgrounds with a horizon to the behavior predicted by random matrix theory. As a result, we confirm that BTZ and de Sitter brick-wall models without averaging exhibit the dip-ramp-plateau structure of the spectral form factor but differ in the structure of the three-level generalized spectral form factor from the one predicted by random matrix theory.
Ageev et al. (2025) studied this question.