  Scrambling in interacting quantum systems out of equilibrium is particularly
effective in the chaotic regime. Under time evolution, initially localized
information is said to be scrambled as it spreads throughout the entire system.
This spreading can be analyzed with the spectral form factor, which is defined
in terms of the analytic continuation of the partition function. The latter is
equivalent to the survival probability of a thermofield double state under
unitary dynamics. Using random matrices from the Gaussian unitary ensemble
(GUE) as Hamiltonians for the time evolution, we obtain exact analytical
expressions at finite $N$ for the survival probability. Numerical simulations
of the survival probability with matrices taken from the Gaussian orthogonal
ensemble (GOE) are also provided. The GOE is more suitable for our comparison
with numerical results obtained with a disordered spin chain with local
interactions. Common features between the random matrix and the realistic
disordered model in the chaotic regime are identified. The differences that
emerge as the spin model approaches a many-body localized phase are also
discussed.
