  We study the behavior of two-time correlation functions at late times for
finite system sizes considering observables whose (one-point) average value
does not depend on energy. In the long time limit, we show that such
correlation functions display a ramp and a plateau determined by the
correlations of energy levels, similar to what is already known for the
spectral form factor. The plateau value is determined, in absence of degenerate
energy levels, by the fluctuations of diagonal matrix elements, which
highlights differences between different symmetry classes. We show this
behavior analytically by employing results from Random Matrix Theory and the
Eigenstate Thermalisation Hypothesis, and numerically by exact diagonalization
in the toy example of a Hamiltonian drawn from a Random Matrix ensemble and in
a more realistic example of disordered spin glasses at high temperature.
Importantly, correlation functions in the ramp regime do not show
self-averaging behaviour, and, at difference with the spectral form factor the
time average does not coincide with the ensemble average.
