  In the time evolution of isolated quantum systems out of equilibrium, local
observables generally relax to a long-time asymptotic value, governed by the
expectation values (diagonal matrix elements) of the corresponding operator in
the eigenstates of the system. The temporal fluctuations around this value,
response to further perturbations, and the relaxation toward this asymptotic
value, are all determined by the off-diagonal matrix elements. Motivated by
this non-equilibrium role, we present generic statistical properties of
off-diagonal matrix elements of local observables in two families of
interacting many-body systems with local interactions. Since integrability (or
lack thereof) is an important ingredient in the relaxation process, we analyze
models that can be continuously tuned to integrability. We show that, for
generic non-integrable systems, the distribution of off-diagonal matrix
elements is a gaussian centered at zero. As one approaches integrability, the
peak around zero becomes sharper, so that the distribution is approximately a
combination of two gaussians. We characterize the proximity to integrability
through the deviation of this distribution from a gaussian shape. We also
determine the scaling dependence on system size of the average magnitude of
off-diagonal matrix elements.
