In order to determine generic non-equilibrium dynamics of the generic local observable O one needs two ingredients. The spectrum of the many-body Hamiltonian and the matrix elements of the operator O in the basis of eigenstates completely specify evolution of the local observable O. Moreover, the matrix elements at small energy differences correspond to the long-time dynamics that may be very hard to access in the thermalizing system.
Using these insights, we studied the properties of the matrix elements across the many-body localization transition in the recent preprint arXiv:1610.02389. In particular, we used matrix elements to define the (many-body) Thouless energy ETh which corresponds to the inverse time scale of relaxation. The behavior of this Thouless energy reveals the critical fan preceding localization transition. In addition, we associate the critical region with the onset of broad distributions and typicality breakdown.