  We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large
coordination numbers Z (i.e., many tunnelling partners). Via a controlled
expansion into powers of 1/Z, we establish a hierarchy of correlations, which
facilitates an approximate analytic solution of the quantum evolution. For the
Bose-Hubbard model, we derive the growth of phase coherence after a quench from
the Mott to the superfluid phase. For a quench within the Mott phase, we find
that various local observables approach a quasi-equilibrium state after a
finite period of time. However, this state is not thermal, i.e., real
thermalisation -- if it occurs -- requires much longer time scales. For a
tilted lattice in the Mott state, we calculate the tunnelling probability and
find a remarkable analogy to the Sauter-Schwinger effect (i.e.,
electron-positron pair creation out of the vacuum due to a strong electric
field). These analytical results are compared to numerical simulations for
finite lattices in one and two dimensions and we find qualitative agreement.
Finally, we generalize these studies to the more involved case of the
Fermi-Hubbard model.
