  In the presence of interactions, periodically-driven quantum systems
generically thermalize to an infinite-temperature state. Recently, however, it
was shown that in random kicked rotors with local interactions, this long-time
equilibrium could be strongly delayed by operating in a regime of weakly
fluctuating random phases, leading to the emergence of a metastable thermal
ensemble. Here we show that when the random kinetic energy is smaller than the
interaction energy, this system in fact exhibits a much richer dynamical phase
diagram, which includes a low-energy pre-thermal phase characterized by a
light-cone spreading of correlations in momentum space. We develop a
hydrodynamic theory of this phase and find a very good agreement with exact
numerical simulations. We finally explore the full dynamical phase diagram of
the system and find that the transition toward full thermalization is
characterized by relatively sharp crossovers.

Quantum Gases;Quantum Physics