  We analyze the unusual slow dynamics that emerges in the bad metal
delocalized phase preceding the Many-Body Localization transition by using
single-particle Anderson Localization on the Bethe lattice as a toy model of
many-body dynamics in Fock space. We probe the dynamical evolution by measuring
observables such as the imbalance and equilibrium correlation functions, which
display slow dynamics and power-laws strikingly similar to the ones observed in
recent simulations and experiments. We relate this unusual behavior to the
non-ergodic spectral statistics found on Bethe lattices. We discuss different
scenarii, such as a true intermediate phase which persists in the thermodynamic
limit versus a glassy regime established on finite but very large time and
length-scales only, and their implications for real space dynamical properties.
In the latter, slow dynamics and power-laws extend on a very large time-window
but are eventually cut-off on a time-scale that diverges at the MBL transition.
