  Using the non-interacting Anderson tight-binding model on the Bethe lattice
as a toy model for the many-body quantum dynamics, we propose a novel and
transparent theoretical explanation of the anomalously slow dynamics that
emerges in the bad metal phase preceding the Many-Body Localization transition.
By mapping the time-decorrelation of many-body wave-functions onto Directed
Polymers in Random Media, we show the existence of a glass transition within
the extended regime separating a metallic-like phase at small disorder, where
delocalization occurs on an exponential number of paths, from a bad metal-like
phase at intermediate disorder, where resonances are formed on rare, specific,
disorder dependent site orbitals on very distant generations. The physical
interpretation of subdiffusion and non-exponential relaxation emerging from
this picture is complementary to the Griffiths one, although both scenarios
rely on the presence of heavy-tailed distribution of the escape times. We
relate the dynamical evolution in the glassy phase to the depinning transition
of Directed Polymers, which results in macroscopic and abrupt jumps of the
preferred delocalizing paths when a parameter like the energy is varied, and
produce a singular behavior of the overlap correlation function between
eigenstates at different energies. By comparing the quantum dynamics on
loop-less Cayley trees and Random Regular Graphs we discuss the effect of
loops, showing that in the latter slow dynamics and apparent power-laws extend
on a very large time-window but are eventually cut-off on a time-scale that
diverges at the MBL transition.
