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Using the non-interacting Anderson tight-binding model on the Bethe latticeas a toy model for the many-body quantum dynamics, we propose a novel andtransparent theoretical explanation of the anomalously slow dynamics thatemerges in the bad metal phase preceding the Many-Body Localization transition.By mapping the time-decorrelation of many-body wave-functions onto DirectedPolymers in Random Media, we show the existence of a glass transition withinthe extended regime separating a metallic-like phase at small disorder, wheredelocalization occurs on an exponential number of paths, from a bad metal-likephase at intermediate disorder, where resonances are formed on rare, specific,disorder dependent site orbitals on very distant generations. The physicalinterpretation of subdiffusion and non-exponential relaxation emerging fromthis picture is complementary to the Griffiths one, although both scenariosrely on the presence of heavy-tailed distribution of the escape times. Werelate the dynamical evolution in the glassy phase to the depinning transitionof Directed Polymers, which results in macroscopic and abrupt jumps of thepreferred delocalizing paths when a parameter like the energy is varied, andproduce a singular behavior of the overlap correlation function betweeneigenstates at different energies. By comparing the quantum dynamics onloop-less Cayley trees and Random Regular Graphs we discuss the effect ofloops, showing that in the latter slow dynamics and apparent power-laws extendon a very large time-window but are eventually cut-off on a time-scale thatdiverges at the MBL transition.

Anomalous dynamics in the ergodic side of the Many-Body Localization transition and the glassy phase of Directed Polymers in Random Media