  We propose a multi-scale diagonalization scheme to study disordered
one-dimensional chains, in particular the transition between many-body
localization (MBL) and the ergodic phase, expected to be governed by resonant
spots. Our scheme focuses on the dichotomy MBL versus {ETH} (eigenstate
thermalization hypothesis). We show that a few natural assumptions imply that
the system is localized with probability one at criticality. On the ergodic
side, delocalization is induced by a quantum avalanche seeded by large ergodic
spots, whose size diverges at the transition. On the MBL side, the typical
localization length tends to a finite universal value at the transition, but
there is a divergent length scale related to the response to an inclusion of
large ergodic spots. A mean field approximation analytically illustrates these
results and predicts as a power-law distribution for thermal inclusions at
criticality.

Disordered Systems and Neural Networks;Statistical Mechanics