  A central theoretical issue at the core of the current research on many-body
localization (MBL) consists in characterizing the statistics of rare long-range
resonances in many-body eigenstates. This is of paramount importance to
understand: (i) the critical properties of the MBL transition and the mechanism
for its destabilization through quantum avalanches; (ii) the unusual transport
and anomalously slow out-of-equilibrium relaxation when the transition is
approached from the metallic side. In order to study and characterize such
long-range rare resonances, we develop a large-deviations approach based on an
analogy with the physics of directed polymers in random media, and in
particular with their freezing glass transition on infinite-dimensional graphs.
The basic idea is to enlarge the parameter space by adding an auxiliary
parameter (which plays the role of the inverse temperature in the directed
polymer formulation) which allows us to fine-tune the effect of anomalously
large outliers in the far-tails of the probability distributions of the
transmission amplitudes between far-away many-body configurations in the
Hilbert space. We first benchmark our approach onto two non-interacting
paradigmatic toy models, namely the single-particle Anderson model on the
(loop-less) Cayley tree and the Rosenzweig-Porter random matrix ensemble, and
then apply it to the study of a class of disordered quantum spin chains in a
transverse field. This analysis shows the existence of a broad disorder range
in which rare, long-distance resonances, that may form only for a few specific
realizations of the disorder and a few specific choice of the random initial
state, destabilize the MBL phase, while the genuine MBL transition is shifted
to much larger values of the disorder than originally thought.
