  A canonical model for many-body localization (MBL) is studied, of interacting
spinless fermions on a lattice with uncorrelated quenched site-disorder. The
model maps onto a tight-binding model on a `Fock-space (FS) lattice' of
many-body states, with an extensive local connectivity. We seek to understand
some aspects of MBL from this perspective, via local propagators for the FS
lattice and their self-energies (SE's); focusing on the SE probability
distributions, over disorder and FS sites. A probabilistic mean-field theory
(MFT) is first developed, centered on self-consistent determination of the
geometric mean of the distribution. Despite its simplicity this captures some
key features of the problem, including recovery of an MBL transition, and
predictions for the forms of the SE distributions. The problem is then studied
numerically in $1d$ by exact diagonalization, free from MFT assumptions. The
geometric mean indeed appears to act as a suitable order parameter for the
transition. Throughout the MBL phase the appropriate SE distribution is
confirmed to have a universal form, with long-tailed L\'evy behavior as
predicted by MFT. In the delocalized phase for weak disorder, SE distributions
are clearly log-normal; while on approaching the transition they acquire an
intermediate L\'evy-tail regime, indicative of the incipient MBL phase.
