  Identifying and measuring the "localization length'' in many-body systems in
the vicinity of a many-body localization transition is difficult. Following
Hatano and Nelson, a recent work (Heuben, White, Refael, PRB 103, 064201
(2021)) introduced an "imaginary vector potential'' to a disordered ring of
interacting fermions, in order to define a many-body localization length
(corresponding, in the non-interacting case, to the end-to-end Green's function
of the hermitian system). We extend these results, by connecting this
localization length to the length scale appearing in the avalanche model of
delocalization. We use this connection to derive the distribution of the
localization length at the MBL transition, finding good agreement with our
numerical observations. Our results demonstrate how a localization length
defined as such probes the localization of the underlying ring, without the
need to explicitly construct the l-bits.
