  In order to elucidate the relationship between rate-independent hysteresis
and metastability in disordered systems driven by an external field, we study
the Gaussian RFIM at T=0 on regular random graphs (Bethe lattice) of finite
connectivity z and compute to O(1/z) (i.e. beyond mean-field) the quenched
complexity associated with the one-spin-flip stable states with magnetization m
as a function of the magnetic field H. When the saturation hysteresis loop is
smooth in the thermodynamic limit, we find that it coincides with the envelope
of the typical metastable states (the quenched complexity vanishes exactly
along the loop and is positive everywhere inside). On the other hand, the
occurence of a jump discontinuity in the loop (associated with an infinite
avalanche) can be traced back to the existence of a gap in the magnetization of
the metastable states for a range of applied field, and the envelope of the
typical metastable states is then reentrant. These findings confirm and
complete earlier analytical and numerical studies.
