  We investigate in detail the phase diagrams of the p-body +/-J Ising model
with and without random fields on random graphs with fixed connectivity. One of
our most interesting findings is that a thermodynamic spin glass phase is
present in the three-body purely ferromagnetic model in random fields, unlike
for the canonical two-body interaction random-field Ising model. We also
discuss the location of the phase boundary between the paramagnetic and spin
glass phases that does not depend on the change of the ferromagnetic bias. This
behavior is explained by a gauge transformation, which shows that
gauge-invariant properties generically do not depend on the strength of the
ferromagnetic bias for the +/-J Ising model on regular random graphs.
