  We study the effect that randomness has on long-range interacting systems by
using the ferromagnetic Ising model with $p$-body interactions in random
fields. The case with p=2 yields a phase diagram similar to that of previously
studied models and shows known features that inequivalence of the canonical and
microcanonical ensembles brings with it, for example negative specific heat in
a narrow region of the phase diagram. When p>2, however, the canonical phase
diagram is completely different from the microcanonical one. The temperature
does not necessarily determine the microcanonical phases uniquely, and thus the
ferromagnetic and paramagnetic phases are not separated in such a region of a
conventional phase diagram drawn with the temperature and field strength as the
axes. Below a certain value of the external field strength, part of the
ferromagnetic phase has negative specific heat. For large values of the
external field strength the ergodicity is broken before the phase transition
occurs for p>2. Moreover, for p>2, the Maxwell construction cannot be derived
in a consistent manner and therefore, in contrast to previous cases with
negative specific heat, the Maxwell construction does not bridge the gap
between the ensembles.
