  The infinite-range-interaction Ising spin glass is considered in the presence
of an external random magnetic field following a trimodal (three-peak)
distribution. The model is studied through the replica method and phase
diagrams are obtained within the replica-symmetry approximation. It is shown
that the border of the ferromagnetic phase may present first-order phase
transitions, as well as tricritical points at finite temperatures. Analogous to
what happens for the Ising ferromagnet under a trimodal random field, it is
verified that the first-order phase transitions are directly related to the
dilution in the fields (represented by $p_{0}$). The ferromagnetic boundary at
zero temperature also exhibits an interesting behavior: for $0<p_{0}<p_{0}^{*}
\approx 0.30856$, a single tricritical point occurs, whereas if
$p_{0}>p_{0}^{*}$ the critical frontier is completely continuous; however, for
$p_{0}=p_{0}^{*}$, a fourth-order critical point appears. The stability
analysis of the replica-symmetric solution is performed and the regions of
validity of such a solution are identified; in particular, the Almeida-Thouless
line in the plane field versus temperature is shown to depend on the weight
$p_{0}$.
