The behavior of the nonlinear susceptibility $chi_3$ and its relation to the spin-glass transition temperature $T_f$, in the presence of random fields, are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue $lambda_{rm AT}$ (replicon) on the random fields is analyzed. Particularly, in absence of random fields, the temperature $T_f$ can be traced by a divergence in the spin-glass susceptibility $chi_{rm SG}$, which presents a term inversely proportional to the replicon $lambda_{rm AT}$. As a result of a relation between $chi_{rm SG}$ and $chi_3$, the latter also presents a divergence at $T_f$, which comes as a direct consequence of $lambda_{rm AT}=0$ at $T_f$. However, our results show that, in the presence of random fields, $chi_3$ presents a rounded maximum at a temperature $T^{*}$, which does not coincide with the spin-glass transition temperature $T_f$ (i.e., $T^* > T_f$ for a given applied random field). Thus, the maximum value of $chi_3$ at $T^*$ reflects the effects of the random fields in the paramagnetic phase, instead of the non-trivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that $chi_3$ still maintains a dependence on the replicon $lambda_{rm AT}$, although in a more complicated way, as compared with the case without random fields. These results are discussed in view of recent observations in the LiHo$_x$Y$_{1-x}$F$_4$ compound.
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