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We investigate the replica symmetry broken (RSB) phase of spin glass (SG)models in a random field defined on Bethe lattices at zero temperature. Fromthe properties of the RSB solution we deduce a closed equation for the extremevalues of the cavity fields. This equation turns out not to depend on theparameters defining the RSB, and it predicts that the spontaneous RSB does nottake place homogeneously on the whole system. Indeed, there exist spins havingthe same effective local field in all local ground states, exactly as in thereplica symmetric (RS) phase, while the spontaneous RSB manifests only on theremaining spins, whose fraction vanishes at criticality. The characterizationin terms of spins having fixed or fluctuating local fields can be extended alsoto the random field Ising model (RFIM), in which case the fluctuating spins arethe only responsible for the spontaneous magnetization in the ferromagneticphase. Close to criticality we are able to connect the statistics of the localfields acting on the spins in the RSB phase with the correlation functionsmeasured in the paramagnetic phase. Identifying the two types of spins on giveninstances of SG and RFIM, we show that they participate very differently toavalanches produced by flipping a single spin. From the scaling of the numberof spins inducing RSB effects close to the critical point and using the$M$-layer expansion we estimate the upper critical dimension $D_U \geq 8$ forSG.

The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase