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

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