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The fully-connected Ising $p$-spin model has for $p >2$ a discontinuous phasetransition from the paramagnetic phase to a stable state with one-step replicasymmetry breaking (1RSB). However, simulations in three dimension do not looklike these mean-field results and have features more like those which wouldarise with full replica symmetry breaking (FRSB). To help understand how thismight come about we have studied in the fully connected $p$-spin model thestate of two-step replica symmetry breaking (2RSB). It has a free energydegenerate with that of 1RSB, but the weight of the additional peak in $P(q)$vanishes. We expect that the state with full replica symmetry breaking (FRSB)is also degenerate with that of 1RSB. We suggest that finite size effects willgive a non-vanishing weight to the FRSB features, as also will fluctuationsabout the mean-field solution. Our conclusion is that outside the fullyconnected model in the thermodynamic limit, FRSB is to be expected rather than1RSB.

Possible instability of one-step replica symmetry breaking in p-spin Ising models outside mean-field theory