Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass

We derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to describe the large deviation function (LDF) L(e) for the speed-N atypical fluctuations of the intensive ground-state energy e of a generic spherical spin-glass in the presence of a random external magnetic field of variance Γ . We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground-state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold e≥ e _t a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the p= 2 spherical spin-glass (Fyodorov and Le Doussal in J Stat Phys 154:466, 2014) and the rigorous results by Dembo and Zeitouni (J Stat Phys 159:1306, 2015) which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy e _c≥ e _t , acting as a “wall”, the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from N to a faster growth. In addition, we show that in the limit Γ → 0 the LDF takes non-trivial scaling forms (i) L(e) ∼ G((e- e _c) / Γ) in the vicinity of the wall (ii) L(e) ∼ Γ ^{η ν}F((e- e _typ) / Γ ^ν) in the vicinity of the typical energy, characterised by two new exponents η≥ 1 and ν characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of typical fluctuations, identified as e- e _typ∼ N ^{- 1 / η} and Γ ∼ N ^{- 1 / ( η ν )} .