## Abstract

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}
^{)} .

Original language | English |
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Article number | 11 |

Journal | Journal of Statistical Physics |

Volume | 191 |

Issue number | 2 |

Early online date | 27 Jan 2024 |

DOIs | |

Publication status | Published - Feb 2024 |