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) \(\mathcal{L}(e)\) $\mathcal{L}(e)$ 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 \(\Gamma \) $\mathrm{\Gamma }$ . 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 \(e\ge e_{t}\) $e\ge e_t$ a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the \(p=2\) $p=2$ 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 \(e_{c}\ge e_{t}\) $e_c\ge e_t$ , 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 \(\Gamma \rightarrow 0\) $\mathrm{\Gamma }\to 0$ the LDF takes non-trivial scaling forms (i) \(\mathcal{L}(e) \sim G((e-e_c)/\Gamma )\) $\mathcal{L}(e)\sim G((e-e_c)/\mathrm{\Gamma })$ in the vicinity of the wall (ii) \(\mathcal{L}(e) \sim \Gamma ^{\eta \nu } F((e-e_{\textrm{typ}})/\Gamma ^{\nu })\) $\mathcal{L}(e)\sim {\mathrm{\Gamma }}^{\eta \nu }F((e-e_{\text{typ}})/{\mathrm{\Gamma }}^{\nu })$ in the vicinity of the typical energy, characterised by two new exponents \(\eta \ge 1\) $\eta \ge 1$ and \(\nu \) $\nu$ characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of typical fluctuations, identified as \(e-e_{\textrm{typ}} \sim N^{-1/\eta }\) $e-e_{\text{typ}}\sim N^{-1/\eta }$ and \(\Gamma \sim N^{-1/(\eta \nu )}\) $\mathrm{\Gamma }\sim N^{-1/(\eta \nu )}$ .