Replica-Symmetry Breaking Transitions in the Large Deviations of the Ground-State of a Spherical Spin-Glass

Abstract

AbstractWe derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to describe the large deviation function (LDF) $$\mathcal{L}(e)$$ 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 $$\Gamma $$ 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 $$e\ge e_{t}$$ e ≥ e t a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the $$p=2$$ 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 ≥ 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 $$\Gamma \rightarrow 0$$ Γ → 0 the LDF takes non-trivial scaling forms (i) $$\mathcal{L}(e) \sim G((e-e_c)/\Gamma )$$ L ( e ) ∼ G ( ( e - e c ) / Γ ) in the vicinity of the wall (ii) $$\mathcal{L}(e) \sim \Gamma ^{\eta \nu } F((e-e_{\textrm{typ}})/\Gamma ^{\nu })$$ L ( e ) ∼ Γ η ν F ( ( e - e typ ) / Γ ν ) in the vicinity of the typical energy, characterised by two new exponents $$\eta \ge 1$$ η ≥ 1 and $$\nu $$ ν 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 typ ∼ N - 1 / η and $$\Gamma \sim N^{-1/(\eta \nu )}$$ Γ ∼ N - 1 / ( η ν ) .