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
)
.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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