On the nature of the low-temperature phase in discontinuous mean-field spin glasses

14 Pages·0.23 MB·English

by  Andrea Montanari

3 On the nature of the low-temperature phase in
0
0
discontinuous mean-field spin glasses
2

n
a
J Andrea Montanari

0
3 Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure∗
24, rue Lhomond, 75231 Paris CEDEX 05, FRANCE

] E-mail: [email protected]
n
n
-
s Federico Ricci-Tersenghi
i
d
Dipartimento di Fisica and SMC and UdR1 of INFM
.
t
a Universita` di Roma ”La Sapienza”
m Piazzale Aldo Moro 2, I-00185 Roma, ITALY
- E-mail: [email protected]
d
n
o
c January 30, 2003
[

1
v
1
9
Abstract
5
1
The low-temperature phase of discontinuous mean-field spin glasses is generally described
0
3 by a one-step replica symmetry breaking (1RSB) Ansatz. The Gardner transition, i.e. a very-
0
low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often
/
t regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the
a
m metastable states which are relevant for the out-of-equilibrium dynamics of such systems are
- always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin
d
spherical model and the random energy model (REM). We also discuss the consequences of
n
o our results for aging dynamics and for local search algorithms in hard combinatorial problems.
c
:
v
i
X
r
a
∗
UMR 8549, Unit´e Mixte de Recherche du Centre National de la Recherche Scientifique et de l’ Ecole Normale
Sup´erieure.
q(x)
Σ(f)
m x
q(x)
m x
f f f ’ f
s G d d
Figure 1: The complexity curve for a generic discontinuous mean-field spin glass. The dashed line
is the 1RSB approximation, while the continuous line is the exact result. The two coincide below
f . The gray region is FRSB. The shape of the Parisi order parameter is shown in the insets.
G
1 Introduction and main results
During the last decade the p-spin spherical spin glass has been thoroughly investigated both
in its statical and in its dynamical behavior [1,2,3,4]. In fact this model is usually considered
the prototypical example of discontinuous mean-field spin glasses. The latter, in turn, have
beenarguedtobecrucialforunderstandingthestructuralglass transition [5]. Inthispaperwe
show that the p-spin spherical model is indeed quite an exceptional case among discontinuous
mean-fieldmodels. Whileourresults donotinvalidate theinsightgained sofar, they addsome
new important feature to the general picture.
The out-of-equilibrium dynamics of the p-spin spherical models is closely related to the
structure of metastable states [6]. Below the dynamical temperature T , the Gibbs measure
d
decomposes among an exponential number of metastable states, whose free-energy densities
lie between two values f and f . The relevant quantity in this regime is the complexity (or
s d
configurational entropy) Σ (f). This is defined in terms of the number (f) of metastable
T T
N
states having free energy density f,
(f) exp NΣ (f) , (1.1)
T T
N ∼ { }
N being the number of degrees of freedom of the system. Σ (f) is strictly positive between
T
f and f , with an absolute maximum at f . The free-energy density f (T) of the pure states
s d d ∗
which dominate the Gibbs measure, results from the balance between energetic and entropic
considerations. While the former would privilege the states at f , the latter favor the states
s
at f .
d
On the other hand, in a typical out-of-equilibrium set up, the system is rapidly cooled
below T from its high-temperature phase [3]. In the thermodynamic limit, the system never
d
equilibrates and its behavior is dominated by the most numerous metastable states, i.e. the
ones at f . This means that, for any single-time observable (t), the following identity holds
d
O
lim lim (t) = lim , (1.2)
t→∞N→∞hO i N→∞hOifd
2
where the average on the left-hand side is taken with respect to thermal histories, while
on the right-hand side it is taken with respect to a constrained Gibbs measure. Outside
the thermodynamic limit, the system eventually equilibrates on an exponential time scale
t (N) = exp O(N) .
erg
{ }
The complexity Σ (f) can be calculated within a 1RSB scheme [7]. This calculation is
T
known to be correct for the p-spin spherical model [1]. This does not rule out the possibility
of further replica-symmetry breakings for more general cases: one would then expect a FRSB
calculation to be necessary. The consequences of FRSB on the above picture have not been
investigated so far. However, it is usually thought that FRSB would play a minor role. The
intuition, as far as we can understand it, goes as follows. As shown in 1984 by Elisabeth
Gardner [8] in the case of the p-spin Ising model, discontinuous spin glasses may have a
FRSB phase which overcomes the 1RSB phase at very low temperature. The common wisdom
associates low temperature to low energy: if any FRSB effect is present, it affects, at most,
the lowest part of the complexity spectrum. As a consequence, out-of-equilibrium dynamics
is unaffected by FRSB. Here we show that this intuition is incorrect and that FRSB plays an
important role also for discontinuous mean-field glasses.
The correct scenario is sketched in Fig. 1. The 1RSB solution is stable with respect to
FRSB up to some value f of the free-energy density. At higher free energies, the 1RSB
G
solution becomes unstable and a FRSB calculation is required. Quite generally f < f
G d
strictly. We know just two exceptions to this rule (both are somehow degenerate cases): the
spherical model and the REM [9]. On the contrary both the situations f < f and f > f
G s G s
are possible. A thermodynamic FRSB phase transition occurs when f crosses f (T). Above
G ∗
f , the FRSB calculation will give a new complexity curve, and, in particular a new threshold
G
free energy f′. Of course, out-of-equilibrium dynamics will be dominated (in the sense of Eq.
d
(1.2)) by the states at f′. Finally, also the nature of aging dynamics will change. We expect
d
relaxation to be characterized by an infinite number of time sectors [10,11] (instead of just
two), although two of them will be most relevant (namely the first and the last one).
In the next Sections we shall illustrate the general scenario with two examples which are,
at the same time, simple and representative. In Sec. 2 we reconsider the fully-connected
Ising p-spin model. The simplicity of this example allows us to study the problem at finite
temperatures. In Sec. 3 we turn to finite connectivity models at zero temperature. In this case
we can compare our results with numerical simulations.
2 Infinite connectivity
The fully-connected p-spin model is defined by the Hamiltonian
(σ) = J σ σ ...σ , (2.1)
H − i1i2...ip i1 i2 ip
(iX1...ip)
where the N variables σ = 1 are Ising spins and the J are quenched random inter-
i ± i1i2...ip
actions extracted from a Gaussian distribution with zero mean and variance p!/(2Np−1). For
illustration we shall often consider the p = 3 case, although our results are qualitatively valid
for any finite p > 2.
Using the standard replica formalism with a 1RSB Ansatz, one obtains the action [8]
β 1 1
φ(β,m) = 1+(p 1)(1 m)qp pqp−1 log2 log z coshm(βλz) , (2.2)
−4 − − − − β − βm D
Z
(cid:2) (cid:3)
3
T T
s d
1
m (T)
1RSB s
0.8
stable
q > 0
region
0.6
m
0.4 m (T)
d
q = 0
m (T) q > 0
0.2 G unstable
region
0
0 0.1 0.2 T 0.3 T 0.4 0.5 0.6 0.7
G
Figure 2: 3-spin fully-connected Ising model: the Parisi parameter m for thermodynamic states
(dotted-dashed curve) and for threshold states (dashed curve) as a function of the temperature. In
the shaded region, 1RSB solutions are unstable with respect to further replica symmetry breakings.
where z e−z2/2dz/√2π, λ p qp−1 and q is determined by the saddle point equation
D ≡ ≡ 2
q
z coshm(βλz)tanh2(βλz)
q = D . (2.3)
z coshm(βλz)
R D
From the action (2.2) we have the usualRparametric representation [7] of the complexity Σ (f)
T
at temperature T = 1/β:
Σ = βm2∂ φ(β,m) , f = ∂ [mφ(β,m)] . (2.4)
m m
The outcome of this formulae agrees with the general picture sketched in Fig. 1, that is Σ (f)
T
is positive for f [f ,f ], which corresponds to m [m ,m ]. Both the extrema of this range
s d s d
∈ ∈
depend on the temperature. The internal energy of metastable states can be calculated using
the identity
1 β
e(β,m) = J σ σ ...σ = 1 (1 m)qp , (2.5)
−N i1i2...iph i1 i2 ipi −2 − −
(i1X,...,ip) h i
where q satisfies Eq. (2.3).
The thermodynamics of the model at temperature T (within the 1RSB approximation)
is obtained maximizing φ(T,m) with respect to m [0,1]. The resulting m (T) is shown in
s
∈
Fig. 2 (dotted-dashed curve): it becomes smaller than 1 at the critical temperature T and it
s
vanishes for T 0 as m µ T. Threshold states, those maximizing the complexity Σ, are
s s
→ ∼
obtained fixing m = m (T), which is shown in Fig. 2 with a dashed curve: m (T) becomes
d d
smaller than 1 at the dynamical critical point T and vanishes as m µ T.
d d d
∼
The stability of the 1RSB solution with respect to a second step (and eventually infinite
steps) of replica symmetry breaking can be evaluated following Ref. [8]. For any given T and
4
m the 1RSB solution is stable provided
2 z coshm−4(βλz)
> D . (2.6)
p(p 1)β2qp−2 zcoshm(βλz)
− R D
InFig.2theregionwherereplicasymmetryshoRuldbebrokenmorethanoncehasbeenshaded.
We call m (T) its boundary. In Ref. [8] Elizabeth Gardner calculated the critical point T
G G
where thermodynamic states are no longer 1RSB (this point was called T in Ref. [8]). Still
2
more interesting is the fact that threshold states are always in the unstable region.
The physical scenario already described in Sec. 1 (see Fig. 1) is confirmed for any temper-
ature below T . In general we find that f is strictly less than f , while both the inequalities
d G d
f < f and f > f are possible, depending on the temperature. For f < f the 1RSB
G s G s G
solution is correct and the Parisi order parameter q(x) has a single step at x = m (see lower
inset in Fig. 1 and please remind that in the absence of an external field q = 0). For f > f ,
0 G
the FRSB calculation will give an order parameter q(x) with a continuous non-trivial part
above the step (see upper inset in Fig. 1), a new complexity curve and, in particular, a new
threshold free energy f′. The geometrical structure of metastable states above f is therefore
d G
the following. There is an exponential number of families of states, each one having a FRSB
structure inside. The typical overlap between two families is 0, while the minimum overlap
between states of the same family is given by the step size in the function q(x).
Let us suggest a natural generalization of the above method, for computing Σ(f) above
f . Given the free-energy functional [12] Φ [T,q(x)], one can parameterize q(x) by the
G FRSB
jump location m and by the continuous function r(x) for x [m,1]. Then one can define
∈
φ(T,m) = max Φ [T,q(x)], by maximizing the free-energy functional for fixed T and
r(x) FRSB
m1. Finally the complexity is obtained by taking the Legendre transform of φ(T,m) as in Eq.
(2.4). Thecomplete calculation can bedoneusinge.g. the numericalmethod of Ref.[13]. Here
we show the result of a 2RSB approximated calculation at zero temperature.
In the T 0 limit, the 1RSB free energy is simply given by
→
µ 1 √p
φ1RSB(µ) = ln 1+erf µ , (2.7)
T=0 −4 − µ 2
(cid:20) (cid:18) (cid:19)(cid:21)
where µ = lim βm and erf(x) 2 xdte−t2/√π. The resulting complexity Σ(e) is shown
β→∞ ≡ 0
in Fig. 3 (dashed curve).
R
The2RSBfreeenergyφ2RSB(µ ,µ )dependsontwonumberswhichparameterize, asinthe
T=0 1 2
1RSB case, the zero temperature limit of the Parisi breaking parameters. The three overlaps
behaves as follows: q = 0, q = q (with 0 < q < 1 strictly) and q 1 ωT. The saddle
0 1 2
≃ −
point equation for q reads
z Iν−2(z,q,µ ) I2(z,q,µ )
q = D + 2 − 2 , (2.8)
z Iν(z,q,µ )
R D + 2
eµ2λz ηµ λz e−µ2λz ηµ λz
R 2 2
I (z,q,µ ) = 1+erf + 1+erf , (2.9)
± 2
2 2 η ± 2 2 − η
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:21)
1This meansthatonehastomaximizeΦFRSB[T,q(x)]amongallthe orderparametersq(x) suchthatq(x)=0for
x<m. Letus callmFsRSB the positionofthe discontinuityinthe staticalFRSBsolutionqs(x). Ifm<mFsRSB,there
is one trivial solution to this maximization problem: q(x) = qs(x). We expect a non-trivial secondary maximum to
exist. On such a maximum the discontinuity in q(x) is located at m. This expectation is indeed confirmed by our
2RSB calculation.
5
0.07
0.06
0.05 1RSB
0.04
Σ
0.03
0.02 2RSB
0.01
0
-0.82 -0.81 -0.8 -0.79 -0.78 -0.77 -0.76 -0.75
f = e
Figure 3: T = 0 complexity as a function of the (free) energy density for the 3-spin fully-connected
Ising model, obtained with 1RSB and 2RSB approximations.
where ν µ /µ , λ pqp−1 and η p(1 qp−1). At the saddle point, the expressions
≡ 1 2 ≡ 2 ≡ −
for the action and the eqnergy simplify to p
µ 1
φ(µ ,µ ) = 2 1+(p 1)(1 ν)qp pqp−1 ln z Iν(z,q,µ ) , (2.10)
1 2 − 4 − − − − µ D + 2
1
h i Z
1
e(µ ,µ ) = pω(q,µ )+µ (µ µ )qp , (2.11)
1 2 2 2 2 1
−2 − −
ω(q,µ2) = √2πhpe−14ν2µ22R Dz I+ν−D1(zzIp+ν(1z−,qq,pµ−i21),q,µ2) , (2.12)
R
where, as usual, q satisfies the saddle point equation (2.8). In the interesting region of pa-
rameters the shape of φ(µ ,µ ) is that of 2 com-penetrating paraboloids (see Fig. 4): On the
1 2
left paraboloid we have that q = 0 and we recover the 1RSB solution with µ = µ , while on
2
the right paraboloid q takes non-trivial values and we find there the absolute maximum of
φ(µ ,µ ), correspondingto the groundstate energy. Please note that, on the right paraboloid,
1 2
the most relevant variable is the smaller breaking parameter µ .
1
We proceed by maximizing φ(µ ,µ ) over µ , and Legendre-transforming the result with
1 2 2
respect to µ . The result is shown in Fig. 3 (continuous line). It is worth mentioning at least
1
two important facts:
The 2RSB threshold energy e2RSB is much lower than the 1RSB one e1RSB.
• d d
Intheregionbetweene2RSB ande2RSB thecomplexitycurvedoesnotchangesignificantly.
• s d
In general we expect Σ(e) not to change below e , but in this case e < e2RSB.
G G s
How does this new physical scenario affect observable quantities, like the internal energy?
In the main panel of Fig. 5 we plot the thermodynamic energy e (T), the threshold states
s
energy e (T), and the internal energy e (T) at the instability point, for the fully-connected
d G
6
φ(µ ,µ )
1 2
-0.81
-0.82
-0.83
-0.84
1
2
1.5
2
µ2 1 µ
3 0.5 1
0
Figure 4: Replicated free-energy for the 3-spin fully-connected Ising model at T = 0 as a function
of the breaking parameters. The bold line is the result of a maximization over µ at fixed µ . It
2 1
ends on the border of the region with q = 0 (left paraboloid). Notice that the trivial maximum in
this region has to be discarded when computing the complexity, since it has ∂ φ(µ ,µ ) = 0.
µ1 1 2
-0.7
small connectivities T
d
-0.72
-0.74
1RSB
e
-0.76 d,T=0
e (T)
d
-0.78
2RSB
e
d,T=0
e (T)
G
-0.8
e (T)
s
-0.82
T T
0 0.1 0.2 G 0.3 0.4 0.5 0.6 s 0.7
Figure 5: Thermodynamic energy (dotted-dashed line) and threshold energy (dashed line) as a
function of the temperature within the 1RSB approximation. States above the full line are unsta-
ble with respect to further replica symmetry breakings. The arrows mark the threshold energies
calculated directly at T = 0. Main panel: 3-spin fully-connected Ising model. Inset: pictorial view
for a 3-spin with small connectivity.
7
3-spin model. The incorrectness of the 1RSB Ansatz for threshold states is clear from the
unphysical behavior of e (T), which is not a monotonously increasing function of T: an an-
d
nealing experiment with a very slow cooling, following 1RSB threshold states, would produce
an increase of energy when temperature is decreased!
The energy obtained with an extremely slow cooling, i.e. that of threshold states, must
lie below e (T) and above both e (T) and e (T), and it must be a monotonously increasing
d s G
functionof T. We can estimate e (T = 0), from thezero-temperature 2RSBcalculation above.
d
The result is marked by an arrow in Fig. 5. As already noticed, this value is much lower than
what predicted by an 1RSB calculations. Such a conclusion has positive consequences on the
use of simulated annealing and related techniques for the search of low energy solutions in
hard combinatorial problems.
However, the situation is not always like the one just described. In some cases, e (T) >
G
e (T)foralltemperaturesbelow T (see thesketch intheinsetofFig. 5). Thisiswhatactually
s d
happens for p-spin models with small connectivities, as will be shown in the next Section.
3 Finite connectivity and numerical simulations
Finite connectivity mean-field models have been the object of intense investigation in the last
years. On one hand, they are thought to share some properties of finite-dimensional models
(namely each spin interact with a finite number of neighbors). On the other hand, they are
closelyrelatedtoextremelyhardcombinatorialoptimizationproblems[14]. Nevertheless, upto
now, the theoretical investigations have been limited to the 1RSB level [15,16]. Here we want
to show how the 1RSB phase becomes unstable with respect to 2RSB fluctuations, and how
to compute the stability threshold. We expect that, as usual, once the 1RSB phase becomes
unstable, FRSB has to be used for properly describing the system.
For sake of simplicity, wepresent our calculation in a particularly simplecase, which allows
a fully explicit derivation. We shall comment later on the generalizations of our results. To
be definite, let us consider a fixed-connectivity Ising model with p-spin interactions (hereafter
p > 2) with Hamiltonian
(σ) = J σ ...σ . (3.1)
H − i1...ip i1 ip
(i1.X..ip)∈G
In the above formula is the hypergraph of interactions, i.e. a set of M among the N
G p
possible p-uples of the N spins. Here we shall take to have fixed connectivity: each spin is
G (cid:0) (cid:1)
supposed to participate to (l +1) interaction terms. Finally we shall consider the couplings
J to take the values 1 with equal probability.
i1...ip ±
Under these hypothesis the 2RSB order parameter is a normalized measure on a space of
probability distributions and does not dependuponthe siteof the sample[17]. Themean-field
equations are most conveniently written in terms of two such measures: [ρ] and [ρ]. At
Q Q
zero temperature they have the form
b b
l
1
[ρ] = d [ρ(α)] z[ ρ(α) ]µ1/µ2δ ρ ρ(0)[ ρ(α) ] , (3.2)
Q Q { } − { }
Z Z αY=1 h i
p−1 b b b b
[ρ] = d [ρ(i)]δ ρ ρ(0)[ ρ(i) ] , (3.3)
Q Q − { }
Z Yi=1 h i
b b b b
8
where isanormalizationconstantand0 < µ µ < arethe2RSBparametersintheβ
1 2
Z ≤ ∞ →
limit. Theprobabilitydistributionρissupportedovertheintegersqsuchthat 1 q +1.
∞ − ≤ ≤
Itisthereforegivenintermsofthreepositivenumbers: ρ = ρ ,ρ ,ρ (withρ +ρ +ρ = 1).
+ 0 − + 0 −
{ }
The distribution ρ is instead supporbted over the integers l q l. However, for our
− ≤ ≤
purposes, we can parametrize it using the three numbebrs ρb b bl ρ ,ρ ,bρ =b −b1 ρ .
{ + ≡ q=1 q 0 − q=−l q}
The “functionals” [ρ] and [ρ] are therefore nothing but distributions over two-dimensional
Q Q P P
simplexes. Finally, the functions ρ(0)[...] and ρ(0)[...] entering in Eqs. (3.2)-(3.3) are defined
as follows: b b
l b l l
1
ρ(0) = ρ(α) exp µ q q , (3.4)
+,0,− z[ ρ(α) ] qi · (− 2" | i|−| i|#)
{ } {Xqα} αY=1 αX=1 αX=1
qα>0,=0,<0 b
b
1 p−P1(ρ(i)+ρ(i))+ p−1(ρ(i) ρ(i)) if q = +,
2 i=1 + − i=1 + − −
 hQ Q i
ρ(q0) =  1− pi=−11(ρ(+i)+ρ(−i)) if q = 0, (3.5)


Q
b 1 p−1(ρ(i)+ρ(i)) p−1(ρ(i) ρ(i)) if q = ,
 2 i=1 + − − i=1 + − − −

and the constant z[ ρ(α) h]Qis such that ρ(0) isQnormalized. i
{ }
The authors of Ref. [17] considered the 1RSB solution to this problem. This is nothing but
the fixed point (ρ∗,ρb∗) of Eqs. (3.4) and (3.5):
ρ∗ = ρ(0)[ρ∗,...,ρ∗]; ρ∗ = ρ(0)[ρ∗,...,ρ∗]. (3.6)
b
The above equations have always a symmetric solution: ρ∗ = ρ∗, ρ∗ = ρ∗, which is the
b b b b + − + −
physical one. The physical stability of the 1RSB phase coincides with the stability of this
solution under the iteration (3.2)-(3.3). We must therefore considerb how bthe 2RSB order
parameters [ρ], [ρ] can reduce to the 1RSB solution (3.6):
Q Q
The first possibility is that [ρ] and [ρ] concentrate around ρ∗, ρ∗:
• b b Q Q
Q[ρ]≈ f(ρ−ρ∗b),b Q[ρ]≈ f(ρ−ρ∗), b (3.7)
where f(·) and f(·) are supported in a neighbbborhoobdbof 0b. The two distributions on the
two-dimensional simplex , tend, in the 1RSB limit, to delta functions in a particular
Q Q
point (ρ∗,ρ∗) ofbthe simplex.
The stability condition of thbe delta-function solution under perturbations of the type
(3.7) is easbily derived. Define the 2 2 matrices L and L by linearizing Eqs. (3.4), (3.5)
×
around (ρ∗,ρ∗):
b
(0) (0)
Lq,bq′ = ∂∂ρρq(1)(cid:12) , Lq,q′ = ∂∂ρρ(q1)(cid:12) , for q,q′ ∈ {+,−}. (3.8)
q′ (cid:12)(cid:12)ρ∗ bq′ (cid:12)(cid:12)ρ∗
(cid:12) b (cid:12)
(cid:12) (cid:12)
If we call λ thebeige(cid:12)nvalue of their produ(cid:12)cbtL L having the maximum absolute value,
MAX
·
we obtain the stability condition
b
l(p 1) λ < 1. (3.9)
MAX
− ·| |
9
InfactthematricesLandLcanbeeasilydiagonalizedbyusingsymmetryconsiderations:
one of the two eigenvectors is symmetric, and the other is antisymmetric under the
exchange + . The antbisymmetric eigenvalue vanishes for p > 2. The symmetric one
↔ −
can be shown to verify always the stability condition (3.9). This type of instability is
therefore irrelevant for the problem under study.
The second possibility is that the functionals [ρ] and [ρ] concentrate around delta-
• Q Q
function distributions:
b b
[ρ] ρ∗f [ρ δ ]+ρ∗f [ρ δ ]+ρ∗f [ρ δ ], (3.10)
Q ≈ + + − + 0 0 − 0 − − − −
and analogously for [ρ]. In the above expression the distributions f [] are supported
q
Q ·
around 0, and δ (respectively δ , δ ) is the distribution 1,0,0 (respectively 0,1,0 ,
+ 0 − { } { }
0,0,1 ). In other wobrdbs the measures [ρ] and [ρ] concentrate over the corners of the
{ } Q Q
simplex, with weights given by the 1RSB solution.
In order to derive the stability condition it is convbenbient to use variables which are small
near the corners of the simplex. One possible choice is to rewrite the distributions f []
q
·
as functions of the variables
ǫ = ρ δ , (3.11)
± ± q,±
−
andanalogouslyforthedistributionsf []. Noticethattwovariablesaresufficientbecause
q
·
of the normalization constraint. Moreover the signs of ǫ , ǫ are fixed by the value of q.
+ −
It is now easy to linearize the equatiobns for f [], f [] in the limit ǫ 1. If we define
q q ±
· · ≪
ǫ (respectively ǫ ), the average of ǫ (ǫ ) with respect to f [] (f []), we get the
± q ± q ± ± q q
h i h i · ·
equations b
b b b bb
ǫ l T ǫ , ǫ (p 1) T ǫ , (3.12)
σ q qσ,qσ σ q σ q qσ,qσ σ q
h i ≈ h i h i ≈ − h i
q,σ q,σ
X X
bb bb b bb b bbb
where the sums overbbq and q run over +,0, , while the ones over σ and σ run over
{ −}
+, . Explicit formulae for the 6 6 matrices T and T are reported in App. A. As
{ −} ×
before, we look for the eigenbvalue of the product T T which has the largesbt absolute
·
value. If we call ω this eigenvalue, the stability criteribon is
MAX
b
l(p 1) ω < 1. (3.13)
MAX
− ·| |
It turns out that ω depends uniquely on the parameter µ , which can be identified
MAX 2
as the 1RSB parameter. This criterion, unlike (3.9), has a non-trivial content which we
shall consider in the following.
The criterion (3.13) select a range of the 1RSB parameter for which the 1RSB solution is
stable. This range has the form µ > µ . Using the relation between µ and the energy e of
G
metastable states [7], see also theprevious Section, onecan convert this condition as an energy
condition of the form e< e . This confirms our general picture summarized in Fig. 1.
G
In Table 1 we exemplify the general situation by considering the case p = 3. In the low
connectivity regime (l+1) 10 we have e < e < e strictly: the ground state is correctly
s G d
≤
described by 1RSB while high-lying metastable states are unstable to FRSB. The expected
temperature dependence of e in this regime of connectivities, is sketched in the inset of Fig.
d
5. At higher connectivities the ground state becomes unstable too: e < e < e . We know
G s d
10