An approach to chaos in some mixed p-spin models

0.19 MB·English

by  Wei-Kuo Chen

An approach to chaos in some mixed p-spin models
Wei-Kuo Chen Dmitry Panchenko
∗ †
January 12, 2012
2
1
0
2

n Abstract
a
J
We consider the problems of chaos in disorder and temperature for coupled
0
copies of the mixed p-spin models. Under certain assumptions on the parameters of
1
the models we will first prove a weak form of chaos by showing that the overlap is

]
concentrated around its Gibbs average depending on the disorder and then obtain
R
P several results toward strong chaos by providing control of the overlap between two
. systems in terms of their Parisi measures.
h
t
a
Keywords: spin glass models, stability, chaos
m
[

1 1 Introduction and main results.
v
8
9 The phenomenon of chaos in disorder and temperature in spin glasses was discovered
1
in [5] and [1] and has been studied extensively in the context of various models in the
2
. physics literature (e.g. see [17] for a recent review). In recent years, several mathematical
1
0 results have also been obtained. An example of chaos in external field for the spherical
2 Sherrington-Kirkpatrickmodelwasgivenin[16], chaosindisorder formixedp-spinmodels
1
with even p 2 and without external field was considered in [2], [3] (among many other
:
v ≥
results) and a more general situation in the presence of external field was handled in [4].
i
X
In this paper we will develop an approach to chaos in disorder and temperature for mixed
r
a p-spin models which is based on a novel application of the Ghirlanda-Guerra identities [6]
used here to derive a new family of identities in the setting of the coupled systems. At
the moment, our approach only works under certain assumptions on the parameters of
the models but these new examples are still welcome considering paucity of the results in
this direction. Given N 1, let us consider pure p-spin Hamiltonians H (σ) for p 1
N,p
≥ ≥
indexed by σ Σ = 1,+1 N,
N
∈ {− }
1
H (σ) = g σ ...σ , (1)
N,p N(p−1)/2 i1,...,ip i1 ip
X
1≤i1,...,ip≤N
∗Department of Mathematics, University of California at Irvine, email: [email protected].
†Department of Mathematics, Texas A&M University, email: [email protected]. Partially
supported by NSF grant.
1
where random variables (g ) are standard Gaussian independent for all (i ,...,i )
i1,...,ip 1 p
and p 1. The covariance of this Gaussian process can be easily computed and is given
≥
by
1
EH (σ1)H (σ2) = R(σ1,σ2) p, (2)
N,p N,p
N
(cid:0) (cid:1)
where quantity R(σ1,σ2) = N−1 N σ1σ2 is called the overlap of spin configurations
i=1 i i
σ1,σ2 Σ . Let us define a mixedPp-spin Hamiltonian by a linear combination
N
∈
H (σ) = β H (σ) (3)
N p N,p
X
p≥1
with coefficients (β ) that decrease fast enough to ensure that the process is well defined,
p
for example, 2pβ2 < . The Gibbs measure G (σ) on Σ is defined by
p≥1 p ∞ N N
P
expH (σ)
G (σ) = N ,
N
Z
N
where the normalizing factor Z is called the partition function. The behavior of the
N
Gibbs measure is intimately related to the computation of the free energy N−1logZ in
N
the thermodynamic limit and, as a result, has been studied extensively since the ground-
breaking work of G. Parisi in [9], [10]. In particular, various physical properties of the
Gibbs measure, such as ultrametricity and lack of self-averaging, implied by the choice of
the replica matrix in the Parisi ansatz were discovered by the physicists in the eighties
(see [8] for detailed account). The chaos problem, or “chaotic nature of the spin-glass
phase” [1], arose from the discovery that, in some models, small changes in temperature
or disorder may result in dramatic changes in the location of the ground state with the
energy maxσHN(σ), as well as the overall energy landscape and the organization of the
pure states of the Gibbs measure G . One very basic way to define such instability of the
N
Gibbs measure is to sample a vector of spin configurations σ fromG and a vector ρ from
N
the measure G′ corresponding to the perturbed parameters and consider their overlap
N
R(σ,ρ). The fact that this overlap behaves very differently than the overlap R(σ1,σ2)
of two replicas σ1,σ2 sampled from the same measure G indicates that the set of con-
N
figurations in Σ on which the Gibbs measure concentrates (the location of pure states)
N
is affected significantly by a small change of parameters. A typical statement that one
is looking for in this case is that the overlap R(σ,ρ) is concentrated near zero when the
model has symmetry or, more generally, near a constant when the symmetry is broken,
for example, in the presence of external field. Indeed, this behavior is quite different
from a typical “lack of self-averaging” when the overlap between σ1 and σ2 can take
many different values for any realization of the disorder in the low temperature phase.
Moreover, even if we could show that the overlap R(σ,ρ) concentrates near its Gibbs’
average which depends on the disorder, this would already indicate some form of chaos for
exactly the same reasons. This is precisely what we will show in the case of perturbations
of the disorder and for some perturbations of the inverse temperature parameters (β ).
p
Furthermore, under additional assumptions on the sequence (β ) we will provide stronger
p
control of the overlap in terms of the Parisi measures of the two systems.
2
We will consider two systems with Gibbs’ measures G1 and G2 corresponding to the
N N
HamiltoniansH1(σ)andH2(ρ)asin(3)forσ,ρ Σ definedintermsofpossibly differ-
N N ∈ N
entsequences ofparameters(β1)and(β2)and, again, possiblydifferentGaussiandisorders
p p
(g1 ) and (g2 ) for p 1. However, we will assume that all pairs (g1 ,g2 )
i1,...,ip i1,...,ip ≥ i1,...,ip i1,...,ip
are jointly Gaussian and independent for all (i ,...,i ) and p 1. We will denote by
1 p
≥
(σl,ρl) an i.i.d. sequence of replicas from the measure G1 G2 and by the Gibbs
l≥1 N × N h·i
average with respect to (G1 G2 )⊗∞.
N × N
Weak forms of chaos. For j 1,2 let us denote
∈ { }
e = p 2N : βj = 0 , o = p 2N 1 : βj = 0
Ij ∈ p 6 Ij ∈ − p 6
(cid:8) (cid:9) (cid:8) (cid:9)
and let = e o. When we talk about chaos in disorder we will assume that the
Ij Ij ∪ Ij
following condition about their correlation is satisfied for at least one p 1,
≥
p and corr(g1 ,g2 ) = t [0,1) (4)
∈ I1 ∩I2 i1,...,ip i1,...,ip p ∈
for all (i ,...,i ). Our first result yields a weak form of chaos in disorder.
1 p
Theorem 1. If (4) holds for some even p 2 then
≥
lim E R(σ,ρ) R(σ,ρ) 2 = 0. (5)
N→∞ | |−h| |i
(cid:10)(cid:0) (cid:1) (cid:11)
If (4) holds for some odd p 1 then
≥
lim E R(σ,ρ) R(σ,ρ) 2 = 0. (6)
N→∞ −h i
(cid:10)(cid:0) (cid:1) (cid:11)
For example, for pure 3-spin model, the overlap R(σ,ρ) is concentrated around its Gibbs
average R(σ,ρ) and for pure 2-spin (SK) model, the absolute value of the overlap
h i
R(σ,ρ) is concentrated around its Gibbs average R(σ,ρ) . If t = 1 in (4) for all
p
| | h| |i
p 1, we can prove a weak form of chaos in temperature under certain assumptions on
≥
the sequences (β1) and (β2). Let us introduce a family of subsets of natural numbers,
p p
= N : linear span of (xp) is dense in C[0,1], . (7)
0 p∈I ∞
C nI ⊆ k·k o
(cid:0) (cid:1)
Let us define the following conditions on the sequences (β1) and (β2):
p p
(Ce) either e e = or there exist e and p e such that and for
1 I2 \I1 6 ∅ A ⊆ I1 0 ∈ I1 \A A ∈ C0
some τ R we have β2 = τβ1 for p and β2 = τβ1 ,
∈ p p ∈ A p0 6 p0
(Co) either o o = or there exist o and p o such that and for
1 I2 \I1 6 ∅ A ⊆ I1 0 ∈ I1 \A A ∈ C0
some τ R we have β2 = τβ1 for p and β2 = τβ1 ,
∈ p p ∈ A p0 6 p0
and let us define (Ce) and (Co) in the same way by flipping indices 1,2 . We will define
2 2 { }
conditions
(Co) = (Co) (Co), (Ce) = ((Ce) (Co)) ((Ce) (Co)). (8)
1 ∧ 2 1 ∨ 1 ∧ 2 ∨ 2
The role of (7) and condition will be to ensure the validity of the extended
0
A ∈ C
Ghirlanda-Guerra identities from the identities for moments. The following weak form of
chaos in temperature holds.
3
Theorem 2. Condition (Ce) implies (5) and condition (Co) implies (6).
Example 1. If = 3 and = 5 then (6) holds. If = 2 and = 4 or
1 2 1 2
I { } I { } I { } I { }
= 3 then (5) holds.
2
I { }
Example 2. If = = 2N, β1 = β2 and τβ1 = β2 for all even p 4 and τ = 1 then (5)
I1 I2 2 2 p p ≥ 6
holds.
Example 3. If = 2 and = 2N+2, then (5) holds.
1 2
I { } I
Toward strong chaos. To formulate the results that provide some strong control of the
overlap R(σ,ρ) we need to recall some consequences of the validity of the Parisi formula
for the free energy in mixed p-spin models, which was proved in [19] for even-spin models
using the replica symmetry breaking interpolation idea from [7] and in [15] in the general
case using ultrametricity result from [14]. The first consequence that was found in [20]
(see [11] or [21] for a simplified proof) states that the Parisi formula is differentiable in
the inverse temperature parameters β for all p 1 which together with convexity implies
p
≥
that for all p ,
1
∈ I
1
lim E Rp(σ1,σ2) = qpdµ (q), (9)
Z 1
N→∞
(cid:10) (cid:11) 0
where µ is any probability measure on [0,1] that achieves the minimum in the variational
1
problem that defines the Parisi formula. Any such µ is called a Parisi measure of the
1
system. Similarly, for all p ,
2
∈ I
1
lim E Rp(ρ1,ρ2) = qpdµ (q) (10)
Z 2
N→∞
(cid:10) (cid:11) 0
foranyParisi measureµ ofthesecond system. Another consequence oftheParisi formula
2
will be the strong form of the Ghirlanda-Guerra identities proved in [12] that will be used
in the next section. In the situations that we consider below the linear span of (xp)
p∈Ij
will be dense in (C[0,1], ) for one or both j = 1,2 in which case (9), (10) imply that
∞
k·k
the Parisi measure µ is unique. In this case let
j
c = infsuppµ
j j
be the smallest point in the support of µ . The following result provides some control of
j
the overlap and points toward strong chaos in disorder.
Theorem 3. If (4) holds for some p 1 and e for j = 1 or j = 2 then
≥ Ij ∈ C0
lim E I R(σ,ρ) > √c = 0. (11)
j
N→∞ | |
(cid:10) (cid:0) (cid:1)(cid:11)
If (4) holds for some odd p 1 and e for both j = 1 and j = 2 then
≥ Ij ∈ C0
lim E I R(σ,ρ) > √c c = 0. (12)
1 2
N→∞ | |
(cid:10) (cid:0) (cid:1)(cid:11)
In particular, if the Parisi measure µ of the system that satisfies e contains zero
j Ij ∈ C0
in its support then the overlap R(σ,ρ) concentrates around zero. Again, if t = 1 in (4)
p
for all p 1, we have a similar result for chaos in temperature.
≥
4
Theorem 4. If e for j = 1 or j = 2 then condition (Ce) implies (11), and if
Ij ∈ C0
e for both j = 1 and j = 2 then condition (Co) implies (12).
Ij ∈ C0
Finally, all our results also hold for the spherical mixed p-spin models when Σ is the
N
sphere of radius √N with uniform measure, as long as 2N 1 . This restriction
1 2
I ∪I ⊆ ∪{ }
is due to the fact that the Parisi formula for the spherical model has so far been proved
only for such models in [18].
2 Ghirlanda-Guerra identities for coupled systems.
In this section we will show how one can use the Ghirlanda-Guerra identities for each
system intheformoftheconcentrationoftheHamiltoniantoobtainanewsetofidentities
for the overlaps of the coupled system. First of all, condition (4) means that the Gaussian
pair (g1,g2) is equal in distribution to
( t g + 1 t z1, t g + 1 t z2)
p p p p
− −
p p p p
for three independent standard Gaussian random variables g,z1,z2 and, therefore, the
pair of processes H1 (σ) and H2 (ρ) is equal in distribution to the pair
N,p N,p
t H (σ)+ 1 t Z1 (σ) and t H (ρ)+ 1 t Z2 (ρ),
p N,p − p N,p p N,p − p N,p
p p p p
where we denote by H ,Z1 and Z2 three independent copies of (1). Let us consider
N,p N,p N,p
the quantities
H1 (σ1) H1 (σ1)
Γ1 = E N,p E N,p ,
p D(cid:12) N − D N E(cid:12)E
(cid:12) (cid:12)
(cid:12)H2 (ρ1) H2 (ρ1) (cid:12)
Γ2 = E N,p E N,p ,
p D(cid:12) N − D N E(cid:12)E
(cid:12) (cid:12)
(cid:12)Z2 (σ1) Z2 (σ1) (cid:12)
∆1 = E N,p E N,p ,
p D(cid:12) N − D N E(cid:12)E
(cid:12) (cid:12)
(cid:12)Z1 (ρ1) Z1 (ρ1) (cid:12)
∆2 = E N,p E N,p .
p D(cid:12) N − D N E(cid:12)E
(cid:12) (cid:12)
(cid:12) (cid:12)
The Ghirlanda-Guerra identities [6] in the form of the concentration of the Hamiltonians
can be stated as follows.
Lemma 1. For all p 1, we have ∆1,∆2,Γ1,Γ2 0.
≥ p p p p →
Proof. Notice that in the definition of ∆1 we are averaging the Hamiltonian Z2 from
p N,p
the second system over the first coordinate σ1, which means that it is independent of
the randomness in . Therefore, if we denote by E′ the expectation with respect to
h·i
the randomness Z2 then E Z2 (σ1) = E E′Z (σ1) = 0 and, using (2) and Jensen’s
N,p h N,p i h N,p i
inequality,
Z2 (σ1) E′ Z2 (σ1) 2 1/2
E N,p E | N,p | N−1/2.
D(cid:12) N (cid:12)E ≤ D N2 E ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
5
We conclude that ∆1 0 and, similarly, ∆2 0. On the other hand, as we mentioned in
p → p →
the introduction, the validity of the Parisi formula for the free energy and the argument
in [12] (see also Chapter 12 in [21]) imply that Γ1 0 and Γ2 0 which is the usual
p → p →
formulation of the Ghirlanda-Guerra identities in the form of the concentration of the
Hamiltonian.
Given replicas (σl,ρl) let us denote by
l≥1
Rl1,l′ = R(σl,σl′), Rl2,l′ = R(ρl,ρl′), Rl,l′ = R(σl,ρl′)
the overlaps within each system and between the two systems. Notice that with these
notations the cross overlap is not symmetric, Rl,l′ = Rl′,l. Given integer n 1, a function
6 ≥
ψ ∈ C[−1,1] and a bounded measurable function f of the overlaps (Rl1,l′)l,l′≤n, (Rl2,l′)l,l′≤n
and (Rl,l′)l,l′≤n on n replicas, we define
n
1 1
Φ (f,ψ) = E fψ(R1 ) E f E ψ(R1 ) E fψ(R1 ) , (13)
1,n h 1,n+1 i− n h i h 1,2 i− n h 1,l i
X
l=2
n
1
Ψ (f,ψ) = E fψ(R ) E fψ(R ) , (14)
1,n 1,n+1 1,l
h i− n h i
X
l=1
n
1 1
Φ (f,ψ) = E fψ(R2 ) E f E ψ(R2 ) E fψ(R2 ) , (15)
2,n h 1,n+1 i− n h i h 1,2 i− n h 1,l i
X
l=2
n
1
Ψ (f,ψ) = E fψ(R ) E fψ(R ) . (16)
2,n n+1,1 l,1
h i− n h i
X
l=1
Throughout the paper we will use the notation
ψ (x) = xp.
p
The following lemma contains a computation based on the Gaussian integration by parts
analogous to the one for the original Ghirlanda-Guerra identities [6] for one system.
Lemma 2. For all p 1 we have,
≥
∆1
sup β2 1 t Ψ (f,ψ ) p, (17)
(cid:12) p − p 1,n p (cid:12)≤ n
kfk∞≤1(cid:12) p (cid:12)
(cid:12) (cid:12) ∆2
sup β1 1 t Ψ (f,ψ ) p, (18)
(cid:12) p − p 2,n p (cid:12)≤ n
kfk∞≤1(cid:12) p (cid:12)
(cid:12) (cid:12) Γ1
sup β1Φ (f,ψ )+β2t Ψ (f,ψ ) p, (19)
(cid:12) p 1,n p p p 1,n p (cid:12) ≤ n
kfk∞≤1(cid:12) (cid:12)
(cid:12) (cid:12) Γ2
sup β2Φ (f,ψ )+β1t Ψ (f,ψ ) p. (20)
(cid:12) p 2,n p p p 2,n p (cid:12) ≤ n
kfk∞≤1(cid:12) (cid:12)
(cid:12) (cid:12)
6
In particular, Lemma 1 implies that all the quantities on the left hand side go to zero and,
under certain assumptions on the parameters of the models, this will imply that some or
all quantities in (13) - (16) go to zero. Equations (13) and (15) will yield the familiar
Ghirlanda-Guerra identities, only now the function f may depend on the overlaps of the
two systems. Furthermore, equations (14) and (16) will provide important additional
information about how the two systems interact with each other.
Proof. We will only show (17) and (19) since the proof of (18) and (20) is similar. As
usual, we begin by writing that for f 1,
∞
k k ≤
Z2 (σ1) Z2 (σ1)
E N,p f E N,p E f ∆1 (21)
(cid:12) D N E− D N E (cid:12) ≤ p
(cid:10) (cid:11)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
H1 (σ1) H1 (σ1)
E N,p f E N,p E f Γ1. (22)
(cid:12) D N E− D N E (cid:12) ≤ p
(cid:10) (cid:11)
(cid:12) (cid:12)
Using (2) and Gaussia(cid:12)n integration by parts we get (cid:12)
Z2 (σ1) n
E N,p f = β2 1 t E (R )pf nE (R )pf .
D N E pp − p(cid:16)X (cid:10) 1,l (cid:11)− (cid:10) 1,n+1 (cid:11)(cid:17)
l=1
and since E Z2 = 0, (21) implies (17). Similarly, using Gaussian integration by parts,
h N,pi
H1 (σ1)
E N,p = β1 1 E (R1 )p
D N E p − 1,2
(cid:0) (cid:10) (cid:11)(cid:1)
and
H1 (σ1) n
E N,p f = β1 E (R1 )pf nE (R1 )pf
D N E p(cid:16)X (cid:10) 1,l (cid:11)− (cid:10) 1,n+1 (cid:11)(cid:17)
l=1
n
+ β2t E (R )pf nE (R )pf .
p p(cid:16)X (cid:10) 1,l (cid:11)− (cid:10) 1,n+1 (cid:11)(cid:17)
l=1
Therefore, (22) implies (19) and this completes the proof.
We will use Lemmas 1 and 2 in combination with the following result.
Lemma 3. Let j 1,2 . Suppose that
∈ { }
lim sup Ψ (f,ψ) = 0 (23)
j,n
N→∞kfk∞≤1| |
holds with ψ = ψ for some p 1. If p 2 is even then (23) also holds for all even
p
≥ ≥
ψ C[ 1,1] and if p 1 is odd then (23) holds for all ψ C[ 1,1].
∈ − ≥ ∈ −
Proof. It suffices to prove the results for j = 1. For all l 2 (using symmetry),
≥
E (R )p (R )p 2 = 2E (R )2p 2E (R )p(R )p = 2Ψ (f,ψ )
1,1 1,l 1,1 1,1 1,2 1,1 p
− − −
(cid:10)(cid:0) (cid:1) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
7
by definition of Ψ in (14) with n = 1 and f = (R )p. If p 2 is even then using that
1,n 1,1
≥
x y p xp yp for all x,y 0 we can write
| − | ≤ | − | ≥
E R R E R R 2p 1/2p (24)
1,1 1,l 1,1 1,l
| |−| | ≤ | |−| |
(cid:10)(cid:12) (cid:12)(cid:11) (cid:0) (cid:10)(cid:12) (cid:12) (cid:11)(cid:1)
(cid:12) (cid:12) E (cid:12)(R )p (R (cid:12))p 2 1/2p = 2Ψ (f,ψ ) 1/2p.
1,1 1,l 1,1 p
≤ − −
(cid:0) (cid:10)(cid:0) (cid:1) (cid:11)(cid:1) (cid:0) (cid:1)
If (23) holds for ψ = ψ , this implies that R R for all l 2 and, therefore, (23)
p 1,l 1,1
| | ≈ | | ≥
holds for all even ψ C[ 1,1]. Whenever (23) holds for ψ = ψ and odd p 1 we use
p
∈ − ≥
the same argument and the fact that x y p 2p−1 xp yp for all x,y R to show that
| − | ≤ | − | ∈
R R for all l 2 and, therefore, (23) holds for all ψ C[ 1,1].
1,l 1,1
≈ ≥ ∈ −
We are ready to state several consequences of Lemmas 1 - 3 under additional assumptions
on the parameters of the models that appear in our main results. First, we consider the
condition (4) that is used to prove weak chaos in disorder.
Proposition 1. Suppose that (4) holds for some p 1. For j 1,2 , if p is even then
≥ ∈ { }
(23) holds for all even ψ C[ 1,1] and if p is odd then (23) holds for all ψ C[ 1,1].
∈ − ∈ −
Proof. Since under (4), β1,β2 = 0 and t < 1, equations (17), (18) and Lemma 1 imply
p p 6 p
that (23) holds with ψ = ψ for both j 1,2 . The statement follows from Lemma 3.
p
∈ { }
One can prove a similar result under the conditions (8) that appear in the results con-
cerning chaos in temperature.
Proposition 2. Suppose that t = 1 for all p 1. For j 1,2 , condition (Ce) implies
p
≥ ∈ { }
(23) for all even ψ C[ 1,1] and condition (Co) implies (23) for all ψ C[ 1,1].
∈ − ∈ −
Proof. The result will follow immediately from the definition of (Ce) and (Co) in (8) if
we can show that
(i) (Ce) implies (23) for j = 1 and even ψ C[ 1,1],
1 ∈ −
(ii) (Co) implies (23) for j = 1 and all ψ C[ 1,1],
1 ∈ −
(iii) (Ce) implies (23) for j = 2 and even ψ C[ 1,1],
2 ∈ −
(iv) (Co) implies (23) for j = 2 and all ψ C[ 1,1].
2 ∈ −
We will only prove (i) since all other cases can be treated similarly. Let us show that (Ce)
1
implies
lim sup Ψ (f,ψ ) = 0 (25)
N→∞kfk∞≤1| 1,n p0 |
for some even p 2 from which (23) for j = 1 and even ψ C[ 1,1] follows from
0
≥ ∈ −
Lemma 3. First, if we suppose that e e = then there exists some even p 2 such
I2 \I1 6 ∅ 0 ≥
that β2 = 0 and β1 = 0, and (25) immediately follows from (19). Next, suppose that
p0 6 p0
there exist e and p e such that and for some τ R we have β2 = τβ1
A ⊆ I1 0 ∈ I1\A A ∈ C0 ∈ p p
8
for p and β2 = τβ1 . Since β1 = 0, let β2 /β1 =: τ′ = τ. Lemma 1 and equation
∈ A p0 6 p0 p0 6 p0 p0 6
(19) imply that
lim sup Φ (f,ψ )+τ′Ψ (f,ψ ) = 0 (26)
N→∞kfk∞≤1| 1,n p0 1,n p0 |
and for p (using that β2 = τβ1 and β1 = 0),
∈ A p p p 6
lim sup Φ (f,ψ )+τΨ (f,ψ ) = 0.
1,n p 1,n p
N→∞kfk∞≤1| |
Since , we can approximate ψ uniformly by ψ for p to obtain
A ∈ C0 p0 p ∈ A
lim sup Φ (f,ψ )+τΨ (f,ψ ) = 0. (27)
N→∞kfk∞≤1| 1,n p0 1,n p0 |
Since τ′ = τ, (26) and (27) again imply (25) and, thus, (Ce) implies (23) for j = 1 and
6 1
even ψ C[ 1,1].
∈ −
Now that we obtained control of quantities Ψ , equations (19) and (20) can be used to
j,n
control Φ .
j,n
Proposition 3. Suppose that (4) holds for some p 1. For j 1,2 , if e then
≥ ∈ { } Ij ∈ C0
lim sup Φ (f,ψ) = 0 (28)
j,n
N→∞kfk∞≤1| |
for all even ψ C[ 1,1].
∈ −
Proof. Let us only consider the case j = 1. By Proposition 1, (23) holds for all even
ψ C[ 1,1] and, therefore, equation (19) and Lemma 1 imply that
∈ −
lim sup Φ (f,ψ ) = 0
1,n p
N→∞kfk∞≤1| |
for all p e. Since e , we can approximate even ψ C[ 1,1] by polynomials with
∈ I1 I1 ∈ C0 ∈ −
powers p e and (28) follows for j = 1.
∈ I1
Exactly the same proof using Proposition 2 instead of Proposition 1 gives the following.
Proposition 4. Suppose that t = 1 for all p 1. For j 1,2 , if e then either
p ≥ ∈ { } Ij ∈ C0
condition (Ce) or (Co) implies (28).
3 Proof of the main results.
As an immediate consequence of Propositions 1 and 2 we get Theorems 1 and 2.
Proof of Theorems 1 and 2. Suppose that either (4) holds for some even p 2 or
≥
condition (Ce) holds. By Propositions 1 and 2, (23) holds for all even ψ C[ 1,1] for
∈ −
both j 1,2 and (24) implies
∈ { }
lim E R R 2 = 0.
1,1 1,2
N→∞ | |−| |
(cid:10)(cid:0) (cid:1) (cid:11)
9
An argument similar to (24) also gives
lim E R R 2 = 0.
2,2 1,2
N→∞ | |−| |
(cid:10)(cid:0) (cid:1) (cid:11)
Equation (5) follows by writing
E R R 2 E R R 2
1,1 1,1 1,1 2,2
| |−h| |i ≤ | |−| |
(cid:10)(cid:0) (cid:1) (cid:11) 2E(cid:10)(cid:0) R R (cid:1) (cid:11)2 +2E R R 2 .
1,1 1,2 2,2 1,2
≤ | |−| | | |−| |
(cid:10)(cid:0) (cid:1) (cid:11) (cid:10)(cid:0) (cid:1) (cid:11)
If either (4) holds for some odd p 1 or condition (Co) holds then, by Propositions 1 and
≥
2, (23) holds for all ψ C[ 1,1] and a similar argument yields (6).
∈ −
Let us denote by µ the distribution of the array of all overlaps
N
(Rl1,l′)l,l′≥1,(Rl2,l′)l,l′≥1 and (Rl,l′)l,l′≥1 (29)
under the annealed Gibbs measure E(G1 G2 )⊗∞. By compactness, the sequence (µ )
N × N N
converges weakly over subsequences but, for simplicity of notation, we will assume that
µ converges weakly to the limit µ. We will still use the notations (29) to denote the
N
elements of the overlap array in the limit and, again, for simplicity of notations we will
denote by E the expectation with respect to measure µ. For example, whenever (28)
holds, the measure µ will satisfy the Ghirlanda-Guerra identities
n
1 1
Efψ(Rj ) = Ef Eψ(Rj )+ Efψ(Rj ) (30)
1,n+1 n 1,2 n 1,l
X
l=2
forallboundedmeasurablefunctionsf oftheoverlapsonnreplicasandevenψ C[ 1,1].
∈ −
Consequently, (30) also holds for all even bounded measurable functions ψ. Similarly, (5)
impliesthatµ-almostsurely Rl,l′ = R1,1 and(6)impliesthatµ-almostsurelyRl,l′ = R1,1
| | | |
for all l,l′ 1.Given µ, let µ ,µ and µ denotethe distributions of R1 , R2 andR
≥ 1 2 1,2 | 1,2| | 1,2| 1,1
underµcorrespondingly(wewillabusethenotationssince, indeed, belowthedistributions
of R1 , R2 will coincide with the Parisi measures in (9), (10)). Given measurable sets
| 1,2| | 1,2|
A ,A [0,1] and A [ 1,1] let us define the events
1 2
⊆ ⊆ −
B = R A, R1 A for l = l′ n, R2 A for l = l′ n (31)
n n 1,1 ∈ | l,l′| ∈ 1 6 ≤ | l,l′| ∈ 2 6 ≤ o
and
C = R A, R1 A for l = l′ n+1, R2 A for l = l′ n (32)
n n 1,1 ∈ | l,l′| ∈ 1 6 ≤ | l,l′| ∈ 2 6 ≤ o
The following lemma will be crucial in the proof of Theorems 3 and 4.
Lemma 4. If µ satisfies (30) for j = 1 and A = [0,1] then
2
µ(C ) µ (A )nµ (A). (33)
n 1 1 1,2
≥
If µ satisfies (30) for j = 2 and A = [0,1] then
1
µ(B ) µ (A )n−1µ (A). (34)
n 2 2 1,2
≥
If µ satisfies (30) for both j = 1 and j = 2 then
µ(B ) (µ (A )µ (A ))n−1µ (A). (35)
n 1 1 2 2 1,2
≥
10