Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound

0.32 MB·English

by  Wei-Kuo Chen

Variational representations for the Parisi functional and the
two-dimensional Guerra-Talagrand bound
Wei-Kuo Chen ∗
University of Chicago and University of Minnesota
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May 16, 2016
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Abstract

3 ThevalidityoftheParisiformulaintheSherrington-Kirkpatrickmodel(SK)wasinitiallyproved
1 by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled

free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It
]
R is believed that this bound and its higher dimensional generalization are highly related to the con-
P jectures of temperature chaos and ultrametricity in the SK model, but a complete investigation
. remainselusive. Motivatedby Bovier-Klimovsky[3] andAuffinger-Chen[2], the aimofthis paper is
h
to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound
t
a in the mixed p-spin models in terms of optimal stochastic control problems. We compute the direc-
m
tional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We
[ demonstrate how our approachprovides a simple and efficient control for the GT bound that yields

severalnew results on Talagrand’spositivity of the overlap[20, Section 14.12]and disorder chaosin
2
v Chatterjee [5] and Chen [6]. In particular, we provide some examples of the models containing odd
5 p-spin interactions.
3
6
6 1 Introduction
0
.
1
In 1979, Parisi [14] suggested an ingenious variational formula for the limiting free energy in the
0
5 Sherrington-Kirkpatrick (SK)model. Its validity was rigorously established by Talagrand [18]following
1 thebeautifuldiscovery ofGuerra’sreplicasymmetrybreakingscheme[9]. Inthemoregeneralsituation,
:
v Parisi’s formula was later shown to be valid in the mixed p-spin models by Panchenko [16]. More
i precisely, for N 1, the Hamiltonian of the mixed p-spin model is defined as
X
≥
r
a N
H (σ)= X (σ)+h σ (1)
N N i
i=1
X
for σ = (σ ,...,σ ) Σ := 1,+1 N, where X is the linear combination of the pure p-spin
1 N N N
∈ {− }
Hamiltonians,
γ
p
X (σ) = β g σ σ
N N(p−1)/2 i1,...,ip i1··· ip
Xp≥2 1≤i1X,...,ip≤N
for i.i.d. standard Gaussian random variables, g , for all 1 i ,...,i N and p 2. In physics,
g ’s are called the disorder, h R is the sit1r,.e..n,ipgth of the≤ext1ernal fipe≤ld and β >≥0 is called the
i1,...,ip ∈
∗[email protected]
1
(inverse) temperature. Here, we assume that the nonnegative sequence (γ ) decays fast enough, e.g.
p p≥2
2pγ2 < , such that the covariance of X can be computed as
p≥2 p ∞ N
P EX (σ1)X (σ2)= Nξ(R )
N N 1,2
for any two spin configurations σ1 = (σ1,...,σ1 ) and σ2 = (σ2,...,σ2 ) from Σ , where
1 N 1 N N
N
1
R := σ1σ2 (2)
1,2 N i i
i=1
X
is called the overlap between σ1 and σ2 and
ξ(s) := β2sp, s [ 1,1] (3)
p ∀ ∈ −
p≥2
X
for β := βγ for all p 2. An important example of ξ is the mixed even p-spin model, i.e., γ = 0 for
p p p
≥
all odd p 3. In particular, the SK model corresponds to ξ(s) = β2s2/2. Denote the Gibbs measure by
≥
expH (σ)
N
G (σ)= , (4)
N
Z
N
where the normalizing factor Z = expH (σ) is called the partition function.
N σ∈ΣN N
The formulation of the Parisi formula is described as follows. Let be the space of all probability
P M
measures on [0,1] and bethecollection of all atomic measures in . Denote by α thedistribution
d µ
M M
function of µ . We endow the space with the metric
∈ M M
1
d(µ,µ′)= αµ(s) αµ′(s)ds. (5)
| − |
Z0
For any µ , let Φ be the solution to the Parisi PDE on [0,1] R,
µ
∈M ×
ξ′′(s)
∂ Φ (s,x) = ∂ Φ (s,x)+α (s)(∂ Φ (s,x))2 ,
s µ xx µ µ x µ
− 2 (6)
Φ (1,x) = logcosh(cid:0)x. (cid:1)
µ
Here, for any µ , this PDE can be explicitly solved by performing the Hopf-Cole transformation.
d
∈ M
As for an arbitrary probability measure µ , the solution Φ is understood in the weak sense (see
µ
∈ M
Jagannath and Tobasco [10]). Define the Parisi functional on by
P M
1 1
(µ) = log2+Φ (0,h) α (s)sξ′′(s)ds.
µ µ
P − 2
Z0
Note that this functional is Lipschitz continuous (see Guerra [9]). The famous Parisi formula says that
1
lim ElogZ = min (µ).
N
N→∞N µ∈MP
Here, the quantity inside the limit of the left-hand side is called the free energy of the model. Recently,
it was shown (see Auffinger and Chen [2]) that the Parisi functional is strictly convex, which implies
the uniqueness of the minimizer. We will call such minimizer the Parisi measure and denote it by µ .
P
In order to classify the structure of µ , we say that the Parisi measure is replica symmetric (RS) if it
P
is a Dirac measure, is k replica symmetry breaking (k-RSB) if it is atomic with exactly k +1 jumps
2
and is full replica symmetry breaking (FRSB) otherwise. In addition, for given sequence (γ ) and
p p≥2
fixed external field h, we define the high temperature regime as the collection of all β > 0 such that
the corresponding Parisi measures are RS and the low temperature regime is set as the complement of
the former. An important quantity associated to the mixed p-spin model is the overlap R between
1,2
two independently sampled spin configurations σ1 and σ2 from the Gibbs measure G . At very high
N
temperature, i.e., when β is exceedingly small, this overlap is concentrated around a constant (see
Talagrand [20, Chapter 13] for the SK model and Jagannath and Tobasco [11] for the mixed p-spin
model), whereas in the low temperature regime, it is typically supported by a set containing more than
one point (see Panchenko [15]).
Arguably, in the past decade, the most important development in the mean-field spin glasses is
Guerra’s replica symmetry breaking bound [9] for the free energy in the mixed even p-spin model. Its
statement reads that any N 1 and µ ,
≥ ∈M
1
ElogZ (µ). (7)
N
N ≤ P
BasedonGuerra’sinterpolationscheme[9],thefirstproofofParisi’sformulawasobtainedintheseminal
work of Talagrand [18], where the central ingredient was played by a two-dimensional extension of
Guerra’s inequality (7) for the coupled free energy with constrained overlaps, which was used to control
the error estimate between the two sides of (7) when µ is very close to the Parisi measure. Later the
fully generalization of Guerra’s inequality (7), called the Guerra-Talagrand (GT) bound throughout
this paper, was presented in [20, Section 15.7]. The two-dimensional GT bound, in particular, has two
important consequences regarding the behavior of the overlap under the Gibbs measure. The first is
known as the positivity of the overlap established by Talagrand [20, Section 14.12] in the mixed even
p-spin model, which says that if the external field is present, h = 0, then the overlap defined above is
6
essentially bounded from below by some positive constant. Note that this behavior is very different
from the one when the external field is absent, h= 0, in which case the overlap R is symmetric with
1,2
respect to the origin.
Another consequence is concerned with the phenomenon of chaos in disorder. It arose from the
observationthatinsomespinglassmodels,asmallperturbationtothedisorderwillresultinadramatic
change to the overall energy landscape (see Rizzo [17] for a recent survey in physics). In the mixed
p-spin model, one typical way to measure such instability is to consider two Hamiltonians,
H1(σ1)= X1 (σ1)+h σ1 and H2(σ2) = X2 (σ2)+h σ2,
N N i N N i
1≤i≤N 1≤i≤N
X X
where X1 and X2 are jointly Gaussian with mean zero and covariance structure,
N N
EX1(σ1)X1 (σ2) = ξ(R ) = EX2(σ1)X2(σ2),
N N 1,2 N N
(8)
EX1(σ1)X2 (σ2) = tξ(R )
N N 1,2
for some t [0,1]. Let σ1 and σ2 be independent samplings from G1 and G2 respectively and let
∈ N N
R be their overlap, which now also depends on t. The case t = 1 means that the two systems are
1,2
the same, H1 = H2 = H , and the overlap has the behavior we described before. From physics
N N N
literature (e.g. Bray-Moore [4], Fisher-Huse [8], Krz¸aka la-Bouchaud [13]), chaos in disorder is defined
by the phenomenon that R is concentrated around a nonrandom number independent of N if the
1,2
two systems are decoupled, i.e., t (0,1). The key point here is that such behavior is predicted to be
∈
true at any temperature. The first rigorous result along this direction was justified in the mixed even
p-spin models without external field in the work of Chatterjee [5] and the situation in the presence of
the external field was carried out in Chen [6].
3
As the above discussionindicates, the Parisi functional andthe GT boundhave played fundamental
roles in the study of the mixed p-spin model. Several challenging conjectures, such as the strong
ultrametricity and temperature chaos (see Talagrand [20, Section 15.7]), rely heavily on the subtle
control of these two objects and their higher dimensional generalization. To this regard, the aim of this
paper is to present a novel approach to analyzing the Parisi functional as well as the two-dimensional
GT bound by means of the optimal stochastic control theory. Ultimately, we hope that this new
method will shed some light on how to tackle the remaining open problems. Our idea is motivated
by the observation that the Parisi PDE solution Φ admits a variational representation (see Theorem
µ
1 below) in terms of an optimal stochastic control problem that corresponds to the Hamilton-Jacobi-
Bellman equation induced by a linear diffusion control problem. This was formerly used in Bovier and
Klimovsky [3] to study the strict convexity of the Parisi functional for some cases of the SK model
with multidimensional spins. Later it was understood that this approach allows to derive the strict
convexity of the Parisi functional in the mixed p-spin models by Auffinger and Chen [2].
Thisarticleconsistsoffourmajorresults. ThefirstpartgivesananalyticstudyoftheParisiformula,
where we compute the directional derivative of the Parisi functional and give equivalent criteria for the
Parisimeasure. Asanapplication, wegeneralize atheorem ofToninelli[21], whichstates thattheParisi
measureintheSKmodelisnotaDiracmeasurewhenthetemperatureandexternalfieldstayabovethe
Almeida-Thouless transition line (see (15) below). In addition, we extend Talagrand’s characterization
[20, Theorem 13.4.1] of the high temperature regime for the SK model to the temperature regime of
k-RSB Parisi measures for any mixed p-spin models. Second, we establish a variational representation
for the two-dimensional Parisi PDE solution in terms of an optimal stochastic control problem and
use this to give a new formulation of the original GT bound. Based on this new form, our last two
results are devoted to demonstrating a self-contained proof to establish the positivity of the overlap
and disorder chaos in the mixed p-spin model. We recover the aforementioned results and furthermore,
extend them to many new examples of the model allowing odd p-spin interactions. Along the way,
we also obtain a nonnegativity principle of the overlap in the mixed p-spin model, which says that
in the absence of the external field, the overlap is basically nonnegative if one adds certain odd p-
spin interactions to the Hamiltonian. As one shall see in Section 5 below, our approach significantly
simplifies and avoids several technicalities in the control of the two-dimensional GT bound compared
to the arguments in Talagrand [20, Section 14.12] and Chen [6]. For instance, the error estimate of
this bound was previously obtained through a quite involved iteration for certain functions of Gaussian
random variables. With the new approach, it now becomes quantitatively simpler in the critical case
(see Proposition 5 below).
This paper is organized as follows. In Section 2, we state the four main results described above
and their proofs are presented in the following three sections. The analytic properties of the Parisi
functional is investigated in Section 3 and the variational representation for the two-dimensional GT
bound is derived in Section 4. Finally, we present the proof for the results on the positivity of the
overlap and disorder chaos in Section 5.
Acknowledgements. TheauthorthanksArnabSenforseveralsuggestions regardingthepresentation
of the paper. This research is partially supported by the AMS-Simons Travel Grant.
2 Main results
2.1 Some properties of Parisi’s functional and measure
First, we recall the variational representation for the Parisi PDE from Auffinger and Chen [2]. Let
(P,F,(F ) ) be a filtrated probability space satisfying the usual condition, i.e., it is complete
r 0≤r≤1
and the filtration is right continuous. Let B = B(r),F ,0 r 1 be a standard Brownian
r
{ ≤ ≤ }
4
motion. For 0 s < t 1, let D[s,t] be the collection of all progressively measurable processes u
≤ ≤
with respect to (F ) satisfying sup u(r) 1. We equip the space D[s,t] with the norm
r s≤r≤t s≤r≤t| | ≤
u = ( tEu(w)2dw)1/2. Let ξ and h be fixed. Set ζ =ξ′′. For µ , we define a functional
k k s ∈M
R Fs,t(u,x) = E[Cs,t(u,x) Ls,t(u)] (9)
µ µ − µ
for u D[s,t] and x R, where letting α be the distribution function of µ,
µ
∈ ∈
t t
Cs,t(u,x) := Φ t,h+ α (w)ζ(w)u(w)dw + ζ(w)1/2dB(w) ,
µ µ µ
(cid:18) Zs Zs (cid:19)
1 t
Ls,t(u) := α (w)ζ(w)u(w)2dw.
µ 2 µ
Zs
The Parisi PDE solution can be expressed as
Theorem 1 ([2,Theorem 3 and Proposition 3]). We have
Φ (s,x) = max Fs,t(u,x). (10)
µ µ
u∈D[s,t]
Here, the maximum is attained by u (r) = ∂ Φ (r,X(r)), where (X(r)) satisfies
µ x µ s≤r≤t
r r
X(r)= x+ α (w)ζ(w)∂ Φ (w,X(w))dw + ζ(w)1/2dB(w). (11)
µ x µ
Zs Zs
In addition, the maximizer is unique if α > 0 on [s,t] and tα (r)dr < 1.
µ s µ
Here and thereafter, the existence of the partial derivatRives of ∂ Φ and ∂ Φ is ensured by [1,
x µ xx µ
Proposition 2]. Letting (s,t) = (0,1) in Theorem 1, the Parisi functional now reads
1 1
(µ) = log2+ max F0,1(u,h) α (w)wζ(w)dw .
P u∈D[0,1](cid:18) µ − 2 Z0 µ (cid:19)
Our first main results below are the computation of the directional derivative of the Parisi functional
and the equivalent criteria for the Parisi measure.
Theorem 2. Let µ . Define µ = (1 θ)µ +θµ for each µ and θ [0,1]. We have
0 θ 0
∈ M − ∈ M ∈
d 1 1
(µ ) = ζ(r)(α (r) α (r))(Eu (r)2 r)dr (12)
dθP θ 2 µ − µ0 µ0 −
(cid:12)θ=0 Z0
(cid:12)
for all µ , where d (µ )(cid:12) is understood as the right derivative at 0 and u is the maximizer
∈ M dθP θ (cid:12)θ=0 µ0
of (10) using µ and (s,t)= (0,1). In addition, the following statements are equivalent
0 (cid:12)
(cid:12)
(i) µ is the Parisi measure.
0
(ii) d (µ ) 0 for all µ .
dθP θ θ=0 ≥ ∈ M
(iii) d (µ )(cid:12) 0 for all Dirac measures µ =δ with q [0,1].
dθP θ (cid:12)θ=0 ≥ q ∈
The equiva(cid:12)lence of (i) and (ii) is mainly due to the strict convexity of the Parisi functional. The
(cid:12)
criterion (iii) essentially says that if one could not lower the Parisi functional by addingone more jump
to µ , then µ must be the Parisi measure. There are two immediate consequences that can be drawn
0 0
from this theorem. For convenience, we set k for k 0 to be the collection of all members in
Md ≥ Md
that have no more than k +1 atoms. In particular, 0 denotes the space of all Dirac measures on
Md
[0,1]. In the first consequence, we extract some information about the support of the Parisi measure.
5
Proposition 1. Let S be the support of µ . For all q S,
P
∈
E∂ Φ (q,X(q))2 = q, (13)
x µP
ζ(q)E∂ Φ (q,X(q))2 1, (14)
xx µP ≤
where (X(s)) satisfies the following stochastic differential equation,
0≤s≤1
s s
X(s) =h+ α (r)ζ(r)∂ Φ (w,X(w))dw + ζ(w)1/2dB(w), s [0,1].
µP x µP ∀ ∈
Z0 Z0
Remark 1. Suppose that µ is a Dirac measure at some q [0,1]. A direct computation gives
P
∈
1(ξ′(1) ξ′(s))+Elogcosh x+z(ξ′(q) ξ′(s))1/2 , if (s,x) [0,q) R,
ΦµP(s,x) = 12(ξ′(1)−ξ′(q))+logcoshx, − if (s,x) ∈[q,1] ×R,
(cid:26) 2 − (cid:0) (cid:1) ∈ ×
for some standard Gaussian random variable z. Since α = 0 on [0,q), Theorem 1 reads
µP
Etanh2 zξ′(q)1/2+h = q,
(cid:16) 1 (cid:17)
ζ(q)E 1. (15)
cosh4 zξ′(q)1/2 +h ≤
Note that if q [0,1] minimizes the Parisi func(cid:0)tional over all(cid:1)choices in 0, then one can get the first
∈ Md
equation (by a direct differentiation, see e.g. [19, Chapter 1]). But if the temperature and external
field are above the Almeida-Thouless line, i.e., (15) is violated, then the Parisi measure can not be RS.
This generalizes Toninelli’s theorem [21], where he established the same statement for the SK model
ξ(s)= β2s2/2.
Remark 2. ConsidertheSKmodelwithoutexternalfield,i.e., ξ(s)= β2s2/2andh = 0.Wenowargue
that the high temperature regime, defined as the collection of all β such that µ is a Dirac measure, is
P
described by β 1. To see this, note that since h = 0, 0 is always in the support of the Parisi measure
≤
by [1, Theorem 1]. Thus, it suffices to show that µ = δ if and only if β 1. If µ = δ and β > 1,
P 0 P 0
≤
we will obtain a contradiction as (15) is violated. Conversely, suppose β 1. A use of Itˆo’s formula
≤
and (6) gives
r r 1
u (r)= β ∂ Φ (w,X(w))dB(w)+u (0) = β dB(w)
δ0 xx δ0 δ0 cosh2X(w)
Z0 Z0
and hence,
r 1 r
Eu (r)2 = β2 dw β2 1dw r.
δ0 cosh4X(w) ≤ ≤
Z0 Z0
Therefore, for all µ ,
∈M
d β2 1
(µ ) = (α (r) 1)(Eu (r)2 r)dr 0
dθP θ (cid:12)θ=0 2 Z0 µ − δ0 − ≥
(cid:12)
and Theorem 2 implies that δ0 is(cid:12)the Parisi measure.
The second consequence of Theorem 2 is a generalization of Talagrand’s characterization [20, The-
orem 13.4.1] of the high temperature regime for the SK model, ξ(s) = β2s2/2, where he showed that
this regime is indeed equal to the set of all β such that inf (µ) = (µ ) for some µ 0. For
µ∈M1dP P 0 0 ∈ Md
any such β, he proved that µ will automatically be the Parisi measure. With the help of Theorem 2
0
(iii), this result can be generalized to any k-RSB Parisi measures.
6
Proposition 2. Consider arbitrary ξ and h. Let k 0 and µ be an optimizer of over k. If
≥ 0 P Md
inf (µ) = (µ ), (16)
0
µ∈Mk+1P P
d
then µ is the Parisi measure.
0
In other words, for fixed sequence (γ ) and external field h, the temperature regime of k-RSB
p p≥2
Parisimeasures is describedby thecollection of all β > 0such thatthe correspondingParisifunctionals
satisfy (16) for some optimizer µ of restricted to k.
0 P Md
It is generally very difficult to compute the Parisi measure as one needs to minimize over all
P
probability measures on [0,1]. In principle, Proposition 2 suggests a heuristic way to simulate k-RSB
Parisi measures. The procedure is based on the observation that if we restrict to k, then it is a
P Md
differentiable function that depends only on 2(k+1) variables on a compact set,
k+1
(q ,...,q ,a ,...,a ): 0 q q 1,0 a , ,a 1, a = 1 ,
1 k+1 1 k+1 1 k+1 1 k+1 i
≤ ≤ ··· ≤ ≤ ≤ ··· ≤
n Xi=1 o
on which one can compute the derivative of and numerically simulate the minimizer of over k.
P P Md
Starting from the case k = 0, if (16) is satisfied, then one can stop and obtain the RS Parisi measure;
otherwiseonemustproceedto thecase k = 1andcontinue this process. If eventually thereisa smallest
integer k 0 such that (16) is obtained, then one gets a k-RSB Parisi measure.
≥
2.2 A variational representation for the two-dimensional GT bound
The two-dimensional GT bound in the setting of [20, Theorem 15.7] is formulated as follows. Let
h ,h R and X1,X2 be jointly Gaussian processes indexed by Σ with mean zero and covariance,
1 2 ∈ N N N
EXNℓ (σ1)XNℓ′(σ2)= Nξℓ,ℓ′(R1,2)
for 1 ℓ,ℓ′ 2 and σ1,σ2 Σ , where R is the overlap between σ1,σ2 defined through (2). Here
N 1,2
≤ ≤ ∈
ξℓ,ℓ′’s are convex functions on [ 1,1] defined in terms of infinite series as ξ in (3). Consider two mixed
−
p-spin Hamiltonians,
Hℓ (σℓ)= Xℓ (σℓ)+h σℓ, ℓ = 1,2. (17)
N N ℓ i
1≤i≤N
X
Denote by S the collection of all possible values that R could attained. Fix q S . Assume that
N 1,2 N
∈
(yℓ) for 1 ℓ 2 are jointly centered Gaussian random variables such that for certain real
seqpu0e≤npc≤eks (ρℓ,ℓ′) ≤ ≤ for 1 ℓ,ℓ′ 2 with
p 0≤p≤k+1
≤ ≤
1,1 2,2 1,2 2,1 1,1 2,2 1,2 2,1
ρ = ρ = ρ = ρ = 0, ρ = ρ = 1, ρ = ρ = q, (18)
0 0 0 0 k+1 k+1 k+1 k+1
we have
Eypℓypℓ′ = ξℓ′,ℓ′(ρℓp,+ℓ′1)−ξℓ′,ℓ′(ρℓp,ℓ′).
Theorem 3 (Guerra-Talagrand). Let (m ) be a sequence with m = 0 < m < < m <
p 0≤p≤k 0 1 k−1
···
m = 1. Under the assumptions stated above, we have that
k
1
F (q) := Elog exp H1(σ1)+H2(σ2)
N N N N
RX1,2=q (cid:0) (cid:1)
(19)
k
≤ 2log2+Y0−λq− 12 mp(θℓ,ℓ′(ρℓp,+ℓ′1)−θℓ,ℓ′(ρℓp,ℓ′)),
1≤ℓ,ℓ′≤2p=0
X X
7
where θℓ,ℓ′(s):= sξℓ′,ℓ′(s)−ξℓ,ℓ′(s) and Y0 is defined as follows. Denote by Ep the expectation with respect
to yℓ,ℓ′. Starting with
p
k k
Y = log cosh h + y1 cosh h + y2 coshλ
k+1 1 p 2 p
(cid:16) (cid:16) Xp=0 (cid:17) (cid:16) Xp=0 (cid:17)
k k
+sinh h + y1 sinh h + y2 sinhλ ,
1 p 2 p
(cid:16) Xp=0 (cid:17) (cid:16) Xp=0 (cid:17) (cid:17)
we define decreasingly Y = m−1E expm Y for 1 p k. Finally, set Y = E Y .
p p p p p+1 ≤ ≤ 0 0 1
The inequality (19) is a two-dimensional extension of Guerra’s replica symmetry breaking bound
(7). Its proof as well as the higher dimensional extension can be found in [20, Section 15.7]. Recall q
from the statement of Theorem 3. Let ι = 1 if q 0 and ι = 1 otherwise. For 1 ℓ,ℓ′ 2, let ρℓ,ℓ′
≥ − ≤ ≤
be nondecreasing continuous functions on [0,1] with
ρ (0) = ρ (0) = ρ (0) = ρ (0) = 0,
1,1 1,2 2,1 2,2
(20)
ρ (1) = ρ (1) = 1, ρ (1) = ρ (1) = q .
1,1 2,2 1,2 2,1
| |
Assume that these functions are differentiable everywhere except at a finite number of points, on which
the right derivatives exist. For any s [0,1], we define
∈
ζ (s) ζ (s) d ξ′ (ρ (s)) d ξ′ (ιρ (s))
T(s)= 1,1 1,2 := ds 1,1 1,1 ds 1,2 1,2 (21)
ζ (s) ζ (s) d ξ′ (ιρ (s)) d ξ′ (ρ (s))
(cid:20) 2,1 2,2 (cid:21) (cid:20) ds 2,1 2,1 ds 2,2 2,2 (cid:21)
In the right-hand side of (21), the derivatives are understood as the ones from the right if one of ρℓ,ℓ′’s
is not differentiable. We suppose that T(s) is positive semi-definite and its operator norm T(s) is
k k
uniformly boundedfrom above by some constant K > 0. For µ , we consider the classical solution
d
∈ M
Ψ to the two-dimensional Parisi PDE,
µ
1
∂ Ψ = T,▽2Ψ +α T▽Ψ ,▽Ψ (22)
s µ µ µ µ µ
−2 h i
(cid:0)(cid:10) (cid:11) (cid:1)
for (λ,s,x) R [0,1) R2 with terminal condition
∈ × ×
Ψ (λ,1,x) = log(coshx coshx coshλ+sinhx sinhx sinhλ). (23)
µ 1 2 1 2
The assumption µ guarantees the existence of the solution by a usual application of Hopf-Cole
d
∈ M
transformation. Onemay refertoLemma3belowforthepreciseformulaofthesolution. Ourfirstmain
result below says that the mapping µ Ψ is Lipschitz with respect to the metric d defined by
d µ
∈ M 7→
(5).
Theorem 4. For any µ,µ′ , we have that
d
∈ M
Ψµ(λ,s,x) Ψµ′(λ,s,x) 3Kd(µ,µ′)
| − | ≤
for (λ,s,x) R [0,1] R2.
∈ × ×
This Lipschitz property allows us to extend Ψ continuously to all µ by using sequences of
µ
∈ M
atomic probability measures. Denote by = (r) = ( (r), (r)),G ,0 r < a two-dimensional
1 2 r
B {B B B ≤ ∞}
Brownian motion, where (G ) satisfies the usual condition. For 0 s < t 1, denote by [s,t] the
r r≥0
≤ ≤ D
8
space of all two-dimensional progressively measurable processes v = (v ,v ) with respect to (G )
1 2 r s≤r≤t
satisfying sup v (r) 1 and sup v (r) 1. Endow the space [s,t] with the norm
s≤r≤t| 1 | ≤ s≤r≤t| 2 | ≤ D
t 1/2
v = E (v (w)2 +v (w)2)dw .
s,t 1 2
k k
(cid:16) Zs (cid:17)
Similar to the formulation of (9), we define a functional
s,t(λ,v,x) = E s,t(λ,v,x) s,t(v)
Fµ Cµ −Lµ
for (λ,v,x) R [s,t] R2, where (cid:2) (cid:3)
∈ ×D ×
t t
s,t(λ,v,x):= Ψ λ,t,x+ α (w)T(w)v(w)dw + T(w)1/2d (w) ,
Cµ µ µ B
(cid:16) Zs Zs (cid:17)
1 t
s,t(v) := α (w) T(w)v(w),v(w) dw.
Lµ 2 µ h i
Zs
The following is an analogue of Theorem 1 for Ψ .
µ
Theorem 5. We have
Ψ (λ,s,x) = max s,t(λ,v,x) v [s,t] . (24)
µ Fµ ∈ D
Herethe maximum of (24)isattained byv (r)=(cid:8)▽Ψ (λ,r,X(cid:12)(r)), where(cid:9)the two-dimensional stochastic
µ µ (cid:12)
process (X(r)) satisfies
s≤r≤t
r r
X(r) = x+ α (w)T(w)▽Ψ (λ,w,X(w))dw+ T(w)1/2d (w). (25)
µ µ
B
Zs Zs
Using the notations introduced above, we can now formulate the GT bound in terms of Ψ .
µ
Theorem 6 (Guerra-Talagrand). Suppose that T is positive semi-definite for all s. Then
F (q) 2log2+EΨ (λ,0,h ,h ) λq
N µ 1 2
≤ −
1 1 (26)
αµ(s) ρℓ,ℓ′(s)ζℓ,ℓ′(s)+ι ρℓ,ℓ′(s)ζℓ,ℓ′(s) ds.
− 2
Z0 ℓ=ℓ′ ℓ6=ℓ′ !
X X
Typically to use this bound, one needs to first find suitable parameters λ and ρℓ,ℓ′ depending on q
such that the right-hand side is less than or equal to 2 (µ ) for any q [ 1,1]. In Section 5, we shall
P
P ∈ −
see that this could be achieved in the case of ξ = ξ and h = h , but the general situation remains
1,1 2,2 1 2
mysterious.
2.3 Some properties of the overlap
Recall the Hamiltonian H and the Gibbs measure G from (1) and (4). Let µ bethe Parisi measure
N N P
associated to H and set η = minsuppµ . It is known (see [7]) that
N P
η = 0 if h = 0 and η > 0 if h= 0. (27)
6
Recall that as we have discussed in the introduction, the overlap R between two independently
1,2
sampled spin configurations from G is symmetric with respect to the origin if the mixed p-spin is even
N
and the external field is absent. The positivity principle of the overlap says that this symmetry will
be broken in such a way that the overlap is essentially bounded from below by η if the external field is
present. More specifically, below is our main result.
9
Theorem 7 (Positivity of the overlap). Assume that ξ is convex on [ 1,1] and is not identically equal
−
to zero. If h = 0, then under any one of the following two assumptions,
6
(i) ξ is even,
(ii) ξ is not even and the function below is nondecreasing on (0,1],
ξ′′(s)
, (28)
ξ′′(s)+ξ′′( s)
−
we have that for any ε >0, there exists a constant K > 0 such that
0
N
EG G (σ1,σ2): R η ε K exp , N 1. (29)
N N 1,2 0
× ≤ − ≤ −K ∀ ≥
0
(cid:16) (cid:17)
(cid:0) (cid:1)
The inequality (29) means that if the external field is present, then the overlap essentially charges
weight only in the interval [η,1] (0,1]. Positivity of the overlap under the condition (i) was initially
⊆
established by Talagrand [20, Section 14.10]. Our main contribution here is the case (ii), where we
allow odd p-spin interactions in the Hamiltonian. Below we describe a concrete example of the case
(ii).
Example 1. Consider ξ(s)= β2(γ2 s2p+γ2 s2p+1) on [ 1,1] with γ and γ satisfying
2p 2p+1 − 2p 2p+1
(2p+1)γ2
2p+1
c := < 1.
(2p 1)γ2
− 2p
It is easy to verify that this condition ensures the convexity of ξ on [ 1,1]. Since
−
ξ′′(s) 1+cs
=
ξ′′(s)+ξ′′( s) 2
−
is nondecreasing on (0,1], condition (ii) in Theorem 7 is satisfied, from which we obtain (29) for any
β > 0.
Our next result shows that in the absence of the external field h = 0, the behavior of the overlap is
also influenced drastically by the odd p-spin interactions in the Hamiltonian, in which case the overlap
will be nonnegative.
Theorem 8 (Nonnegativity of the overlap). Assume that ξ is convex on [ 1,1] and is not identically
−
equal to zero. If h = 0 and the assumption (ii) in Theorem 7 holds, then for any ε > 0, there exists a
constant K > 0 such that
0
N
EG G (σ1,σ2) :R ε K exp , N 1. (30)
N N 1,2 0
× ≤ − ≤ −K ∀ ≥
0
(cid:16) (cid:17)
(cid:0) (cid:1)
2.4 Chaos in disorder
Recall the Hamiltonians H1 and H2 from (17). Assume that the Gaussian parts of the Hamiltonians,
N N
X1 and X2 , have the following covariance structure,
N N
ξ = ξ = ξ, ξ = ξ = ξ (31)
1,1 2,2 1,2 2,1 0
for some series ξ defined in a similar way as ξ and that the external fields satisfy
0
h = h =h. (32)
1 2
10