J Stat Phys DOI 10.1007/s10955-015-1264-3

Disorder Chaos in the Spherical Mean-Field Model Wei-Kuo Chen1 · Hsi-Wei Hsieh2 · Chii-Ruey Hwang2 · Yuan-Chung Sheu3

Received: 12 January 2015 / Accepted: 11 April 2015 © Springer Science+Business Media New York 2015

Abstract We study the problem of disorder chaos in the spherical mean-field model. It concerns the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra’s replica symmetry breaking scheme, we establish this at the levels of the free energy and the Gibbs measure. Keywords

Disorder chaos · Replica symmetry breaking · Mean-field model

Mathematics Subject Classification

60K35 · 82B44

1 Introduction and Main Results This paper concerns the chaos problem in mean-field spin glasses. It arose from the discovery that in some models, a small perturbation to the external parameters will result in a dramatic change to the overall energy landscape and the organization of the pure states of the Gibbs measure. Over the past decades, physicists have intensively studied chaos phenomenon at

B

Wei-Kuo Chen [email protected] Hsi-Wei Hsieh [email protected] Chii-Ruey Hwang [email protected] Yuan-Chung Sheu [email protected]

1

Department of Mathematics, University of Chicago, Chicago, IL, USA

2

Institute of Mathematics, Academia Sinica, Taipei, Taiwan

3

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

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the free energy level utilizing the replica method, where most related works were discussed in models with Ising spin. We refer readers to the survey of Rizzo [10] and the references therein along this direction. Recently, mathematical results also have been obtained in the Ising-spin mixed even-spin model. Chaos in disorder without external field was considered in Chatterjee [1] and more general situations with external field were handled in Chen [2,5]. Some special cases of temperature chaos were obtained in Chen and Panchenko [4] and Chen [6]. More recently, chaos in temperature in the generic even-spin models was studied by Panchenko [8]. The aim of this work is to investigate the problem of disorder chaos in the spherical meanfield model. Our approach is based on Guerra’s replica symmetry breaking bound for the coupled free energy with overlap constraint. This methodology was adapted in Chen [2] to establish chaos in disorder for Ising-spin mixed even-spin model with external field, where many estimates were highly involved due to the nature of the Ising spin. The goal of this paper is twofold. First, we illustrate how the same method may as well be applied to the spherical model and clarify several ideas behind the proof sketch of Research Problem 15.7.14. about disorder chaos problem in Talagrand [12] and Chen [2] with more explicit and simpler computations. It should be mentioned that in the absence of external field, an alternative way of proving disorder chaos is via the Ornstein–Uhlenbeck semi-group techniques for the moments of the overlaps as was shown in Chatterjee [1]. However, when the external field is presented, it is unclear how to apply the same argument to this situation since now the symmetry in the Gibbs measure is broken. In this paper, we shall see that Guerra’s replica symmetry breaking scheme works equally well both in the presence and absence of the external field and leads to chaos in disorder. Second, we show that disorder chaos is a stronger effect comparing with temperature chaos. In fact, in Panchenko and Talagrand [9], the same approach as the present paper was formerly used to discuss the conjectures of ultrametricity and chaos in temperature for spherical pure even-spin model, where it has been pointed out that these problems can not be achieved at the level of the free energy. Our main results here establish chaos in disorder at the levels of the free energy and the Gibbs measure in the mixed even-spin model. We now state our main results. For each N ∈ N, let X N be a centered Gaussian process indexed by the configuration space    S N = σ = (σ1 , . . . , σ N ) ∈ R N : σi2 = N i≤N

and equipped with the covariance structure       E X N σ 1 X N σ 2 = N ξ R1,2 , where R1,2 = N −1 σ 1 · σ 2 is called the overlap between two configurations σ 1 , σ 2 ∈ S N and ξ : [0, 1] → R is an even convex function with ξ  (x) > 0 for x > 0 and ξ  (x) ≥ 0 for x ≥ 0. The spherical model is defined on S N and its Hamiltonian takes the form, −H N (σ ) = X N (σ ) + h

N 

σi .

i=1

Set the corresponding Gibbs measure, dG N (σ ) =

123

  1 exp −H N (σ ) dλ N (σ ), ZN

Disorder Chaos in the Spherical Mean-Field Model

where dλ N is the uniform probability measure on S N and the normalizing factor Z N is called the function. An important example of ξ is the mixed even-spin model, partition 2 2 p for any sequence of real numbers (β ) p 2 ξ(x) = p p≥1 with p≥1 β p x p≥1 2 β p < ∞. −1 Denote by p N = N E log Z N the limiting free energy. The Parisi formula of the spherical model is formulated as lim p N = inf P (x, b).

N →∞

x,b

(1.1)



1 Here for any distribution function x on [0, 1] and b > max 1, 0 ξ  (s)x(s)ds , 

 1  1 h2 ξ  (q) 1 P (x, b) := ξ  (q)x(q)dq , + dq + b − 1 − log b − 2 b − d(0) 0 b − d(q) 0 (1.2)  1  where d(q) := q ξ (s)x(s)ds. The formula (1.1) was firstly verified by Talagrand [11] and later generalized to the spherical mixed p-spin model including odd p in Chen [3]. A key fact of the variational formula (1.1) is the existence and uniqueness of the optimizer, which are guaranteed by Talagrand [11, Theorem 1.2 and Sect. 4]. In the problem of disorder chaos, we are interested in understanding how the system changes when the disorder is perturbed. To this end, we shall consider two copies X 1N and X 2N of X N with covariance       E X 1N σ 1 X 2N σ 2 = t N ξ R1,2 for some t ∈ [0, 1]. In the same manner as H N , G N and Z N , we denote by H N1 , H N2 the Hamiltonians, G 1N , G 2N the Gibbs measures and Z 1N , Z 2N the partition functions corresponding to (X 1N , h) and (X 2N , h), respectively. Let · denote the Gibbs expectation with respect to the product measure dG 1N (σ 1 ) × dG 2N (σ 2 ). If t = 1, these two systems are identically the same in which case the limiting distribution of the overlap R1,2 under the measure E · is typically non-trivial in the replica symmetry breaking region. Contrary to the situation t = 1, our main results on disorder chaos stated in the following theorems say that the system will change dramatically at the levels of the free energy and the Gibbs measure if the two systems are decoupled 0 < t < 1. Theorem 1.1 For u ∈ [−1, 1] and α > 0, define the coupled partition function,           Z N ,u,α = exp − H N1 σ 1 − H N2 σ 2 dλ N σ 1 dλ N σ 2 . |R1,2 −u|<α

and set the coupled free energy, p N ,u,α =

1 E log Z N ,u,α . N

(1.3)

If 0 < t < 1, there exists some u ∗ ∈ [0, 1) such that for all u  = u ∗ , lim sup lim sup p N ,u,α < 2 inf P (x, b). α↓0

N →∞

x,b

(1.4)

In other words, there is free energy cost if u  = u ∗ for 0 < t < 1. Here the determination of u ∗ is a technical issue, which is described through an equation related to the Parisi formula as well as the associated optimizer. We shall leave the details to Sect. 3. Roughly speaking, u ∗ is equal to zero if h = 0 and it stays positive if h  = 0. As an immediate application of

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the Gaussian concentration of measure, Theorem 1.1 yields the concentration of the overlap near the constant u ∗ in the following theorem. Theorem 1.2 If 0 < t < 1, then there exists some u ∗ ∈ [0, 1) such that for any ε > 0, 

  N (1.5) E 1{|R1,2 −u ∗ |>ε} ≤ K exp − K for all N ≥ 1, where K is a constant independent of N . This paper is organized as follows. Our approach is based on a two-dimensional extension of the Guerra replica symmetry breaking bound for (1.3) and a sketch of the proof for disorder chaos in the Ising-spin mixed even-spin model as was outlined in Talagrand [12, Sect. 15.7] and later implemented in Chen [2]. In Sect. 2, using Guerra’s bound, we will compute explicitly a manageable upper bound for the coupled free energy (1.3). In Sect. 3, we will establish the determination of u ∗ and conclude Theorem 1.1. Finally we carry out the proof of Theorem 1.2.

2 Guerra’s Interpolation The main goal of this section is to derive the following upper bound for the coupled free energy (1.3), which is an extended version of [11, Proposition 7.8]. Proposition 2.1 For any distribution function x on [0, 1], λ ∈ R and b > |λ|, we have that for any u ∈ [−1, 1],   lim sup lim sup p N ,u,α ≤ Pu x, b, λ , α↓0

1 0

ξ  (s)x(s)ds +

N →∞

where the functional Pu (x, b, λ) is defined as follows. Set d(q) = φu (q) = d(u) +

1 q

(2.1)

ξ  (s)x(s)ds and

 1−t d(q) − d(u) . 1+t

Define  Pu (x, b, λ) :=

where

  Tu x, b, λ +   Tu x, b, λ +





|u|

h2 b−λ−d(0) , h2 b−λ−φ|u| (0) ,

if u ∈ [0, 1], if u ∈ [−1, 0),

ξ  (s) 1−t Tu (x, b, λ) = log ds + b − ηλ − d(s) 2 0  1  1   ξ (s) ξ (s) 1 1 ds + ds + 2 |u| b − λ − d(s) 2 |u| b + λ − d(s)  1 − λu + b − 1 − log b − ξ  (q)x(q)dq. b2 1+t + b2 − λ2 2

 0

|u|

ξ  (s) ds b+ηλ − φ|u| (s)

(2.2)

0

We mainly follow the procedure of the proof of [11, Theorem 5.2] to prove Proposition 2.1. Fix u ∈ [−1, 1] and η ∈ {1, −1} with u = η|u|. It suffices to prove (2.1) only for discrete

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Disorder Chaos in the Spherical Mean-Field Model

x. For k ≥ 0, consider two sequences of real numbers m = (m )0≤ ≤k and q = (q )0≤ ≤k+1 that satisfy 0 = m 0 ≤ m 1 ≤ · · · ≤ m k ≤ m k = 1, 0 = q0 ≤ q1 ≤ · · · ≤ qk+1 ≤ qk+1 = 1.

(2.3)

Let x be a distribution function on [0, 1] associated to this triplet (k, m, q), that is, x(q) = m

for q ∈ [q , q +1 ) and 0 ≤ ≤ k and x(1) = 1. Without loss of generality, we may assume that qτ = |u| for some 0 ≤ τ ≤ k + 1. We consider further independent pairs of centered Gaussian random vectors (y 1p , y 2p )0≤ p≤k that possess covariance  j 2     E y p = ξ  q p+1 − ξ  q p , 0 ≤ p ≤ k, j = 1, 2,   E y 1p y 2p = ηtξ  (q p+1 ) − ξ  (q p ) , 0 ≤ p < τ, y 1p , y 2p are independent, τ ≤ p ≤ k.

(2.4)

Let (yi,1 p , yi,2 p )0≤ p≤k be independent copies of (y 1p , y 2p )0≤ p≤k for 1 ≤ i ≤ N and be independent of X 1N , X 2N . Following Guerra’s scheme, we define the interpolated Hamiltonian H N ,a (σ 1 , σ 2 ) for a ∈ [0, 1], 

−H N ,a σ , σ 1

2



 N 2   j √  2   √  1  1 j 2 1−a yi, p + h σi . = a XN σ + XN σ + j=1 i=1

Let

0≤ p≤k

(n p )kp=0

satisfy 0 ≤ n 1 ≤ · · · ≤ n k ≤ 1. Define         Fk+1 (a) = log exp −H N ,a σ 1 , σ 2 dλ N σ 1 dλ N σ 2 . |R1,2 −u|<α

1 , y 2 ) and define Denote by E p the expectation in the random variables (yi,1 p , yi,2 p ), . . . , (yi,k i,k recursively for 0 ≤ p ≤ k,  1 log E p exp n p F p+1 (a), if n p  = 0, F p (a) = n p E p F p+1 (a), if n p = 0.

Finally set φ(a) = N −1 E F0 (a) and denote F0 = φ(0). Following essentially the same proof as either [9, Theorem 5] or [11, Theorem 7.1], one can prove that the interpolated free energy φ yields Proposition 2.2 Let α > 0 and x correspond to (k, m, q). We have that       p N ,u,α ≤ F0 − (1 + t) n p θ (q p+1 ) − θ (q p ) − n p θ (q p+1 ) − θ (q p ) + R 0≤ p≤τ

τ < p≤k

(2.5) where θ (q) = qξ  (q) − ξ(q) and lim sup N →∞ |R| = 0. In order to prove (1.4), one needs to find proper parameters (n p )kp=0 such that the righthand side of (2.5) recovers 2P (x, b) as N tends to infinity. From the second and third terms in (2.5), we shall take, throughout the rest of the paper, m1 m τ −1 0 = n0, n1 = (2.6) , . . . , n τ −1 = , nτ = m τ , . . . , nk = m k , 1+t 1+t

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as this choice allows us to handle    n p θ (q p+1 ) − θ (q p ) + (1 + t) 0≤ p<τ

=



mp



 θ (q p+1 ) − θ (q p )



  n p θ (q p+1 ) − θ (q p )

τ ≤ p≤k+1

0≤ p≤k



=

1

ξ  (q)x(q)dq.

(2.7)

0

The main difficulty here comes from the first term F0 . We remark that Talagrand’s proof of the Parisi formula [11] uses the inequality (2.5) with the choice t = 1 in (2.6), which simplifies the covariance structure of (y 1p ) and (y 2p ) in (2.4). Indeed, these random variables satisfy that y 1p = y 2p for 0 ≤ p < τ and y 1p , y 2p are i.i.d. for τ ≤ p ≤ k. As a result, applying λ = 0 in Lemma 2.1 below leads to lim sup N →∞ p N ,u,α ≤ 2P (x, b) directly. In our case, since (y 1p ) and (y 2p ) are governed by the coupling parameter 0 < t < 1, it is not immediately clear why the same inequality would still be valid. We now turn to the control of F0 with a careful tracking of the iterative procedure in terms of t as follows. For b > 1, let ν Nb be the probability measure of N i.i.d. Gaussian random variables with mean zero and variance b−1 , that is,

N

 b 2 b b dν N ( y) = exp −  y2 d y. 2π 2 Let τ Nb = −N −1 log ν Nb ({σ : σ 2 ≥ N }). Without ambiguity, we simply write ν b for ν1b . Given a number λ, we define the function          Bk+1 x 1 , x 2 , λ = log exp x 1 · σ 1 + x 2 · σ 2 + λσ 1 · σ 2 dν Nb σ 1 dν Nb σ 2 for x 1 , x 2 ∈ R N and recursively, for 1 ≤ p ≤ k, ⎧     ⎨ 1 log E exp n p B p+1 x 1 + y1p , x 2 + y2p , λ , np 1 2   Bp x , x , λ = ⎩ E B p+1 x 1 + y1 , x 2 + y2 , λ , p p j

if n p  = 0, if n p = 0,

j

where y p = (yi, p )1≤i≤N for j = 1, 2 and 0 ≤ p ≤ k. Let h = (h, . . . , h). Following the same argument as in the [11, proof of Lemma 7.1], we obtain 1 Lemma 2.1 Let u ∈ [−1, 1], α > 0 and λ ∈ R. If b > 0 ξ  (s)x(s)ds + |λ|, then   1 F0 ≤ −λu + |λ|α + 2τ Nb + E B1 h + y10 , h + y20 , λ . (2.8) N To compute the term N −1 E B1 (h + y10 , h + y20 , λ), we will need a technical lemma. Lemma 2.2 For x 1 , x 2 ∈ R, we define

 1 2 2 x√ +x

1   2   x + x 2 b−λ 1 1 1 2 1 ρ = Jk+1 x , x , λ = log exp ρ , dν1 √ 2(b − λ) 2  1 2 2 x√ −x

1    2   − x x 2 b+λ 2 2 1 2 2 dν1 ρ = , Jk+1 x , x , λ = log exp ρ √ 2(b + λ) 2 

123

(2.9)

Disorder Chaos in the Spherical Mean-Field Model

and recursively for 1 ≤ p ≤ k and j = 1, 2, ⎧   1 + y1 , x 2 + y2 , λ , ⎨ 1 log E exp n p J j x   p p p+1 np j   Jp x 1, x 2, λ = 1 + y1 , x 2 + y2 , λ , ⎩ EJ j x p p p+1 then

if n p  = 0, if n p = 0,

  1 E B1 h + y10 , h + y20 , λ N      b2 + E J11 h + y01 , h + y02 , λ + E J12 h + y01 , h + y02 , λ . = log 2 2 b −λ

(2.10)

Proof For x 1 , x 2 ∈ R and 1 ≤ p ≤ k + 1, we define the following functions          k+1 x 1 , x 2 , λ = log exp x 1 σ 1 + x 2 σ 2 + λσ 1 σ 2 dν b σ 1 dν b σ 2 ,     1 log E exp n p  p+1 x 1 + y 1p , x 2 + y 2p , λ , 1 ≤ p ≤ k.  p x 1, x 2, λ = np Since (σ11 , σ12 ), . . . , (σ N1 , σ N2 ) are independent under the measure ν Nb ×ν Nb , we see recursively j that B p (x 1 , x 2 , λ) = i≤N  p (xi1 , xi2 , λ), where x j = (xi )i≤N for j = 1, 2. Consequently,     1 E B1 h + y10 , h + y20 , λ = E1 h + y01 , h + y02 , λ . N Now by making change of variables σ1 =

ρ1 + ρ2 ρ1 − ρ2 , σ2 = √ √ 2 2

and noting that ρ 1 , ρ 2 are i.i.d. Gaussian with mean zero and variance b−1 , we obtain   k+1 x 1 , x 2 , λ   2 2       ρ x1 + x2 1 x1 − x2 2 (ρ 1 )2 ρ +λ ρ + √ = log exp dν b ρ 1 dν b ρ 2 −λ √ 2 2 2 2   1 2     ρ x1 + x2 1 ρ +λ = log exp dν b ρ 1 √ 2 2   2 2     ρ x1 − x2 2 ρ −λ + log exp dν b ρ 2 √ 2 2      b2 1 2 x 1 , x 2 , λ + Jk+1 x 1, x 2, λ . + Jk+1 = log 2 2 b −λ Since y 1p + y 2p and y 1p − y 2p are independent, starting with (2.9), an iterative argument implies 1 (x 1 + y 1 , x 2 + y 2 ) and J 2 (x 1 + y 1 , x 2 + y 2 ) are independent of each other, that J p+1 p p p p p+1 which yields        b2 1 1 2 2 1 2  p x 1 , x 2 , λ = log x x + J , x , λ + J , x , λ p p b2 − λ2 for 1 ≤ p ≤ k + 1 and hence (2.10).

 

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Proof of Proposition 2.1 The proof is essentially based on an explicit calculation of the right hand-side of (2.10). To lighten notations, we set v p = ξ  (q p+1 ) − ξ  (q p ) if 0 ≤ p ≤ k,  qτ  1 n v = ξ  (s)x(s)ds if 0 ≤ p ≤ τ − 1, dτ = 0, d p = 1 + t qp p≤ ≤τ −1  1  dp = n v = ξ  (s)x(s)ds if τ ≤ p ≤ k, dk+1 = 0. qp

p≤ ≤k

Recall (2.4). It is straightforward to obtain that for τ ≤ p ≤ k   2 2 y 1p + y 2p y 1p − y 2p E =E = vp √ √ 2 2 and for 0 ≤ p < τ, 2 2   y 1p + y 2p y 1p − y 2p     E = 1 + ηt v p , E = 1 − ηt v p . √ √ 2 2 Combining these with the formula that for a standard Gaussian random variable z, nv < L and y ∈ R,  y2 1 L √ 2 1 n  + 2n log L−nv , if n > 0, 2(L−nv) y + vz = y 2 log E exp v n 2L + , if n = 0, 2L

2L

an iterative procedure leads to 



2h 2  2 b − λ − dτ + (1 + ηt)d0   τ −1 b − λ − dτ + (1 + ηt)d p+1 1 1   + log 2 n b − λ − dτ + (1 + ηt)d p p=0 p

E J11 h + y01 , h + y02 , λ = 

+



k b − λ − d p+1 1 1 log 2 p=τ n p b − λ − dp

(2.11)

and E J12



h+

y01 , h

+



y02 , λ

  τ −1 b + λ − dτ + (1 − ηt)d p+1 1 1   = log 2 n b + λ − dτ + (1 − ηt)d p p=0 p +

k b + λ − d p+1 1 1 log . 2 p=τ n p b + λ − dp

Now recall from the statement of Proposition 2.1, d(q) = φu (q) = d(qτ ) +

123

1 q

ξ  (s)x(s)ds and

 1−t d(q) − d(qτ ) . 1+t

(2.12)

Disorder Chaos in the Spherical Mean-Field Model

Since d p = d(q p ) − d(qτ ) and dτ + (1 ± ηt)d p = d(qτ ) +

 1 ± ηt  d(q p ) − d(qτ ) = 1+t

we have 



2h 2

 = 

2 b ∓ λ − dτ + (1 ± ηt)d0



d(q p ), φu (q p ),

if u ≥ 0, if u < 0,

⎧ ⎪ ⎨ 

if u ≥ 0,

⎪ ⎩

if u < 0.

2h 2 , 2 b∓λ−d(0) 2  2h , 2 b∓λ−φu (0)

(2.13)

Using the fundamental theorem of calculus and the fact that x(s) = m p for s ∈ [q p , q p+1 ),   τ −1  b ∓ λ − dτ + (1 ± ηt)d p+1 1   log n b ∓ λ − dτ + (1 ± ηt)d p p=0 p =

τ −1       1   log b ∓ λ − dτ + (1 ± ηt)d p+1 − log b ∓ λ − dτ + (1 ± ηt)d p np p=0

 (1 + t) (1 ± ηt) q p+1 ξ  (s)x(s)   ds mp 1+t b ∓ λ − dτ + (1 ± ηt)d p qp p=0  q ξ  (s) (1 ± t) 0 τ b∓λ−d(s) ds, if u ≥ 0,  qτ ξ  (s) = (1 ∓ t) 0 b∓λ−φu (s) ds, if u < 0,

=

τ −1 

(2.14)

and k k      b ∓ λ − d p+1 1 1   log b ∓ λ − d(q p+1 ) − log b ∓ λ − d(q p ) log = n b ∓ λ − d n p p=τ p p=τ p

 q p+1 k  1 ξ  (s)x(s) ds m p qp b ∓ λ − d(s) p=τ  1 ξ  (s) ds. = qτ b ∓ λ − d(s)

=

(2.15)

Plugging (2.13), (2.14) and (2.15) into (2.11) and (2.12), the Eq. (2.2), Lemmas 2.1, 2.2 and Proposition 2.2 together complete our proof by taking N → ∞ and α ↓ 0 in (2.8) and noting the usual large deviation principle lim N →∞ τ Nb = 2−1 (b − 1 − log b).  

3 Proofs of Main Results We now turn to the proofs of the main results. Throughout this section, (x, b) stands for the 1 optimizer in (1.1). Denote d(q) = q ξ  (s)x(s)ds and φu (q) = d(u) +

 1−t d(q) − d(u) . 1+t

First of all, we start with a proposition that is used to determine the value u ∗ stated in Theorems 1.1 and 1.2. Let u x be the smallest value of the support of x. A crucial fact about u x is that it must satisfy the following equation,

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h 2 + ξ  (u x ) = ux . (b − d(0))2

(3.1)

This can be seen from the [11, proof of Theorem 7.2 ]. In particular, (3.1) implies u x > 0 if h  = 0. Proposition 3.1 For t ∈ (0, 1), define the function f (u) =

h 2 + tξ  (u) −u (b − d(0))2

(3.2)

for u ∈ [−u x , u x ]. Then f (u) = 0 has a unique solution u ∗ . Moreover, u ∗ = 0 when h = 0 and u ∗ ∈ (0, u x ) when h  = 0. Proof Note that ξ  is an odd function. This implies that f is convex on [0, u x ] and is concave on [−u x , 0]. Assume that h  = 0. In this case, since f (0) > 0 and f (u x ) < 0 by (3.1), the intermediate value theorem and the convexity of f on [0, u x ] conclude that f (u) = 0 has a unique solution u ∗ on [0, u x ] and it satisfies u ∗ ∈ (0, u x ). In addition, since from (3.1), f (−u x ) = −

−h 2 + tξ  (u x ) h 2 + ξ  (u x ) + u > − + u x = 0, x (b − d(0))2 (b − d(0))2

the concavity of f on [−u x , 0] and f (0) > 0 imply that f (u) = 0 has no solution on [−u x , 0]. This finishes the proof for the case h  = 0. The situation for h = 0 is essentially identical. If u x = 0, obviously u ∗ = 0. If u x  = 0, we still have f (−u x ) > 0 > f (u x ), but now f (0) = 0. The convexity and concavity of f on [0, u x ] and [−u x , 0] respectively conclude that 0 is the unique solution to f (u) = 0 on [−u x , u x ].   Proof of Theorem 1.1



Pu (x, b, 0) =

Tu (x, b, 0) + Tu (x, b, 0) +

h2 b−d(0) , h2 b−φ|u| (0) ,

if u ∈ [0, 1], if u ∈ [−1, 0),

(3.3)

where from (2.2),   1 − t |u| ξ  (s) 1 + t |u| ξ  (s) ds + ds 2 b − d(s) 2 b − φ|u| (s) 0 0  1  1 ξ  (s) + ξ  (q)x(q)dq. ds + b − 1 − log b − |u| b − d(s) 0

Tu (x, b, 0) :=

(3.4)

First we consider |u| > u x . Since x(q) > 0 for q ∈ (u x , |u|), we have that for all s ∈ [0, |u|),  |u|  2t  d(s) − d(|u|) = d(s) − φ|u| (s) = ξ  (q)x(q)dq > 0 (3.5) 1+t s and from (3.3), Pu (x, b, 0) ≤ Tu (x, b, 0) +

h2 b − d(0)

for any u ∈ [−1, 1]. In addition, from (3.5), the first line of (3.4) is strictly bounded above by  |u|   ξ  (s) 1 − t |u| ξ  (s) 1 + t |u| ξ  (s) ds + ds = ds 2 b − d(s) 2 b − d(s) b − d(s) 0 0 0

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Disorder Chaos in the Spherical Mean-Field Model

and as a result, these inequalities together with the Eq. (1.2) lead to Pu (x, b, 0) < 2P (x, b). This completes the proof for (1.4) with |u| > u x by using Proposition 2.1. As for the case |u| ≤ u x , since x(q) = 0 for q ∈ [0, |u|), we have that for all s ∈ [0, |u|],  1 d(s) = ξ  (q)x(q)dq = φ|u| (s). ux

This allows us to write



b2 h2 + 2 −λ b − λ − d(0)  |u|  ξ  (s) ξ  (s) 1+t 1 − t |u| + ds + ds 2 b − ηλ − d(s) 2 b + ηλ − d(s) 0 0   1 1 ξ  (s) ξ  (s) 1 1 ds + ds + 2 |u| b − λ − d(s) 2 |u| b + λ − d(s)  1 − λu + b − 1 − log b − ξ  (q)x(q)dq

Pu (x, b, λ) = log

b2

0

for all u ∈ [−u x , u x ]. A direct computation gives that Pu (x, b, 0) = 2P (x, b), ∂λ Pu (x, b, 0) = f (u)

and moreover, for λ in a small open neighborhood of 0, |∂λλ Pu (x, b, λ)| ≤ L , where L is a positive constant independent of λ. Consequently, applying the Taylor theorem and taking λ = −δ f (u)/L for sufficiently small δ > 0, if u ∈ [−u x , u x ] and u  = u ∗ , then Proposition 3.1 yields lim sup lim sup p N ,u,α ≤ Pu (x, b, 0) + ∂λ Pu (x, b, 0)λ + α↓0

N →∞

= 2P (x, b) −

L 2 λ 2

δ δ f (u)2  1− L 2

< 2P (x, b). This proves (1.4) for |u| ≤ u x with u  = u ∗ .

 

At the end of this section, we prove Theorem 1.2. It will need an inequality of Gaussian concentration of measure from the [7, Appendix] stated below. Lemma 3.1 Let ν be a finite measure on R N and g(z) be a centered Gaussian process indexed on R N such that Eg(z)2 ≤ a for z in the support of the measure ν. If  X = log exp g(z)dν(z), RN

then for all s > 0, 



P |X − E X | ≥ s ≤ 2 exp −

s2 4a

 .

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W.-K. Chen et al.

Proof of Theorem 1.2 Let ε > 0 and set Sε = [−1, 1] \ (u ∗ − ε, u ∗ + ε). Using (1.1) and (1.4), for any u ∈ Sε , there exist αu > 0 and Nu ∈ N such that p N ,u,αu < 2 p N − ε for all N ≥ Nu . Set Iu = (u − αu , u + αu ). Since {Iu : u ∈ Sε } forms an open covering of Sε , the n I covers S . Letting compactness of Sε implies that there exist u 1 , . . . , u n such that ∪i=1 ui ε N0 = max{Nu i : 1 ≤ i ≤ n}, we obtain that p N ,u i ,αui < 2 p N − ε

(3.6)

for all 1 ≤ i ≤ n and N ≥ N0 . Next, applying Lemma 3.1, the event A N such that   1 1 ε min log Z 1N , log Z 2N ≥ p N − N N 8 and

 1 ε log Z N ,u i ,αui : i = 1, . . . , n ≤ p N ,u,α + max N 8 

has probability at least 1 − K exp(−N /K ), where K > 0 is independent of N . On A N , these inequalities and (3.6) lead to n      1{ R1,2 ∈Sε } ≤ 1{R1,2 ∈Iui } i=1

≤

n 

  exp log Z N ,u i ,αui − log Z 1N − log Z 2N

i=1

  ε  exp −N p N ,u i ,αui − 2 p N + 4 i=1  3εN  ≤ n exp − . 4

≤

Therefore,

n 

   3εN   E 1{ R1,2 ∈Sε } ≤ n exp − P A N + P AcN 4



 3εN N ≤ n exp − + K exp − , 4 K 

which completes our proof.



 

Acknowledgments The authors thank anonymous referees for the careful reading and giving several suggestions regarding the presentation of the paper. Wei-Kuo Chen and Yuan-Chung Sheu also thank CMMSC(NCTU, Taiwan) and NCTS(Taiwan) for the partial supports during the early stage of the project.

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