Cavity method in the spherical SK model

www.imstat.org/aihp 2009, Vol. 45, No. 4, 1020–1047

DOI: 10.1214/08-AIHP193

© Association des Publications de l’Institut Henri Poincaré, 2009

Cavity method in the spherical SK model ¹

Dmitry Panchenko

Department of Mathematics, Texas A&M University, Mailstop 3386, Room 209, College Station, TX 77843, USA.

E-mail: panchenk@math.tamu.edu

Received 13 February 2008; revised 6 September 2008; accepted 15 September 2008

Abstract. We develop a cavity method for the spherical Sherrington–Kirkpatrick model at high temperature and small external field. As one application we compute the limit of the covariance matrix for fluctuations of the overlap and magnetization.

Résumé. Nous développons la méthode de la cavité pour le modèle sphérique de Sherrington–Kirkpatrick à haute température et champs externe faible. Nous illustrons la méthode par le calcul de la matrice de covariance des fluctuations des recouvrements et de la magnétisation.

MSC:60K35; 82B44

Keywords:Sherrington–Kirkpatrick model; Cavity method

1. Introduction

The cavity method in the Sherrington–Kirkpatrick model [3], as described for example in [5], is one of the most important tools developed for the analysis of the model at high temperature. Among many applications of the cavity method, the most important include self-averaging and Gaussian fluctuations of the overlap (see [4] or [2]). The present paper was motivated by an attempt to find an analogue of the cavity method for the spherical model and, surprisingly, the task turned out to be more challenging than expected. Of course, one can anticipate technical difficulties due to the fact that uniform measure on the sphere is not a product measure and decoupling one coordinate from the others, which is the idea of the cavity method, is not straightforward. As we shall see, the suggested interpolation has some new features compared to [5].

After we develop the cavity method, as one application we will study the fluctuations of the overlap and magne- tization and compute their covariance matrix in the thermodynamic limit. We stop short of proving a central limit theorem since our goal is to give a reasonably simple illustration of the cavity method.

We consider a spherical SK model with Gaussian Hamiltonian (process)H_N(σ)indexed byspin configurationsσ on the sphereS_N of radius√

NinR^N. We will assume that 1

NEH_N σ¹

H_N σ²

=ξ(R_1,2), (1.1)

1Supported in part by NSF grant.

whereR_1,2=N⁻¹

i≤Nσ_i¹σ_i²isthe overlapof configurationsσ¹,σ²∈S_Nand where the functionξ(x)is three times continuously differentiable. A classical example of such Gaussian process, corresponding to the choice ofξ(x)=x^p for integerp≥1,is thep-spin Hamiltonian

H_N(σ)= 1 N^(p−1)/2

1≤i₁,...,i_p≤N

g_i1,...,ipσ_i1· · ·σ_ip,

where(g_i1,...,ip)are i.i.d. standard Gaussian random variables. One could also consider a linear combination ofp-spin Hamiltonians.

This model was studied in [1] and mathematically rigorous results were proved in [6]. Under the additional as- sumptions onξ,

ξ(0)=0, ξ(x)=ξ(−x), ξ(x) >0 ifx >0, (1.2)

the limit of the free energy F_N= 1

NElog

S_N

exp

βH_N(σ)+h

i≤N

σ_i

λ_N(σ) (1.3)

was computed in [6] for arbitraryinverse temperatureβ >0 andexternal fieldh∈R.Hereλ_N denotes the uniform probability measure onS_N.The main results of the present paper will be proved for small enough parametersβandh, i.e. for very high temperature and small external field, andwithoutthe assumptions in (1.2). For example, our results will apply top-spin Hamiltonian for both even and oddp≥1.

To provide some motivation, let us first describe some implications of the results in [6]. For smallβ andh,they imply that under (1.2) the limit of the free energy takes a particularly simple form:

Nlim→∞F_N= inf

q∈[0,1]

1 2

h²(1−q)+ q

1−q +log(1−q)+β²ξ(1)−β²ξ(q)

. (1.4)

The critical point equation for the infimum on the right hand side of (1.4) is h²+β²ξ(q)= q

(1−q)². (1.5)

For small enoughβ the infimum in (1.4) is achieved atq=0 ifh=0 and at the unique solutionq of (1.5) ifh =0.

Theorem 1.2 in [6] suggests that the distribution of the overlapR_1,2with respect to the Gibbs measure is concentrated nearq and by analogy with the classical SK model (see Chapter 2 in [5] or [2]) one expects that the distribution of

√N (R1,2−q) is approximately Gaussian. In the setting of the classical SK model this is proved using the cavity method and the main goal of the present paper will be to develop the analogue of the cavity method for the spherical SK model. Many computations will be more involved and, as a result, instead of using the cavity method to prove a central limit theorem we will only compute the covariance matrix of the overlaps and other related quantities. This main application is formulated in Theorem 5 in Section 3.

We note that our results also imply (1.4) without the assumption (1.2). Namely, since we will prove that for small βandhthe overlapR_1,2is concentrated near the unique solutionqof (1.5), it is a simple exercise to show that in this case

Nlim→∞F_N=1 2

h²(1−q)+ q

1−q +log(1−q)+β²ξ(1)−β²ξ(q)

. (1.6)

To prove this, one only needs to compare the derivatives of both sides with respect toβ.Finally, we suggest to an interested reader not familiar with the original cavity method to learn about it in Chapter 2 in [5], because we believe that one can gain much more by first learning a technically simpler case from which all the main ideas originate.

2. Cavity method

For specificity, from now on we assume thath =0 andβis small enough so thatqis theuniquesolution of (1.5). The case ofh=0 is also significantly simpler because one can use the second moment method as in [5], Section 2.2. All the results below are proved without the assumption (1.2). Given a configurationσ∈S_N,we will denoteε=σ_N and fori≤N−1 denote

ˆ σi=σi

N−ε²

N−1,

so that a vector σˆ =(σˆ₁, . . . ,σˆ_N−1)∈S_N−1,i.e. | ˆσ| =√

N−1. We consider a Gaussian HamiltonianH_N−1(σˆ) independent ofH_N(σ)such that

1

N−1EHN−1

σˆ¹ HN−1

σˆ²

=ξ(Rˆ1,2), (2.1)

whereRˆ1,2=(N−1)⁻¹

i≤N−1σˆ_i¹σˆ_i².We define theinterpolating Hamiltonianby H_t(σ)=√

t βH_N(σ)+√

1−t βH_N−1(σˆ)+h

i≤N−1

ˆ σ_i

1+t

N−ε²

N−1 −1

+hε+√

1−t εzβ

ξ(q)−1

2(1−t )ε²b, (2.2)

wherezis a standard Gaussian r.v. independent ofHN andHN−1and

b=h²(1−q)+β²(1−q)ξ(q). (2.3)

Define thepartition functionalong the interpolation by Z_t=

S_N

expH_t(σ)dλN(σ)

and for a functionf:S_Nⁿ →Rdefine theGibbs averageoff with respect to the Hamiltonian (2.2) by ft= 1

Z_tⁿ

S_Nⁿ

fexp

l≤n

Ht

σ^l

dλⁿ_N. (2.4)

Letν_t(f )=Eftbe the annealed Gibbs average.

The main purpose of this interpolation is to devise a way of computingν(f ):=ν₁(f )for some particular choices of f – typically, functions of the overlaps – and finding the right expression forH_tis always motivated by the properties that such interpolation should have in order to make these computations work.

First of all, any cavity method aims to decouple the last coordinate ε from the other coordinates to make the computation ofν0(f )easier, in some sense. Notice the special form of the Hamiltonian (2.5) att=0,

H₀(σ)=βH_N−1(σˆ)+h

i≤N−1

ˆ

σ_i+εa−1

2ε²b, (2.5)

where we introduced the notation a=zβ

ξ(q)+h. (2.6)

The first two terms depend only onσˆ∈S_N−1and, therefore, the Gibbs average (2.4) att=0 for functions of the type f1(σˆ)f2(ε)will decouple (see (2.15) below). In other words, att=0 we can think ofεandσˆas independent variables,

even thoughσˆ formally depends onε.This explains a somewhat unusual dependence ofH_N−1on(σˆ₁, . . . ,σˆ_N−1)in the above interpolation instead of the more expected dependence on(σ1, . . . , σN−1).This is probably the main new feature compared to the classical SK model on the hypercube{−1,+1}^N.

Another desired property of the interpolation is thatν_t(f )does not change much fort∈ [0,1]so thatν(f )can indeed be approximated byν₀(f ).The proof of this will have several ingredients and, of course, the appropriate choice of parameterqin (1.5) will play an important role at some point. The main ingredients, however, are the computation and control of the derivative ofν_t(f )with respect tot. This computation will be a first step toward clarifying the role of all the terms in (2.2). Forqin (1.5) we define

r=h(1−q). (2.7)

Given(σ^l)_l≥1we denoteε_l=σ_N^l ,Rˆ_l=(N−1)⁻¹

i≤N−1σˆ_i^land definea_l anda_l,l by a_l=1−ε²_l, 2al,l=ξ(q)−1

2

ε_l²+ε_l²

qξ(q)+ξ(q)

+ε_lε_lξ(q). (2.8)

Theorem 1. We have ν_t(f )=h

2

l≤n

νt

f al(Rˆl−r)

−h 2nνt

f an+1(Rˆn+1−r)

+2β²

1≤l<l≤n

ν_t

f a_l,l(Rˆ_l,l−q)

−2nβ²

l≤n

ν_t

f a_l,n+1(Rˆ_l,n+1−q) +n(n+1)β²ν_t

f a_n+1,n+2(Rˆ_n+1,n+2−q)

+ν_t(fR), (2.9)

where the remainderRis bounded by

|R| ≤ L N

β²+h

1+

l≤n+2

ε⁴_l

+Lβ²

1≤l =l≤n+2

1+ε_l²

(Rˆ_l,l−q)².

This computation will be carried out in Section 4. In order for the interpolation to be useful, the above derivative should be small. Notice that all the terms in (2.9) have factors of four typesa_l, a_l,l, (Rˆ_l−r)or(Rˆ_l,l−q)which is why our goal will be to show that the last coordinateεis of order one and the overlapRˆ_l,l and magnetizationRˆ_lare concentrated near constantsq andr. The corresponding results will be proved in Sections 5 and 6.

Theorem 2. Ifβandhare small enough,we can find a constantL >0such that νt

exp 1

Lε²

≤L (2.10)

for allt∈ [0,1].

Theorem 3. Ifβandhare small enough then for anyK >0we can findL >0such that ν_t

I

| ˆR_1,2−q| ≥L logN

N

1/4

≤ L

N^K, (2.11)

νt

I

| ˆR1−r| ≥L logN

N

1/4

≤ L

N^K (2.12)

for allt∈ [0,1].

A step in the proof of Theorem 3, where the conditions (1.5) and (2.7) that define parametersqandrfinally appear, will further clarify the role of all the terms in (2.2). In some sense, this is where all the pieces will fall together.

Finally, let us explain what happens at the end of the interpolation att=0.We start by writing the integration over S_N as a double integral over εand(σ1, . . . , σ_N−1).Letλ^ρ_N denote the area measure on the sphereS_N^ρ of radiusρ inR^N,and let|S_N^ρ|denote its area, i.e.|S_N^ρ| =λ^ρ_N(S_N^ρ).Then,

SN

f (σ)dλN(σ)= 1

|S

√N N |

S

√N N

f (σ1, . . . , σ_N)dλ

√N

N (σ1, . . . , σ_N)

= 1

|S

√N N |

^√N

−√ N

dε

1−ε²/N

S

√N−ε2 N−1

f (σ1, . . . , σ_N−1, ε)dλ

√N−ε²

N−1 (σ1, . . . , σ_N−1)

= ^√_N

−√ N

|S

√N−ε² N−1 |

|S

√N N |

dε

1−ε²/N

SN−1

f

ˆ σ1

N−ε²

N−1, . . . ,σˆN−1

N−ε²

N−1, ε

dλN−1(σˆ)

=a_N ^√_N

−√ N

dε

1−ε² N

(N−3)/2

S_N−1

f

ˆ σ

N−ε²

N−1, ε

dλN−1(σˆ), (2.13) whereaN= |S_N¹₋₁|/(|S_N¹|√

N )→(2π)^−1/2as can be seen by takingf =1.In particular, if f (σ)=f1(ε)f2(σˆ),

then

SN

f (σ)dλN(σ)=a_N ^√N

−√ N

f1(ε)

1−ε² N

(N−3)/2

dε

SN−1

f2(σˆ)dλN−1(σˆ). (2.14) Since the Hamiltonian (2.5) decomposed into the sum of terms that depend only onεor only onσˆ,(2.14) implies that

f0= f10f20, (2.15)

where f1(ε)

0= 1 Z₁

^√_N

−√ N

f1(ε)

1−ε² N

_(N−3)/2

exp

aε−1 2bε²

dε, (2.16)

Z₁= ^√_N

−√ N

1−ε²

N

(N−3)/2

exp

aε−1 2bε²

dε and

f2(σ)ˆ

0= 1 Z₂

SN−1

f2(σˆ)exp

HN−1(σ)ˆ +h

i≤N−1

ˆ σi

dλN−1(σˆ), (2.17)

Z2=

SN−1

exp

HN−1(σˆ)+h

i≤N−1

ˆ σi

dλN−1(σˆ).

As we mentioned above, this decoupling is a crucial feature of the cavity method. Using (2.15), (2.16), we will be able to compute the momentsν₀(ε₁^k1· · ·ε_n^kn)for integer k_i ≥0, which is an important part of the second moment computations and of the cavity method in general. The following holds. Let us recall (2.3), (2.6) and defineγ0= 1, γ1=a/(b+1)and, recursively, fork≥2

γ_k= a

b+1γ_k−1+k−1

b+1γ_k−2. (2.18)

Theorem 4. For small enoughβ >0, ν0

ε₁^k1· · ·ε_n^kn

−Eγk1· · ·γkn≤ L

N, (2.19)

where the constantLis independent ofN.

This theorem will be proved in Section 5 below.

3. Second moment computations

We are now ready to demonstrate the promised application of the cavity interpolation, namely, the computation of the covariance matrix for the fluctuations of a certain collection of overlaps and magnetizations. Let us introduce the following seven functions

f1=(R1,2−q)², f2=(R1,2−q)(R1,3−q), f3=(R1,2−q)(R3,4−q),

f4=(R1,2−q)(R1−r), f5=(R1,2−q)(R3−r), f6=(R1−r)², (3.1) f7=(R1−r)(R2−r)

and letv_N=(ν(f₁), . . . , ν(f₇)).In this section we will compute the vectorNv_Nup to the terms of order o(1). As we mentioned above, it is likely that with more effort one can prove the central limit theorem for the joint distribution of

√N (R_1,2−q), √

N (R_1,3−q), √

N (R_3,4−q), √

N (R₁−r), √

N (R₂−r),

so the computation of this section identifies the covariance matrix of the limiting Gaussian distribution. To describe our main result let us first summarize several computations based on Theorem 4. The definition (2.18) implies that

γ₁= a

b+1, γ₂= a

b+1 2

+ 1

b+1, γ₃= a

b+1 3

+ 3a

(b+1)². (3.2)

The definition (2.3) and (1.5) imply that 1

b+1 = 1

1+(1−q)(β²ξ(q)+h²)=1−q.

Therefore, E a

b+1 =(1−q)Ea=(1−q)h=r, (3.3)

E a

b+1 2

=(1−q)²Ea²=(1−q)²

β²ξ(q)+h²

=q, (3.4)

where we used (1.5) again, and W:=E

a b+1

3

=(1−q)³Ea³=(1−q)³

3β²ξ(q)h+h³

, (3.5)

U:=E a

b+1 4

=(1−q)⁴Ea⁴=(1−q)⁴

h⁴+6β²h²ξ(q)+3β⁴ξ(q)²

. (3.6)

For simplicity of notations let us write x∼y ifx=y+O

N⁻¹ .

Then it is trivial to check that Theorem 4 and (3.2)–(3.6) imply the following relations:

ν₀(ε₁)∼r, ν₀(ε₁ε₂)∼q, ν₀(ε²₁)∼1, ν₀(ε₁ε₂ε₃)∼W, ν0(ε1ε²₂)∼W+h(1−q)², ν0(ε₁³)∼W+3h(1−q)²,

(3.7) ν0(ε₁²ε²₂)∼U+1−q², ν0(ε1ε2ε₃²)∼U+q−q²,

ν0(ε1ε³₂)∼U+3q−3q², ν0(ε1ε2ε3ε4)∼U.

Let us recall the definitionsalanda_l,lin (2.8). Using relations (3.7) it is now straightforward to compute the following nine quantities

ν0

a1,2(ε1ε2−q)

∼Y1, ν0

a1,3(ε1ε2−q)

∼Y2, ν0

a3,4(ε1ε2−q)

∼Y3, ν0

a1(ε1ε2−q)

∼Y4, ν0

a3(ε1ε2−q)

∼Y5, ν0

a1,2(ε1−r)

∼Y6, (3.8)

ν₀

a_2,3(ε₁−r)

∼Y₇, ν₀

a₁(ε₁−r)

∼Y₈, ν₀

a₂(ε₁−r)

∼Y₉,

whereY1, . . . , Y9are functions ofq, r, h, U, W.We omit the explicit formulas forY_j’s since they do not serve any particular purpose in the sequel. Let us define a(7×7)-matrixMthat consists of four blocks

M=

M₁ O₁ O₂ M₂

, (3.9)

whereO1is a(3×2)-matrix andO2is a(4×3)-matrix both entirely consisting of zeros, M₁=

⎛

⎝2β²Y1 −8β²Y2 6β²Y3 hY4 −hY5

2β²Y2 2β²(Y1−2Y2−3Y3) 6β²(−Y2+2Y3) ^h₂(Y4+Y5) ^h₂(Y4−3Y5) 2β²Y₃ 8β²(Y₂−2Y3) 2β²(Y₁−8Y2+10Y3) hY₅ h(Y₄−2Y5)

⎞

⎠,

M₂=

⎛

⎜⎝

2β²(Y₁−2Y2) 2β²(−2Y2+3Y3) (h/2)Y4 (h/2)(Y4−2Y5) 2β²(2Y2−3Y4) 2β²(Y₁−6Y2+6Y3) (h/2)Y5 (h/2)(2Y4−3Y5)

−2β²Y₆ 2β²Y₇ (h/2)Y8 −(h/2)Y9

2β²(Y₆−2Y7) 2β²(−2Y6+3Y7) (h/2)Y9 (h/2)(Y8−2Y9)

⎞

⎟⎠.

Finally, we define a vectorv=(v1, . . . , v7)by

v1=(1−q)U+1−4q²+3q³, v2=(1−q)U+q(1−q)(1−2q), v3=(1−q)U−q²(1−q), v4=W−1

2rU+1 2r

2−6q+3q² ,

(3.10) v5=W−1

2rU+1 2r

−2q+q²

, v6= −1

2rW+1+1

2r²(−4+3q), v7= −1

2rW+q+1

2r²(−2+q).

We are now ready to formulate the main result of this section.

Theorem 5. For small enoughβandhwe have (I−M)v^T_N= 1

Nv^T+o N⁻¹

. (3.11)

Herev^Tdenotes the transpose of vectorv. In the statement of the Theorem we implicitly assumed thatβ andhare small enough so that(I−M)is invertible. Notice that each entry in the matrixMhas either a factor ofβ²orhand, therefore, for small enoughβ andhthe matrix(I−M)will indeed be invertible. Theorem 5 implies

v^T_N= 1

N(I−M)⁻¹v^T+o N⁻¹

.

In the remainder of this section we will prove Theorem 5.

For each functionf_l in (3.1), we will definefˆ_l by replacing each occurrence ofR byR,ˆ i.e.fˆ₁=(Rˆ_1,2−q)², fˆ2=(Rˆ1,2−q)(Rˆ1,3−q), etc. Next, we introduce functions

f₁=(ε₁ε₂−q)(R_1,2−q), f₂=(ε₁ε₂−q)(R_1,3−q), f₃=(ε₁ε₂−q)(R_3,4−q),

f₄=(ε1ε2−q)(R1−r), f₅=(ε1ε2−q)(R3−r), f₆=(ε1−r)(R1−r), (3.12) f₇=(ε1−r)(R2−r).

As in the classical cavity method in [5], we introduce these functions because, first of all, by symmetry, ν(f_l)=ν

f_l

(3.13) and, second of all, emphasizing the last coordinate inf_lis perfectly suited for the application of the cavity method. As above, for each functionf_lwe will definefˆ_lby replacing each occurrence ofRbyR,ˆ i.e.fˆ₁=(ε1ε2−q)(Rˆ1,2−q), etc.

To simplify the notations we will writex≈y whenever

|x−y| =o 1

N +ν₀

(Rˆ_1,2−q)² +ν₀

(Rˆ₁−r)²

. (3.14)

The proof of Theorem 5 will be based on the following.

Theorem 6. For small enoughβandh,for alll≤7, ν₀(fˆ_l)≈ν₀

f_l

+ν₀fˆ_l

. (3.15)

We will start with a couple of lemmas.

Lemma 1. Iff≥0andf_∞is bounded independently ofN then for anyK >0we can findL >0such that νt(f )≤L

N^−K+ν0(f )

. (3.16)

Proof. The derivativeν_t(f )in (2.9) consists of a finite sum of terms of the typeν_t(fp_εg)wherep_εis some polyno- mial in the last coordinates(ε_l)andgis one of the following:

Rˆ_l,l−q, Rˆ_l−r, (Rˆ_l,l−q)², N⁻¹. (3.17)

Theorem 2 and Chebyshev’s inequality imply ν_t

I

|ε_l| ≥logN

≤LN^−K

and combining this with Theorem 3 yields that for anygin (3.17), νt

I

|pεg| ≥N^−1/8

≤LN^−K. Therefore, one can control the derivative

ν_t(f )≤LN^−K+LN^−1/8ν_t(f )≤L

N^−K+ν_t(f )

(3.18)

and (3.16) follows by integration.

Lemma 2. For small enoughβ andhand alll≤7we have ν(fl)≈ν0(fˆl) and ν₀

f_l

≈ν₀fˆ_l

. (3.19)

Proof. We will only consider the casel=1, f1=(R_1,2−q)², since other cases are similar. We have ν

(R_1,2−q)²

−ν₀

(R_1,2−q)²≤sup

t

ν_t

(R_1,2−q)²

≤sup

t

LN^−K+LN^−1/8νt

(R1,2−q)²

≤

LN^−K+LN^−1/8ν₀

(R_1,2−q)²

, (3.20)

where in the second line we used (3.18) and then (3.16). We will use that

R_1,2= ˆR_1,2+s(ε₁, ε₂)Rˆ_1,2+N⁻¹ε₁ε₂, (3.21)

where

s(ε1, ε2)=

1−ε²₁ N

1−ε²₂

N

−1.

Since √

1+x−1−x 2

≤Lx² forx∈ [−1,1] (3.22)

we have

s(ε1, ε2)+ 1 2N

ε²₁+ε²₂≤ L N²

ε⁴₁+ε⁴₂

. (3.23)

Since by (3.21)

R_1,2−q=(Rˆ_1,2−q)+

1−ε₁² N

1−ε₂²

N

−1

Rˆ_1,2+ 1

Nε₁ε₂, (3.24)

squaring both sides and using (3.23) yields (R1,2−q)²−(Rˆ1,2−q)²≤ 1

Np_ε| ˆR1,2−q| + 1 N²p_ε, where from now onp_εdenotes a quantity such that

|p_ε| ≤L

1+

l

ε_l⁴

. Therefore,

ν₀

(R_1,2−q)²

−ν₀

(Rˆ_1,2−q)²≤ 1 Nν₀

p_ε| ˆR_1,2−q| + 1

N²ν₀(p_ε)=o N⁻¹

by Theorems 2 and 3. Thus, (3.20) implies the first part of (3.19). To prove the second part of (3.19) we notice that f₁− ˆf₁=(ε1ε2−q)(R1,2− ˆR1,2)≤ 1

Np_ε

by (3.21) and (3.23). Since each term in the derivativesν₀(f₁)andν₀(fˆ₁)will contain another factor from the list

(3.17), Theorems 2 and 3 imply the result.

Proof of Theorem 6. We start by writing ν

f_l

−ν0

f_l

−ν₀

f_l≤sup

t

ν_t f_l. If we can show that

sup

t

ν_t

f_l≈0 (3.25)

and, thus,ν(f_l)≈ν0(f_l)+ν₀(f_l),then Lemma 2 and (3.13) will imply ν0(fˆ_l)≈ν(f_l)=ν

f_l

≈ν0

f_l +ν₀

f_l

≈ν0

f_l

+ν₀fˆ_l ,

which is precisely the statement of Theorem 6. To prove (3.25) we note that by (2.9) the second derivativeν_t(f_l)will consist of the finite sum of terms of the typef_lp_εg₁g₂whereg₁, g₂are from the list (3.17). Clearly,

|g1g2| ≤L 1

N² +(Rˆ_l,l−q)²+(Rˆ_l−r)²

and since eachf_lcontain another small factor(R_l,l−q)or(Rl−r),Theorems 2 and 3 imply (3.25).

We are now ready to prove Theorem 5.

Proof of Theorem 5. Let us first note thatν₀(fˆ_l)is defined exactly the same way asν(f_l)forN−1 instead ofN.In other words,

v⁰_N:=

ν₀(fˆ₁), . . . , ν₀(fˆ₇)

=v_N−1

and, therefore, it is enough to prove that (I−M)v^0T_N = 1

Nv^T+o N⁻¹

. (3.26)

Replacing 1/N by 1/(N−1) on the right hand side is not necessary since the difference is of orderN⁻². Each equation in the system of Eqs (3.26) will follow from the corresponding equation (3.15). Namely, we will show that

ν0

f₁ , . . . , ν0

f₇

≈ 1

Nv and ν₀fˆ₁

, . . . , ν₀fˆ₇T

≈Mv^0T_N. (3.27)

Then (3.15) will imply thatv^0T_N ≈N⁻¹v^T+Mv^0T_N. However, since the definition (3.14) means that the error in each equation is of order o(N⁻¹+ν0(fˆ1)+ν0(fˆ6)), this system of equation can be rewritten as

(I−M−EN)v^0T_N = 1 Nv+o

N⁻¹ ,

where the matrixENis such thatEN =o(1). Therefore, whenever the matrixI−Mis invertible (for example, for smallβandh) we have forN large enough

v^0T_N =(I−M−EN)⁻¹ 1

Nv^T+o N⁻¹

= 1

N(I−M)⁻¹v^T+o N⁻¹

.

Hence, to finish the proof we need only to show (3.27). We will only carry out the computations forl=1 since all other cases are similar. Let us start by proving thatν0((ε1ε2−q)(R1,2−q))≈v1.Using (3.24) and (2.15), we write

ν0

(ε1ε2−q)(R1,2−q)

= 1 Nν0

ε1ε2(ε1ε2−q)

+ν0(ε1ε2−q)ν0(Rˆ1,2−q)

+qν0

(ε1ε2−q)

1−ε²₁ N

1−ε²₂

N

−1

+ν0

(ε1ε2−q)

1−ε²₁ N

1−ε₂²

N

−1

ν0(Rˆ1,2−q).

Using (3.23), one can bound the last term by 1

Nν0(pε)ν0(Rˆ1,2−q)=o N⁻¹ by Theorems 2 and 3. The term

ν0

(ε1ε2−q) ν0

(Rˆ1,2−q)

=o N⁻¹

by Theorem 3 and the second relation in (3.7), i.e.ν0(ε1ε2−q)∼0.Finally, we use

1−ε₁² N

1−ε²₂

N

−1+ ε²₁ 2N + ε²₂

2N ≤ 1

N²p_ε to observe that

qν₀

(ε₁ε₂−q)

1−ε²₁ N

1−ε₂²

N

−1

≈ − 1 2Nqν₀

(ε₁ε₂−q)

ε₁²+ε₂²

= −1 Nqν₀

(ε₁ε₂−q)ε₁² by symmetry and, therefore,

ν₀

(ε₁ε₂−q)(R_1,2−q)

≈ 1 N

ν₀

ε₁ε₂(ε₁ε₂−q)

−qν₀

(ε₁ε₂−q)ε₁²

= 1 N

ν0

ε₁²ε₂²

−qν0(ε1ε2)−qν0

ε₁³ε2

+q²ν0

ε²₁

≈v1

by using (3.7) and comparing with the definition ofv1in (3.10).

Next, we need to show the second part of (3.27) forl=1,i.e.

ν₀

(ε1ε2−q)(Rˆ1,2−q)

≈ Mv^0T_N

1,

where(·)1denotes the first coordinate of a vector. We use (2.9) forn=2 to writeν₀((ε1ε2−q)(Rˆ1,2−q))as hν0

a1(ε1ε2−q)(Rˆ1,2−q)(Rˆ1−r)

−hν0

a3(ε1ε2−q)(Rˆ1,2−q)(Rˆ3−r) +2β²hν0

a1,2(ε1ε2−q)(Rˆ1,2−q)²

−8β²hν0

a1,3(ε1ε2−q)(Rˆ1,2−q)(Rˆ1,3−q) +6β²hν0

a3,4(ε1ε2−q)(Rˆ1,2−q)(Rˆ3,4−q) +ν0

(ε1ε2−q)(Rˆ1,2−q)R

≈2β²Y1ν0(fˆ1)−8β²Y2ν0(fˆ2)+6β²Y3ν0(fˆ3)+hY4ν0(fˆ4)−hY5ν0(fˆ5)+ν0

fˆ₁R

= Mv^0T_N

1+ν0

fˆ₁R ,

where in second to last line we used (3.8) and the last line follows by comparison with the definition ofMin (3.9).

Finally, since clearlyν0(fˆ₁R)≈0 by Theorems 2 and 3, this finishes the proof of Theorem 5.

4. Derivative along the interpolation

In this section we prove Theorem 1. We start by writing ν_t(f )=E

f

l≤n

∂

∂tHt

σ^l

t

−nE

f ∂

∂tHt(σⁿ⁺¹)

t

(4.1)

and

∂

∂tH_t(σ)= β 2√

tH_N(σ)− β 2√

1−tH_N−1(σˆ)+h

i≤N−1

ˆ σ_i

N−ε²

N−1 −1

− 1 2√

1−tεzβ

ξ(q)+1

2ε²b. (4.2)

In order to use a Gaussian integration by parts (see, for example, (A.41) in [5]) we first compute the covariance Cov

H_t

σ¹ , ∂

∂tH_t σ²

=β² 2

N ξ(R1,2)−(N−1)ξ(Rˆ1,2)−ε1ε2ξ(q)

by (1.1) and (2.1). We will rewrite this using Taylor’s expansion ofξ(R_1,2)nearRˆ_1,2.By assumption,ξis three times continuously differentiable and (3.21), (3.23) imply

ξ(R1,2)−ξ(Rˆ1,2)−ξ(Rˆ1,2)(R1,2− ˆR1,2)≤ L N²

ε⁴₁+ε⁴₂

and

ξ(R1,2)−ξ(Rˆ1,2)+ 1 2N

ε²₁+ε²₂Rˆ1,2ξ(Rˆ1,2)− 1

Nε1ε2ξ(Rˆ1,2) ≤ L

N²

ε⁴₁+ε⁴₂ . Therefore,

N ξ(R_1,2)−(N−1)ξ(Rˆ_1,2)=ξ(Rˆ_1,2)−1 2

ε²₁+ε²₂Rˆ_1,2ξ(Rˆ_1,2)+ε₁ε₂ξ(Rˆ_1,2)+R1, (4.3)

where from now onR1will denote a quantity such that

|R1| ≤L N

1+

l≤n+2

ε_l⁴

.

Sinceξ is three times continuously differentiable,

ξ(Rˆ1,2)−ξ(q)=ξ(q)(Rˆ1,2−q)+R2, ξ(Rˆ1,2)−ξ(q)=ξ(q)(Rˆ1,2−q)+R2, Rˆ_1,2ξ(Rˆ_1,2)−qξ(q)=

ξ(q)+qξ(q)

(Rˆ_1,2−q)+R2, whereR2denotes a quantity such that

|R2| ≤L(Rˆ_1,2−q)².

Using this in (4.3) and recalling the definition ofa_l,l in (2.8) we get Cov

H_t

σ^l , ∂

∂tH_t σ^l

=β² 2

2al,l(Rˆ_l,l−q)−1 2

ε²_l +ε²_l

qξ(q)+ξ(q)

+β²R3, (4.4)