N dependence of upper bounds of critical temperatures of 2D O(N) spin models

N dependence of upper bounds of critical temperatures of 2D O(N) spin models

著者 Ito K. R., Tamura Hiroshi

journal or

publication title

Communications in Mathematical Physics

volume 202

number 1

page range 127‑168

year 1999‑04‑01

URL http://hdl.handle.net/2297/1770

N Dependence of Upper Bounds of Critical Temperatures of 2D O(N ) Spin Models

K. R. Ito

Department of Mathematics and Physics, Setsunan University, Ikeda-Naka, Neyagawa 572, Japan

H. Tamura

Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa 920-11, Japan

Abstract

We investigate critical temperature of the classical O(N ) spin model in two dimensions. We show that if N is large and there is a phase transition in the system, the critical inverse temperature β

obeys the bound β

(N ) >

const. N log N .

Running Head: Critical Temperature of 2D O(N ) Spin Model

I. INTRODUCTION

Quark confinement in 4 dimensional non-abelian lattice gauge thoeries and spontaneous mass generations in two dimensional (2D) non-abelian sigma models are widely believed

∗

E-mail : [email protected]

also at : Division of Mathematics, College of Human and Environmental Studies, Kyoto University, Kyoto 606, Japan.

†

E-mail:[email protected]

[18]. These models exhibit no phase transitions in the hierarchical model approximation of Wilson-Dyson type or Migdal-Kadanov type [10], but we still do not have a rigorous proof for the real system.

We recently considered a block-spin-type transformation of random walk which appears in the O(N ) spin models [3,4], and showed that [11] the correlation functions are represented by self-avoiding walks on Z

. This considerably improves our previous estimates for the inverse critical temperature β

of the system

β

N ≥ µ

µ

− 1 , as N → ∞ (1.1)

where µ

∈ (ν, 2ν −1) is the connective constant of self-avoiding walk on Z

(µ

= 2.653 · · ·).

In this paper, we amalgamate our previous methods with the idea of the N

expansion [14,15] and the cluster expansion [5,9,13,16], the technology to represent quantities of infinite volume limit by finite volume quantities. In a spirit, our single block cluster expansion is similar to that in [1]. Our main conclusion in this paper is

Main Theorem The critical inverse temperature β

(N ) of the two-dimensional O(N ) Heisenberg Model obeys the following bound for large N :

β

(N ) > const. N log N (1.2)

where const. > 0 is independent of N .

This result is announced in [12]. As will be discussed, for the dimension ν > 2, we have G

(0) ≥ β

(N )

N ≥ 1 µ

(1.3) where G

(x) is the lattice Green’s function on the ν dimensional lattice Z

. Therefore a strong deviation exists in the N dependence of the critical temperature of the 2D O(N ) Heisenberg model. We expect a combination of the present method and renormalization group type argumemts will establish our longstanding conjecture on the 2D sigma model.

The ν dimensional O(N ) spin (Heisenberg) model is defined by the Gibbs measure

< F >≡ 1 Z

(β)

Z

F (φ) exp[−H

(φ)]

i

δ(φ

− 1)dφ

. (1.4)

Here Λ ⊂ Z

is the large square with its center at the origin. Moreover φ(x) = (φ(x)

, · · · , φ(x)

) is the vector valued spin at x ∈ Λ, Z

is the partition function defined so that < 1 >= 1. H

is the Hamiltonian given by

H

≡ − β(N ) 2

X

|x−y|₁=1

φ(x)φ(y), (1.5)

where |x − y|

=

|x

− y

| and β(N ) is the inverse temperature. To appeal to the 1/N expansion [15], we set

β(N ) = N β. (1.6)

We organize the paper as follows: in Sect.2, we represent the theory in terms of a determinant by introducing an auxialiary field ψ and integrating out the spin variables. We discuss the reason why phase transitions may not occur in two-dimensional systems which have O(N ) symmetries. In Sect.3, we argue the polymer expansion when |ψ(x)| are all small.

Sect.4 is the main part of this paper in which we prove that the contributions from large fields are small and negligible. Since ψ(x) can get large, we decompose Λ into two regions, the large and the small field regions and we estimate their contributions separately. The polymer expansion will be done combining these two regions. In Sect. 5, we represent the free energy by the convergent polymer expansion, from which analyticity of the free energy follows. We discuss some related problems in Sect. 6.

In Appendixes, we calculate decay rates and inverses of Green’s functions used in this paper. We also discuss polymer expansions of Green’s functions and Gaussian measures restricted to subsets of Z

.

II. DETERMINANT REPRESENTATION

We substitute the identity δ(φ

− 1) =

exp[−ia(φ

− 1)]da/2π into eq.(1.4) with the condition [3,4] that Ima

≤ −νN β. We set

Im a

= −N β(ν + m

2 ), Re a

= √

N βψ

, (2.1)

where m

≥ 0 will be determined soon. Thus we have Z

= c

Z

· · ·

Z

exp[− N β

2 < φ, (m

− ∆ + 2i

√ N ψ)φ > +

j

i √

N βψ

]

dφ

dψ

2π

= c

Z

· · ·

Z

det(m

− ∆ + 2i

√ N ψ)

exp[i √

N β

j

ψ

]

dψ

2π

= c

det(m

− ∆)

Z

· · ·

Z

F (ψ)

dψ

2π , (2.2)

where c are constants which may be different on lines, ∆

= −2νδ

+ δ

is the lattice laplacian and

F (ψ) = det(1 + 2iG

√ N ψ)

exp[i √

N β

j

ψ

]. (2.3)

Moreover G = (m

− ∆)

is Green’s function (matrix) discussed later. In the same way, the two point functions are given by

< φ

φ

> = 1 Z ˜

Z

· · ·

Z

(m

− ∆ + 2i

√ N ψ)

F (ψ)

dψ

2π , (2.4)

where ˜ Z is the obvious normalization constant. We choose m ≥ 0 so that G(0) = β, where G(x) =

Z π

−π

· · ·

Z π

−π

g (p)e

ν

Y

i=1

dp

2π , (2.5)

g(p) ≡ 1

m

+ 2

(1 − cos p

) ∈ [ 1

m

+ 4ν , 1

m

]. (2.6)

This choice is possible for any β ( and N ) if and only if ν ≤ 2, that is, if and only if G

(0) ≡ G(0)|

= ∞. In other words, we can rewrite eq.(2.3) as

F (ψ) = det

(1 + 2iG

√

N ψ)

exp[−Tr(Gψ)

] (2.7)

for any β, only for ν ≤ 2, where det

(1 + A) = det[(1 + A)e

].

The factor exp[i √

N β

ψ

] in (2.3) is the reminiscence of the double-well potential

Q

δ(φ

− 1) which is responsible for phase transitions. Then roughly speaking, the disap- pearance of exp[i √

N β

ψ

] in ( 2.7) means absence of the effect of the double-well potential

and is consistent with absence of phase transitions [2].

An explicit calculation shows that m

= β

( √

1 + 4β

− 2β) for ν = 1. For ν = 2, G(0) is expressed by the complete elliptic integral of the first kind F (k, π/2) =

dϕ(1 − k

sin

ϕ)

:

G(0) = 1 2π

Z π 0

dp

q

(1 + 2ε − cos p)(3 + 2ε − cos p)

= k

2π F (k, π/2) = 1

2π [O(ε) + 3

2 log 2 + 1 2 log 1

ε ],

where ε = m

/4 and k = (1 + ε)

. Then the condition G(0) = β implies that

m

∼ 32e

as β → ∞ (2.8)

which is consistent with the renormalization group arguments, see [6] and references therein.

If ν ≥ 3, such an m ≥ 0 exists if β ≤ G

(0). If β > G

(0), there exists spontaneous magnetization in the system [7]. That is N G

(0) > β

(N ) > N/µ

for ν > 2.

If m is chosen so that G(0) = β, det

(1 + 2iGψ/ √

N )

is almost equal to exp[4iTr(Gψ)

/(3 √

N)] and is regarded as a small perturbation to the Gaussian measure

∼ exp[−Tr(Gψ)

]

dψ. Namely F (ψ) looks like |F (ψ)| = det(1 + 4GψGψ/N )

which is strictly positive. If this is justified, then from eq.(2.4), we have exponential decay of the correlation functions :

< φ

φ

> ∼ 1 Z ˜

Z

· · ·

Z

(m

− ∆ + 2i

√ N ψ)

|F (ψ)|

dψ

2π

≤ | sup

ψ

(m

− ∆ + 2i

√ N ψ)

|

≤ (m

− ∆)

∼ e

.

III. POLYMER (CLUSTER) EXPANSION IN SMALL FIELD

A. Polymer Expansion

Let

dµ

(ψ) = det

[C

] exp[− < ψ, C

ψ >]

dψ(x)

√ π (3.1)

be the Gaussian probability measure of mean zero and covariance

C where C

≡ G

and G

is the matrix given by G

(x, y ) = G(x − y)

. The partition function Z

is given by

Z

= Z

Z

det

(1 + 2i

√ N Gψ)dµ

(ψ), (3.2)

Z

≡ det

[C

] = det

[C], (3.3)

up to a non-important multiplicative factor. Our purpose is to discuss analyticity of the free energy α

= − lim log Z

/|Λ| in β. Since m is analytic in β ≥ 0, the assertion is trivial if there is no determinant. In the present case where we have the determinant, which is quite non-linear and non-local in ψ(x), we represent Z

in terms of polymers:

Theorem 1 The partition function Z

is represented by polymers ρ

, X ⊂ Λ:

Z

= Z



 X

p

1 p!

X

∪^p₁Xi=Λ

Y

i

ρ





, (3.4)

where X

are unions of squares ∆ ⊂ Λ of size L × L (L >> 1 is determined later ) and X

∩ X

= ∅, (i 6= j). Given β > 0, if N is chosen large, N ≥ exp[const.β], there exist strictly positive constants δ

and m

such that

|ρ

| ≤ exp[−δ

n

log N − m

L(X)], (3.5)

where n

is the number of squares ∆

in X and L(X) is the length of the shortest connected tree graph over centers of ∆

⊂ X. The free energy is the convergent series of ρ

.

Each ρ

is analytic in β. Thus the Main Theorem follows from Theorem 1 since α

is represented by the convergent series of ρ

. The proof of this theorem is, however postponed until Sect.5. Here we restrict ourselves to the small field case where the expansion can be easily done by the N

expansion.

B. Small and Large Fields

We let ˜ G ≡ [G

]

. Then C and ˜ G have the following Fourier expansions:

C =

Z π

−π

Z π

−π

e

˜ g

(p)

2

Y

i=1

dp

2π , (3.6)

G ˜ =

Z π

−π

Z π

−π

e

˜ g(p)

2

Y

i=1

dp

2π , (3.7)

˜ g(p) =

"

Z π

−π

Z π

−π

g(p − k)g(k)

2

Y

i=1

dk

2π

#1/2

∈ [ c

m

+ 8 , c

m ]. (3.8)

Here and below, c stands for generic constant independent of β which may change from place to place even in the same equations, and c

, c

, · · · stand for similar constants which are kept in the same equations. The following lemma is proved in Appendix A:

Lemma 2 For m < 1, the kernels G, G, ˜ G ˜

and C exhibit the followng exponential decay:

G(x, y) ≤ c log(1 + 1

m ) exp[−m

|x − y|], (3.9)

| G(x, y)| ≤ ˜ c log(1 + 1

m ) exp[−m|x − y|], (3.10)

| G ˜

(x, y)| ≤ c(1 + m

) exp[−m|x − y|], (3.11)

|C(x, y)| ≤ c(1 + m

) exp[−m|x − y|] (3.12) where |x| =

x

+ x

and m

> 0 is a constant defined by 2 cosh(m

) = 2 + m

.

We introduce the notion of large field region R and small filed region K :

R = {x; N

≤ |ψ (x)|}, K = Λ − R (3.13)

where N = N (β) and a positive constant δ < 1/2 is chosen so that if |ψ(x)| ≤ N

for all x, then N

||G

ψG

|| << 1. Then the determinant is perturbatively expanded and the higher order terms are negligible. Since spec G ∈ [(8 + m

)

, m

] and m

∼ (32)

e

, these conditions are satisfied if exp[12πβ] < N for large β. The following is one of the most typical choices satisfying these conditions (though they are not optimal ) :

δ = 1

12 , N(β) = exp[400πβ]. (3.14)

Remark 1 For matrices A and B , we define A ◦ B by (A ◦ B)(x, y) = A(x, y)B(x, y). This

is called the Hadamard product of A and B. It is easy to see that A ◦ B ≥ 0 if A ≥ 0 and

B ≥ 0.

Remark 2 The kernel functions C(x), G(x) ˜ and G ˜

(x) decay faster than exp[− √

2m|x|], see Appendix. Of course, m

< m, m

= m − O(m

). However since m

is almost equal to m in the present problem where m << 1, we use m for m

for notational simplicity in the remaining part of the paper. If β < O(1), it is enough to choose L (the size for the expansion) and N larger than some constants for the convergence. So it suffices to consider the case β >> 1.

Remark 3 In this paper, we use free boundary conditions for Green’s function G and its inverse, and we assume that the ψ field distributes only in the large square region Λ ⊂ Z

. Other boundary conditions can be easily adopted without changing the main estimates in the present paper.

C. Polymer Expansion in Small Field Region

We first consider the case of R = ∅. In this case, we decompose Λ ⊂ Z

into squares (denoted ∆ or ∆

below ) of size L × L whose centers are at Λ ∩ LZ

. Collections of these squares are called paved sets. We also define L

<< L, where L and L

are chosen so that

L << N << e

, G(L

) = N

. (3.15) For this to be satisfied, we take L slightly larger than m

. Typically we may take L = 20m

log N so that e

= N

, in which case L

= L/10. These satisfy the conditions on L and N .

Let τ(ψ) be an even, positive and decreasing (in |ψ| ) C

function such that τ (ψ) =











1 for |ψ| < N

0 for |ψ| > N

+ h

. (3.16)

We may take the limit h → 0 after all calculations (lim

τ (ψ) = θ(N

− |ψ|)), but we can keep h as a non-zero constant (say 1).

We multiply 1 =

K⊂Λ

τ(ψ

)τ

(ψ

) (3.17)

to dµ

, where τ

(ψ) = 1 − τ (ψ), R = K

= Λ − K and τ(ψ

) ≡

τ (ψ(x)), τ

(ψ

) ≡

Q

x∈R

τ

(ψ(x)). We call K the small field region and R = K

the large field region. Then Z

≡ Z

X

R

Z (R), (3.18)

Z(R) ≡

Z

det

(1 + 2i

√ N Gψ)τ

(ψ

)τ (ψ

)dµ

(ψ). (3.19) We put Z

(R) = Z

Z (R) and we first consider the case R = ∅:

Z

(R = ∅) ≡ Z

Z

η

dµ

(ψ), (3.20)

η

≡ det

(1 + 2i

√ N Gψ)

x∈Λ

τ (ψ(x)), (3.21)

We introduce interpolation parameters s

into dµ

(ψ) to expand the measure [5,16]. Let Y ⊂ Λ be a paved set consisting of p squares {∆

, · · · , ∆

}. Let {∆

, · · · , ∆

} be any permutation of them such that ∆

= ∆

and let a be a map from {1, · · · , p − 1} into itself such that a(k) ≤ k. Then we have a set of ordered links {(j

, j

); i = 1, · · · , p − 1} which is regarded as a tree graph T

over {∆

} with root ∆

. Let

C

= χ

Cχ

, (3.22)

where χ

is the charcteristic function of Y . For a given permutation and a function a = a

, we define

C

({s}) = [

p−1

Y

i=1

((1 − s

)P

+ s

)]C

, (3.23)

M

=

p−1

Y

k=1 k−1

Y

i=a(k)

s

, (3.24)

where P

are operators which bisect paved sets: P

C

= C

+ C

, X

≡ ∪

∆

. See Appendix C for the construction and for the proof of next theorem, see [5,16]:

Theorem 3 Z

(R = ∅) have the cluster expansion Z

(R = ∅) = Z



 X

n

1 n!

X

∪ⁿ₁Yi=Λ

Y

i

S





η

, (3.25)

where Y

are paved sets such that ∪

Y

= Λ and Y

∩ Y

= ∅ for i 6= j. Let Y = ∪

∆

be one of Y

. Then S

is the differential and integral operator given by

S

=

T⁰

Z 1 0

ds

· · · ds

M

(s)

Z

dµ

({s}, ψ)

×

p−1

Y

k=1





 X

xk∈∆_ja(k)

X

yk+1∈∆_jk+1

1

2 C(x

, y

) ∂

∂ψ(x

)∂ψ(y

)







, (3.26)

where

is the sum over all tree graphs T

= {(j

, j

)} over {j

, j

, · · · , j

} (j

= 1) and dµ

({s}, ψ) = det

[C

(s)] exp[− < ψ, C

({s})ψ >]

x∈Y

dψ(x)

√ π . (3.27)

Here C

({s}) is given by (3.23) and depends on permutations only.

There are many graphs T

which have the same links and vertices but belong to different permutations {j

, j

, · · · , j

} of {1, · · · , p}. The following lemma is well known [5,16]:

Lemma 4 The measure M

ds

is the probability measure in the following sense:

X

T⁰:T(T⁰)=T

Z 1 0

M

p−1

Y

1

ds

= 1, (3.28)

where

means the sum over tree graphs T

which have the same links with T .

Let

A

= 2i

√ N Gψ (3.29)

for simplicity, and let Λ = ∪

Y

be one of the partitions which appear in eq.(3.25). Since {ψ

} are coupled in the determinant, we introduce interpolation parameters s

and set

A

=

A

+

i<j

(A

+ A

) → A + B(s), (3.30) A ≡

A

, B(s) ≡

i<j

s

(A

+ A

), (3.31) in the determinant, where

A

= χ

A

χ

, A

= χ

A

χ

. (3.32)

We iteratively apply the identity f (1) =

ds∂

f (s) + f (0) to det

(1 + A + B(s)). If all s

are set zero, then the determinant is factorized with respect to ψ

. We thus have :

Z

(R = ∅) = Z



 X

n

1 n!

X

∪ⁿ₁Xi=Λ

Y

i

ρ





.

Here {X

}

are partitions of Λ into polymers, X

∩ X

= ∅, (i 6= j), ∪X

= Λ and ρ

=

p

1 p!

X

Y1∪···∪Yp=X

Y

S





 X

γ∈T˜({Yi})

Z

ds

∂







η

({Y

}), (3.33) η

({Y

}) = det

(1 +

i

A

+

i<j

s

(A

+ A

))τ (ψ

), (3.34) where S

is the interpolation operators on Y defined by (3.26) and

1. ∪Y

= X and Y

are mutually disjoint paved sets,

2. ˜ T ({Y

}) is the set of connected graphs (not necessarily trees) over {Y

}

, 3. ds

=

ds

and ∂

=

(∂/∂s

), (put s

= 0 if (i, j ) ∈ / γ ).

In the rest of this section, we prove the following theorem which ensures that the free energy log Z

(R = ∅) is the convergent series of ρ

[13], if N is chosen large:

Theorem 5 Assume that R = ∅ and let n be the number of ∆ in X ⊂ Λ. If N ≥ N (β), there exist strictly positive constants δ

and m

such that

|ρ

| ≤ exp[−nδ

log N − m

L(X)], n ≥ 2 (3.35)

ρ

= exp[−W

], n = 1 (3.36)

where L(X) is the length of the shortest tree graph connecting all centers of squares ∆

⊂ X, and W

is the single square activity defined later.

To prove this, we first set η

({Y

}) ≡ exp[− N

2

2

X

i=1

V

(A, B )]

x∈X

τ (ψ(x)), (3.37)

V

(A, B ) = 1 2 Tr

B

− (B 1 1 + A )

+ 1 3 Tr

1 1 + A B

3

, (3.38)

V

(A, B ) = log det

(1 + A) + log det

(1 + 1

1 + A B). (3.39)

The derivatives of V

with respect to s

can be done by the contour integrals:

Y

∂

∂s

!

η

(s) =

Z

C

η

(t)

Q

(t

− s

)

Y

dt

2πi

where C is the product of the circles |t

− s

| = r

on C with their radiuses r

given by r

= N

exp[ 4

5 mdist(Y

, Y

)], where ˜ δ > 0. (3.40) Put B

= 2it

(G

ψ

+ G

ψ

)/ √

N . Then for |t

| < r

+ 1 , we find that

|B

(x, y)| ≤ const. log(1 + m

)N

exp[− m

5 |x − y|], N |Trχ

B

χ

| ≤ N

|X|,

ε

≡ −2.1 × log m/ log N (∼ 1/100 if N ∼ e

), (3.41) where ε

is chosen slightly larger than −2 log m/ log N so that N

> cm

log(1 + m

) and some trivial constants can be absorbed by N

. We choose ˜ δ > 0 so that

δ ˆ ≡ 1

2 − 3(δ + ˜ δ) − 2ε

> 0. (3.42)

For example, we can choose as δ = 1/12, ˜ δ = 1/16, ˆ δ = 1/16 − 2ε

. Thus we have:

Lemma 6 If N is chosen so large that (3.42) holds, then

|

(ij)∈γ

∂/∂s

η

| ≤ exp[−n δ ˜ log N − m

X

(ij)∈γ

dist(Y

, Y

)]||η

|| (3.43) where m

= 4m/5 and γ are connected tree graphs over {Y

⊂ X}, and n is the number of the bonds in the graph γ. Moreover

||η

|| ≡ sup

{|t_ij|≤r_ij+1}

|η

(t)| ≤ exp[N

|X|]. (3.44)

Lemma 7 Let ∪

∆

= Y and let x

∈ ∆

. Then

|

Z

dµ

(s, ψ)

n

Y

1

∂

∂ψ(x

) η

(ψ)| < exp[−n δ ˜ log N + N

|Y |]. (3.45)

Proof. Each derivative acts either on det

(· · ·) or on τ(ψ). If it acts on det

(· · ·), it yields the factor bounded by N

. (We can get a much smaller factor N

this case. ) On the other hand, if ∂/∂ψ(x) acts on τ(ψ(x)),

∂

∂ψ(x) τ (ψ(x)) = 0 unless N

< |ψ(x)| < N

+ h.

Note that

dµ

(s) →

exp[−z(x)

] dz(x)

√ π by the linear transformation ψ(x) = ( ˜ G

z)(x), where ˜ G

= √

C

. Since C

= G

and C

(s) is a convex linear combination of {C

}, we see

X

x∈Y

z

=< ψ, χ

C

(s)

χ

ψ >≥ 1 (8 + m

)

X

x∈Y

ψ

(x).

If |ψ(x)| > N

, then {y : |z(y)| > N

, |x − y| < L

} 6= ∅ since |ψ(x)| = |

G ˜

(x, y)z(y)|

and | G ˜

(x)| < c(1 + m

)e

. Thus the contributions from the derivatives of τ are exponentially smaller than those from the derivatives of det

(· · ·). Q.E.D.

The single square activity ρ

= e

is defined by ρ

=

Z

det

(1 + A

)τ (ψ

)dµ

(ψ). (3.46)

Since | log det

(1 + A

)| = O(N | TrA

|), we have W

= O(N

) which is inde- pendent of locations of ∆ (|∆| = L

< N

).

Let d

be the number of lines which connect ∆

with other ∆

in the tree graph, i.e. d

the incidence number. Then

d

= 2n − 2, where n is the number of squares ∆

in Y . In this case there can appear d

derivatievs ∂

/∂ψ(x)

, x ∈ ∆

in eq.(3.26). By integration by parts, we can shift the action of ∂/∂ψ from τ to det

(· · ·) or to exp[− < ψ, C

ψ >].

Lemma 8 [16] With the notation of (3.26) in Theorem 3 (with p replaced by n), let F (x

, y

, · · · , y

) ≡ |

n−1

Y

i=1

C(x

, y

)

Z

dµ

(s, ψ)

n−1

Y

1

∂

∂ψ(x

)∂ψ(y

) η

(ψ)|

where x

∈ ∆

, y

∈ ∆

. Let γ is the tree graph defined by a(·). Then

X

{x_k,y_k+1}

F (x

, y

, · · · , y

) ≤ exp[−n(˜ δ − 4ε

) log N − 4m

5 L

(X) + N

|X|] (3.47)

where x

∈ ∆

, y

∈ ∆

and L

(X) =

dist(∆

, ∆

).

Proof. Without loss, we assume {j

= k}

. Let d

≥ 1 be the incidence number of the vertex ∆

. Since #{∆

: dist(∆

, ∆

) < 2, i 6= j} = 8,

|x

− y

| is larger than

4 5

X

i

|x

− y

| + 1 10

X

i

X

x∈∆_i

X

y:(x,y)∈γ

|x − y| ≥ 4 5

X

i

|x

− y

| + L 10

X

i

[ d

9 ]

, where [x] = the maximal integer not larger than x. By integration by parts, we see that

|F (x

, y

, · · ·)| = |

n−1

Y

i=1

C(x

, y

)

Z

dµ

(s, ψ)ΦΨ| (3.48)

where relabelling {x

, y

} as {x

, x

, · · · , x

}

, x

∈ ∆

, Ψ =

n

Y

i=1

∂

∂ψ(x

) η

(ψ), (3.49)

Φ = (−1)

e

n

Y

i=1 di−1

Y

j=1

∂

∂ψ(x

) e

, (3.50)

H = < ψ

, C

(s)ψ

> . (3.51)

Rewriting {x

} as {ξ

}

, we put Φ = e

n−2

Y

i=1

∂

∂ψ(ξ

) e

=

I

(−1)

i∈I

H

(

P⊂I^c

Y

(j,k)∈P

H

), (3.52) where I are subsets of {1, · · · , n − 2}, P are sets of unordered pairs of elements in I

and

H

= 2

ζ

C

(ξ, ζ)ψ(ζ), H

= 2C

(ξ

, ξ

). (3.53) The number of partitions I ⊂ {1, · · · , n − 2} is 2

(|I

| must be even) and note that

X

P⊂I^c

Y

(j,k)∈P

H

=

Z Y

i∈I^c

φ(ξ

)dν

(φ)

where dν

(φ) is the Gaussian measure of mean zero and covariance H = 2G

. We first estimate the first term of Φ, I = {1, · · · , n − 2}:

X

{ζi}

Y

2|C(x

, y

)|

i

|C

(ξ

, ζ

)|

"

Z

dµ

(s, ψ)

i

|ψ(ζ

)||Ψ|

#

≤ M

Z

dµ

(s, ψ)Ψ

¹

2

where the integral of Ψ

is bounded by Lemma 7 ( easily extended to Ψ

) and M ≡

ζi

Y

2|C(x

, y

)|

|C

(ξ

, ζ

)|

Z

dµ

(s, ψ)

ψ(ζ

)

¹₂

. (3.54)

We take the sum over {ξ

}

⊂ {x

, y

}

and put

X

ξ∈∆_a(k)

X

ξ⁰∈∆_k+1

2|C(ξ, ξ

)||C

(ξ, x ˜

)||C

(ξ

, y ˜

)| ≡ m

δf (∆

, ∆

)(˜ x

, y ˜

).

Then δf (∆

, ∆

)(˜ x

, y ˜

) is bounded by

exp[−m{dist(∆

, ∆

) + dist(∆

, x ˜

) + dist(∆

, y ˜

)}] (3.55) except for a coefficient O(log

(1 + m

)) which originates from C

= G

. Here the constraints ˜ x

∈ ∆

and ˜ y

∈ ∆

do not hold anymore. For x

or y

not con- tained in {ξ}

, we put ˜ x

= x

or ˜ y

= y

and put δf (∆

, ∆

)(˜ x

, y ˜

) = 2C(˜ x

, y ˜

)χ

(˜ x

)χ

(˜ y

). This again satisfies the bound (3.55).

Assume that ˜ ∆

⊂ Λ contains ˜ d

points of {ζ

}. If d

points in ˜ ∆

couple with d

points in ˜ ∆

(the same points appear twice in

ψ(ζ)

),

d

= 2 ˜ d

and we have the factor

_{2 ˜d}

i

dij

_{2 ˜d}

j

dij

d

! ( 2d

for (i, i).) Since

(d

)! < (2 ˜ d

)!, we find that

Z

dµ

ψ(ζ

)

≤

[(2 ˜ d

)!]

i



 X

{d_ij}_j

(2 ˜ d

)!

d

! · · · d

!

Y

j

|C(dist( ˜ ∆

, ∆ ˜

))|





≤

[(2 ˜ d

)!]

i



 X

j

|C(dist( ˜ ∆

, ∆ ˜

))|





2 ˜di

≤ c

[(2 ˜ d

)!]

(3.56)

where c

= O(1). Since (2d)! ≤ e

and

δf (∆

, ∆

)(˜ x

, y ˜

) is bounded by exp[− 4m

5

X

k

{dist(∆

, ∆

) + dist(∆

, x ˜

) + dist(∆

, y ˜

)} − mL 10

X

i

[ d ˜

9 ]

]], we see that (2 ˜ d

)! are compensated and the sum over {˜ x

, y ˜

} yields m

.

The coefficients

φ(ξ)dν

of

H

are again bounded by (3.56) by replacing c

by c

log(1 + m

) and 2 ˜ d

by corresponding incidence numbers. Thus the total contribution of Φ is bounded by 2

times of the result of I = {1, · · · , n − 2}. Q.E.D.

We introduce mass parameters m

for later conveniences : 0 < m

< m ˜

< m

= m

10 < m

= 4m

5 < m. (3.57)

where Lm

∼ O(β) >> 1. The following lemmas are well-known to experts [5,8,16]:

Lemma 9 ( [16], Lemma A.5 ) For a paved set X consisting of n squares {∆

}, let T (X) denote the set of tree graphs γ over ∆

and L(X) denote the length of the shortest tree graph over centers of ∆

⊂ X. Let dist

(∆

, ∆

) be the distance from the center of ∆

to that of

∆

. Then there exist constants K

= o(1) and K

= o(1) such that

(1)

X30

X

γ∈T(X)

exp[− m ˜

(ij)∈γ

dist

(∆

, ∆

)] < K

, (3.58)

(2)

X30

exp[− m ˜

L(X)] < K

. (3.59)

Proof. (1) Interchange the order of

and

, and take the sum over positions of ∆

for each γ. If ∆

are distinguishable, the result is bounded by K

where K = o(1) since ∆

are squares of size L × L and e

<< 1. However the same configuration is counted n!

times. Then

X

X30

exp[− m ˜

(ij)∈γ

dist

(∆

, ∆

)] < K

n! .

We finally note that the number of tree graphs is n

< n!e

to take the sum over γ.

(2) This is clear from exp[− m ˜

L(X)] ≤

exp[− m ˜

(ij)∈γ

dist

(∆

, ∆

)]. Q.E.D.

Lemma 10 ( [5], Appendix C) Let X be a paved set consisting of n

squares ∆

⊂ X.

Let f (Y ) be functions satisfying the bounds

|f (Y )| ≤ exp[−n

δ ˜

log N − m ˜

L(Y )],

where n

is the number of squres ∆

in Y and L(Y ) is the length of shortest tree graph over centers of ∆

⊂ Y . Then there exist strictly positive constants δ

(∼ δ ˜

) and m

(∼ m ˜

) such that

| 1 p!

X

Y1∪···∪Yp=X

Y

f (Y

)| ≤ exp[−n

δ

log N − m

L(X)], (3.60)

where {Y

: i = 1, · · · , p} are paved sets such that X cannot be devided into two disconnected

parts without bisecting some Y

.

Proof. We first extract the tree decay factor exp[−n

δ

log N − m

L(X)] from

f(Y

) choosing δ

and m

slightly less than ˜ δ

and ˜ m

. We show that the remaining sum con- verges. By Cayley’s theorem on the number of the tree graphs with fixed incidence numbers d

, · · · , d

, we have

|

T

(·)| = |

{d_i}

X

T,{d_i}fixed

(·)| ≤

d1,···,dp

(p − 2)!

Q

(d

− 1)! sup

(T,d):fixed

|(·)|,

and take the sum over the Y

’s starting from the end branches of the tree. Let Y

be one of the end branches and let Y

be the ancestor. Fix ∆

⊂ Y

∩ Y

and take the sum over Y

. The sum is convergent and is bounded by

|f (Y

)|. Next take the sum over ∆

⊂ Y

, which yields (n

)

. Repeating this, we see that the sum is bounded by n

[

|f(Y )|e

]

since

n

/d ! ≤ e

. e

is compensated by a fraction of exp[−n

δ ˜

log N ] in f(Y ). See

also [5,16] for the detail. Q.E.D.

Proof of Theorem 5. We obtain f(Y ) in Lemma 10 from Lemma 8 by taking the sum over T

in (3.26). This yields a constant less than 1. Thus we obtain f (Y ) in Lemma 10.

We determine the parameters ˜ δ

and ˜ m

. In Lemma 8, X may be single squares ∆, and they do not have tree decay factors. Moreover ∆

and ∆

may be nearest neighbour each other and dist(∆

, ∆

) = 1. Then we put ˜ δ

≡ (˜ δ − 4ε

)/2 and borrow N

from N

in eq.(3.47) in Lemma 8 to extract the factor exp[− m ˜

L(∆

∪ ∆

)] = e

this case. Namely

˜

m

≡ δ ˜

log N

L (∼ δ ˜ m

40 if L = 20 log N/m). (3.61)

Let T ({Y

}) be the set of tree graphs (no loops) over {Y

} such that ∪Y

= X. Thus applying (3.43) and (3.47) to (3.33), we have from (3.33) that

|ρ

| ≤

n

X

p=1

1 p!

X

∪Y_i=X p

Y

1

A(Y

)e



 X

T

Y

(ij)∈T

b





,

where A(Y ) ≤ exp[−n

˜ δ

log N − m ˜

L(Y ) + c

N

|Y ]|, (c

= O(1)), Y

∩ Y

= ∅ for i 6= j, and b

≡ exp[− δ ˜

log N − m ˜

dist

(Y

, Y

)] comes from ∂/∂s

and

dist

(Y

, Y

) = min

∆i⊂Y_i,∆j⊂Y_j

dist

(∆

, ∆

). (3.62)