# Phase diagrams of one-dimensional models under strong random external field

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Vol. 17, No. 30 (2003) 5781–5794 c

World Scientific Publishing Company

PHASE DIAGRAMS OF ONE-DIMENSIONAL MODELS UNDER STRONG RANDOM EXTERNAL FIELD

AZER KERIMOV

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey kerimov@fen.bilkent.edu.tr

We consider one-dimensional models of classical statistical physics and prove that at each fixed value of the temperature for all realizations of additional sufficiently strong random external field the limiting Gibbs state is unique.

1. Introduction

It is well-known that in one-dimensional models the phenomenon of phase transi-tion essentially depends on the decay rate of the potential: in models with a pair potential U (|x − y|) satisfying the condition

X

y∈Z1

|x − y|U (|x − y|) < ∞

the phase transition is absent.13 Other models certainly may exhibit a phase transition; for example, the one-dimensional ferromagnetical Ising model with the Hamiltonian

H(φ) = − X

x,y∈Z1

|x − y|−1−αφ(x)φ(y)

where spin variables φ(x), φ(y) take values 1 and −1 and 0 < α < 1 at low temper-atures, has at least two limiting Gibbs states corresponding to the constant ground states φ = 1 and φ = −1.4,5

In this paper we investigate one-dimensional models of classical statistical physics without specifying the interaction potential and prove rather a natural result: at any fixed value of the temperature and under sufficiently strong random external field the set of all the limiting Gibbs states has at most one element.

Let us consider a model on Z1 with the formal Hamiltonian H0(φ) =

X

B⊂Z1

U (φ(B)) (1)

5781

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where the spin variables φ(x) ∈ Φ, Φ is a finite subset of the real line R, φ(B) denotes the restriction of the configuration φ to the set B, on which the potential U (φ(B)) is not necessarily translationally invariant.

On the potential U (φ(B)) we impose a natural condition, necessary for the existence of the thermodynamic limit:

X

B⊂Z1:x∈B

|U (φ(B))| < C0 (2)

where the constant C0does not depend on x and the configuration φ.

Now we consider random perturbations of the model (1), namely a model with the Hamiltonian H(φ) = H0(φ) + 1 X x∈Z hxφ(x) (3)

where {hx, x ∈ Z1} is a random external field.

The main result of the present paper is the following:a

Theorem 1. For any model (1) and any fixed value of the inverse temperature β there exists a constant h0 such that for all realizations of the random external field {hx, x ∈ Z1} satisfying |hx| > h0, x ∈ Z1 the model (3) has atmost one limiting Gibbs state.

Let P1and P2be two extreme limiting Gibbs states corresponding to the bound-ary conditions φ1and φ2. It is well known that P1and P2are singular or coincide.6,7 We prove the uniqueness of the limiting Gibbs states of model (3) by showing that P1 and P2 are not singular.

Let VN be an interval with the center at the origin and with the length of 2N . We will denote by Φ(N ) the set of all configurations φ(VN). Suppose that the boundary conditions φi, i = 1, 2 are fixed.

The concatenation of the configurations φ(VN) and φi(Z1− VN) we denote by χ: χ(x) = φ(x), if x ∈ VN and χ(x) = φi(x), if x ∈ Z1− VN. Let us define

HN(φ|φi) =

X

B⊂Zν:B∩VN6=∅

U (χ(B)) .

If the expression |HN(φ|φi)| is bounded uniformly with respect to N , φ and φi then the non-singularity of P1 and P2 directly follows. This simple but rather useful idea was first used in Ref. 5 for the proof of the absence of phase transition in one-dimensional models with long-range interaction. But in our more general case |HN(φ|φi)| need not to be bounded and we use more sophisticated approach.

Due to Lemma 1 below the configuration with minimal energy at fixed N and boundary conditions φi is unique and independent of φiif |h

x| > h0, x ∈ Z1, where h0 is sufficiently large: min φ∈Φ(N )HN(φ|φ i) = H N(φmin,iN |φ i) where φmin,i N = φ min N . a

Later in Section 2 we give the proof.

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Below, HN(φ|φi, φminN ) denotes the relative energy of a configuration φ (with respect to φmin N ): HN(φ|φi, φminN ) = HN(φ|φi) − HN(φminN |φ i) . Let Pi

N be Gibbs distributions on Φ(N ) corresponding to the boundary condi-tions φi, i = 1, 2 defined using the relative energies of configurations. Take M < N and let Pi

N(φ0(VM) be the probability of the event that the restriction of the con-figuration φ(VN) to VM coincides with φ0(VM).

In order to show that P1and P2are not singular, we prove that there exist two positive constants const1and const2, such that for any M and φ0(VM) there exists a number N0(M ) such that for any N > N0(M ), we have

const1< P1N(φ0(VM))/P2N(φ0(VM)) < const2.

The first important point is the introduction of the contour model common for boundary conditions φi, i = 1, 2 (a contour is a connected subconfiguration not coinciding with the ground state). After that, using a well-known trick8 we come to “noninteracting” clusters from interacting contours (a cluster is a collection of contours connected by interaction bonds).

The second important point is combinatorial Lemma 3,9 which allows us to reduce the dependence of the expression

P1N(φ(VM))/P2N(φ(VM))

on the boundary conditions φ1 and φ2 to the sum of statistical weights of some 2-clusters connecting the cube VM with the boundary (so-called long 2-clusters; since the statistical weight of 2-cluster is not necessarily positive, we estimate the sum of absolute values of statistical weights of long 2-clusters). Finally it turns out that if the additional random field is strong enough the sum of statistical weights of 2-clusters connecting VM with the boundary is negligible and the expression P1N(φ(VM))/P2N(φ(VM)) is bounded.

2. Proof of Theorem 1

Let ϕmin,1V ∈ Φ(V ) be a configuration with the minimal energy at fixed boundary conditions φ1. The following simple lemma describes the structure of the configu-ration φmin,1V .

Lemma 1. For any model (3) there exists a positive constant h0 such that for all realizations of the random field {hx, x ∈ Z1} satisfying |hx| > h0 the configuration

φmin,1V is unique and independent of the boundary conditions (φ1).

Proof. The lemma is a straightforward consequence of the condition (2).

Let P1and P2be two extreme limiting Gibbs states corresponding to the bound-ary conditions φ1 and φ2.

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Theorem 2. At any fixed value of the inverse temperature β there exists a constant h0 such that for all realizations of the random external field {hx, x ∈ Z1} satisfying |hx| > h0, x ∈ Z1, the limiting Gibbs measures P1 and P2 are not singular. Proof. In order to prove this theorem it is enough to show that there exists two positive constants c1and C1 such that for any M and φ0(VM) we have

c1≤ P1(φ0(VM))/P2(φ0(VM)) ≤ C1. (4) Since the limiting Gibbs states P1 and P2 are the weak limits of the measures P1N and P2N when N → ∞, for establishing (4) we need to prove that there exists two positive constants c1 and C1 such that for any M and φ0(VM) there exists a number N0(M ) such that for any N > N0(M ), we have

c1< P1N(φ0(VM))/P2N(φ0(VM)) < C1. (5) Suppose that the boundary conditions φ1 are fixed. Consider the probability P1N of the event that the restriction of the configuration φ(VN) to VM coincides with φ0(V M): P1N(φ0(VM)) = P φ(VN):φ(VM)=φ0(VM)exp(−βHN(φ(VN)|φ 1, φmin,1 N )) P φ(VN)exp(−βHN(φ(VM)|φ 1, φmin,1 N )) = exp(−βH in M(φ0(VM)))Y (φ0(VM), VN, φ1)Ξ(VN− VM|φ1, φ0(VM), φmin,1N ) P φ00(V M)exp(−βH in M(φ00(VM)))Y (φ00(VM), VN, φ1)Ξ(VN− VM|φ1, φ00(VM), φmin,1N ) = exp(−βH in M(φ0(VM)))Y (φ0(VM), VN, φ1)Ξφ 10 P φ00(VM)exp(−βHMin(φ00(VM)))Y (φ00(VM), VN, φ1)Ξφ 100 (6)

where the summation in P

φ00(V

M) is taken over all possible configurations of

φ00(VM). Hence HMin(φ0(VM)) = X B⊂VM U (φ0(B)) − U (φmin,1N ) and HMin(φ00(VM)) = X B⊂VM U (φ00(B)) − U (φmin,1N )

are interior relative energies of φ0(V

M) and φ00(VM). Ξφ 10

and Ξφ100

de-note the partition functions corresponding to the boundary conditions φ1(Z1 VN), φ0(VM), φ00(VM): Ξφ10 = Ξ(VN− VM|φ1, φ0(VM), φmin,1N ) , Ξφ100 = Ξ(VN− VM|φ1, φ00(VM), φmin,1N ) . (7)

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The factor Y (φ(VM), VN, φ1) is defined as Y (φ(VM), VN, φ1) = Y A⊂Z1 :A∩VM 6=∅; A∩Z1 −VN 6=∅; A∩VN −VM =∅

exp(−β(U (φ(A)) − U (φmin,1N (A)))) (8)

where φ in Eq. (8) is equal to φ0 for x ∈ V

M and is equal to φ1 for x ∈ Z1− VN. The expression (8) gives the “direct” interaction of φ(VM) with the boundary conditions φ1(Z1− V

N). The probability P1V(ϕ0(VM)) is given by Eq. (6). We can express P2

V(ϕ0(VM)) in just the same way.

In order to prove the inequality (5) it is enough to establish inequalities (9) and (10): 0.9 < Y (φ(VM), VN, ϕi) < 1.1 , i = 1, 2 (9) and 1/S2≤ Ξφ100 Ξφ10 ! , Ξφ200 Ξφ20 ! ≤ 1/s2 (10) for arbitrary ϕ00(V M), where S2= (1.1/0.9)2S and s2= (0.9/1.1)2s1. Indeed, if the inequalities (9) and (10) hold, then

1/(1/s1) ≤ P1V(ϕ0(VM))/P2V(ϕ0(VM)) ≤ 1/(1/S1) since the quotient of (Pn

i=1ai)/(P

n

i=1bi) lies between min(ai/bi) and max(ai/bi).

Now we start to prove the inequalities (9) and (10).

The inequality (9) is a direct consequence of the condition that the potential is a decreasing function: For each fixed M there exists N0, such that if N > N0, then 0.9 < Y (φ(VM), VN, φi) < 1.1 for i = 1, 2.

So, in order to complete the proof of Theorem 2 we have to establish the following inequality [which is just the transformed inequality (10)]:

1 S2 ≤ Ξ φ100 Ξφ20 Ξφ200 Ξφ10 ≤ 1 s2 . (11)

Now we show that for each fixed interval VM, there exists a number N0(M ), which depends on M only, such that if N > N0(M )

s2≤ Ξφ10

Ξφ200

Ξφ100

Ξφ20 ≤ S2 (12)

for two positive constants s2 and S2independent of M , φ1, φ2, φ0 and φ00. Let us consider the partition functions

Ξφ1,φ00 = Ξ(VN− VM|φ1, φ00(VM), φmin,1N )

(6)

corresponding to the boundary conditions φ1(Z1− V

N), φ00(VM) and Ξφ2,φ0 = Ξ(VN− VM|φ2, φ0(VM), φmin,2N ) corresponding to the boundary conditions φ2(Z1− V

N), φ0(VM) as in Eq. (7). Now define a super partition function

(Ξφ100

Ξφ20

)

=Xexp(−βHN(φ3(VN)|φ1, φ00, φVmin,1)) exp(−βHN(φ4(VN)|φ2, φ0, φmin,2N )) where the summation is taken over all pairs of configurations φ3(V

N) and φ4(VN), such that φ3(V

M) = φ00(VM), φ4(VM) = φ0(VM). Consider the partition of Z1 into V

x which is an interval with the length of edge 1 and with the center at x = 1/2 + k (k is an integer). A configuration φgr is said to be a ground state of the model (3) if

H( ¯φgr) − H(φgr) ≥ 0

for all finite perturbations φgr (the set {x : ¯φgr(x) 6= φgr(x)} should be finite) of the configuration φgr. Due to Lemma 1 if h

0 is sufficiently large and |hx| > h0, the model (3) has a unique ground state φgr.

Let us consider an arbitrary configuration φ. We say that a cube Vx is not regular, if φ(Vx) 6= φgr(Vx). Two non-regular cubes are called connected provided their intersection is not empty. The connected components of non-regular segments defined in such a way are called supports of contours and are denoted by supp(K). A pair K = (supp(K), φ(supp(K))) is called a contour. Obviously for each contour K, there exists a configuration ψK such that the only contour of the configuration ψK is K (ψK on Z1− supp(K) coincides with φgr).

Let us define the weight of contour K by the formula:

γ(K) = H(ψK) − H(φgr) . (13)

The statistical weight of contour K is

w(Ki) = exp(−βγ(Ki)) . (14)

Suppose that the contours of the configuration φ(VN) are K1, . . . , Kn. The value of the interaction of contours K1, . . . , Kn between themselves and with the bound-ary conditions φ1 we denote by G(K

1, . . . , Kn). This expression naturally decom-poses into the interaction of single contours with the boundary conditions, pairs of contours between themselves and with the boundary conditions, and so on:

G(K1, . . . , Kn) = n X k=1 X Ki1,...,Kik G(Ki1, . . . , Kik) (15)

(at each fixed k the summation is taken over all possible non-ordered collections Ki1, . . . , Kik) and

G(Ki1, . . . , Kik) =

X B

(U (φ(B)) − U (φgr(B)))

(7)

where the summation is taken over all B ⊂ Z1 such that B ∩ supp(K

ij) 6= ∅

for all j = 1, . . . , k. We say that B is an interaction element λ = λ(i1, . . . , ik) corresponding to the the term G(Ki1, . . . , Kik) if U (φ(B)) − U (φ

gr(B)) 6= 0. The set of all interaction elements λ corresponding to the terms G(Ki1, . . . , Kik) in the

double sum (15) will be denoted by IG.

The following equation is a straightforward consequence of the formulae (14) and (15): exp(−βHN(φ|φ1, φmin,1N ) = n Y i=1 w(Ki) exp(−βG(K1, . . . , Kn)) . (16) The interaction between Ki1, . . . , Kik arises due to the fact that the weight of

the contour Kij, j = 1, . . . , k was calculated under the assumption that the

con-figuration outside supp(Kij) coincides with the ground state. Now we can rewrite

(16) as: exp(−βHN(φ|φ1, φmin,1N )) = n Y i=1 w(Ki) Y λ∈IG (exp(−βG(λ))) = n Y i=1 w(Ki) Y λ∈IG (1 + exp(−βG(λ) − 1)) . (17)

From Eq. (17) we have

exp(−βH(φ|φ1, φmin,1N ) = X I0⊂I n Y i=1 w(Ki) Y λ∈I0 g(G(λ)) (18)

where the summation is taken over all subsets I0 (including the empty set) of the set I, and g(G(λ)) = exp(−βG(λ)) − 1.

Consider an arbitrary term of the sum (18), which corresponds to the subset IG0⊂ IG. Let the interaction element λ ∈ IG0. Consider the set K of all contours such that for each contour K ⊂ K, the set supp(K) ∩ λ is nonempty. We call any two contours from K neighbors in IG0 interaction. The set of contours K0 is called connected in IG0 interaction if for any two contours K

p and Kq there exists a collection (K1= Kp, K2, . . . , Kn= Kq) such that any two contours Kiand Ki+1, i = 1, . . . , n − 1, are neighbors.

The pair D = [(Ki, i = 1, . . . , s); IG0], where IG0 is some set of interaction elements, is called a cluster provided there exists a configuration φ containing all Ki; i = 1, . . . , s; IG0 ⊂ IG; and the set (Ki, i = 1, . . . , s) is connected in IG0 interaction. The statistical weight of a cluster D is defined by the formula

w(D) = s Y i=1 w(Ki) Y λ∈IG0 g(G(λ)) . Note that w(D) is not necessarily positive.

Two clusters D1 and D2 are called compatible provided any two contours K1 and K2 belonging to D1 and D2, respectively, are compatible. A set of clusters is

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called compatible provided any two clusters of it are compatible. If D = [(Ki, i = 1, . . . , s); IG0], then we say that Ki∈ D; i = 1, . . . , s. If [D1, . . . , Dm] is a compatible set of clusters and ∪m

i=1supp(Di) ⊂ VN, then there exists a configuration φ which contains this set of clusters. For each configuration φ we have

exp(−βHN(φ|φ1, φmin,1N )) = X

IG0⊂IG

Y w(Di)

where the clusters Di are completely determined by the set IG0. The partition function is

Ξ(φ1) =Xw(D1) · · · w(Dm)

where the summation is taken over all non-ordered compatible collections of clusters.

Thus, we come to suitable noninteracting clusters from awkward interacting contours.8

The following generalization of the definition of compatibility allows us to rep-resent (Ξφ1,φ00Ξφ2,φ0) as a single partition function.

A set of clusters is called 2-compatible provided any of its two parts coming from two Hamiltonians is compatible. In other words, in 2-compatibility an intersection of supports of two clusters coming from different partition functions is allowed.

If [D1, . . . , Dm] is a 2-compatible set of clusters and ∪mi=1supp(Di) ⊂ VN− VM, then there exist two configurations φ3and φ4which contain this set of clusters. For each pair of configurations φ3 and φ4 we have

exp(−βHN(φ3|φ1, φmin,1N ) exp(−βHN(φ4|φ2, φmin,2N ) =

X

IG0⊂IG,IG00⊂IG

Y w(Di)

where the clusters Di are completely determined by the sets IG0 and IG00. The double partition function is

Ξφ10020

= Ξφ100

Ξφ20

=Xw(D1) · · · w(Dm)

where the summation is taken over all non-ordered 2-compatible collections of clus-ters.

Let w(D1) · · · w(Dm) be a term of the double partition function Ξφ 1

,φ00,φ2,φ0. The connected components of the collection [supp(D1), . . . , supp(Dm)] are the supports of the superclusters. A supercluster SD is a pair (supp(SD), φ(supp(SD)).

A 2-cluster SD = [(Di, i = 1, . . . , m); IG0, IG00] is said to be long if the in-tersection of the set (∪m

i=1supp(Di)) ∪ IG0∪ IG00 with both VM and Z1− VN is non-empty. In other words, a long 2-cluster, by use of its contours and bonds, con-nects the boundary with the interval VM. A set of 2-clusters is called compatible provided the set of all clusters belonging to these 2-clusters are 2-compatible. Lemma 2. There exists a number (0 <  < 1) such that for each fixed interval VM, there exists a number N0 = N0(M ), which depends on M only such that if N > N0 then

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(1 − )Ξφ1,φ0,φ2,φ00 < Ξφ1,φ0,φ2,φ00,(n.l.)

=Xw(SD1) · · · w(SDm) < (1 + )Ξφ

10200

(19) where the summation is taken over all non-long, non-ordered compatible collec-tions of 2-clusters [SD1, . . . , SDm], ∪mi=1supp(SDi) ⊂ VN − VM corresponding to the boundary conditions {φ1(Z1− V

N), φ2(Z1− VN); φ0(VM) and φ00(VM)}. In other words, in models with not-long 2-clusters property the statistical weights of long 2-clusters are negligible.

Proof. Let us define a partition function Ξφ10200,(l.)

asP w(SD1) · · · w(SDm) where the summation is taken over all terms of Ξφ10200,

which are not included into Ξφ10200

,(n.l.). By dividing of both sides of the equality

Ξφ10200 = Ξφ10200,(n.l.) + Ξφ10200,(l) by Ξφ10200 , we get 1 = Ξ φ10200 ,(n.l.) Ξφ10200 + Ξφ10200 ,(l) Ξφ10200 . Consider Ξφ10200,(l) Ξφ10200 = Ξφ10200,(l) Ξφ10200,(n.l.) + Ξφ10200,(l.).

If we replace each term belonging to Ξφ10200,(l.)

by its absolute value, then Ξφ10200

,(l.) turns to Ξφ10200

,(l.,abs.).

Since the sign of Ξφ1,φ0,φ2,φ00,(l.) is not definite, we have (under crucial assump-tion that Ξφ10200

,(n.l.)> Ξφ10200

,(l.,abs.), which will follow below from (20)):

− Ξ φ10200,(l.,abs.) (Ξφ10200,(n.l.) − Ξφ10200,(l.,abs.) ) ≤ Ξφ10200,(l.) (Ξφ10200,(n.l.) + Ξφ10200,(l.) ) ≤ Ξ φ10200 ,(l.,abs.) (Ξφ10200,(n.l.) + Ξφ10200,(l.,abs.) ). It can be easily shown that the inequality (19) follows from the following inequality:

Ξφ1,φ0,φ2,φ00,(l.,abs.)

(Ξφ10200,(n.l.)

+ Ξφ10200,(l.,abs.)

)< /2 . (20)

The expression Ξφ10200,(l.abs)

/Ξφ10200(abs)

naturally can be interpreted as an “absolute probability” of the event that there is at least one long 2-cluster. Lemma 3. There exists a number (0 <  < 1) such that for each fixed interval VM, there exists a number N0 = N0(M ), which depends on M only, such that if N > N0 then (20) is valid.

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We say that a 2-cluster SD connects configurations the φ0(V

M) and φ00(VM) with φ1 and φ2 if the support of SD connects V

M with Z1− VM. By definitions, supports of long 2-clusters connect φ0(V

M) and φ00(VM) with φ1 or φ2. In order to prove lemma, it is sufficient to show that at large values of h0the probability that there is at least one 2-cluster connecting φ(Z1− V

M) and φ(VM) is less than 1, for some 1 < 1. By definitions, the support of any 2-cluster is the union (connected by interaction elements) of contours or heap of intersected contours some sitting on others. Below we call these contours and heaps of contours 2-contours and denote them by SK.

We prove the stronger result asserting that at large values of h0 the absolute probability of the event that there is a 2-contour connected to VM by interaction elements is less than 2 for some 2 < 1. First of all suppose that the support of 2-cluster SD consists of a simple contour K. Then for each t > 0 there exist a value of the constant h0 from Theorem 1 such that

Pabs(K) < exp(−βt|supp(K)|) . This is a straightforward consequence of Peierls argument.

Now suppose that the support of 2-cluster SD consists of only 2-contour SK (without interaction elements) including two contours K1 and K2. We define |supp(SK)| = |supp(K1) ∪ supp(K2)| consistently with above definitions. Similarly, for each s > 0, there exist a value of the constant h0 such that

Pabs(SD) < exp(−βs|supp(SK)|) . (21) Now we are going to estimate the absolute probability of the event that there is at least one 2-cluster connecting φ(−∞, −N ) and φ0(V

M). Suppose that the 2-cluster SD is connected to φ(VM). Let SK be the 2-contour closest to VM which belong to SD (if there are two we choose one of them). We say that a 2-contour K0 is a neighbor of the first order of SK and write SK ↔ SK0 if SK and SK0 are connected by interaction element. A 2-contour SK00is called a neighbor of the qth order of SK provided

SK ↔ SK1↔ SK2↔ · · · ↔ SKq−1↔ SK00 and there is no such diagram with fever arrows.

Lemma 4. Let SK0 be a 2-contour of order k and suppose that for all 2-contours of order k + 1w(SK) < exp  −1 2βs|supp(SK)|  . Then X SD:SD=(SK0,SK,IG0,IG00) w(SD) < exp  −1 2βs|supp(SK0)|  .

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Lemma 4 states that if we fix a 2-contour and take the summation over all its neighbors then the constant s in statistical weight of this 2-cluster worsens at most to s/2. Proof. By definitions X SD:SD=(SK0,SK,IG0,IG00) w(SD) = X SD:SD=(SK0,SK,IG0,IG00) w(SK0)w(SK) Y λ∈IG0,λ∈IG00 g(G(λ)) (22) where g(G(λ)) = exp(−βG(λ)) − 1 by (18).

Now we can estimate the right hand side of last equality X SD:SD=(SK0,SK,IG0,IG00) w(SK0)w(SK) Y λ∈IG0,λ∈IG00 g(G(λ)) ≤ w(SK0) Y x∈supp(SK0) 1 + X λ:x∈λ |g(G(λ))|(1 + Q) !2 (23)

where Q is the sum of statistical weights of all 2-contours passing through fixed point: Q = P

SK:y∈supp(SK)w(SK). Explanation of formula (22): interaction

ele-ment may intersect (or not) any point x ∈ supp(SK0), we have squared the last factor since in 2-contour there are two supports one sitting one the other.

Now note that due to inequality (21) and the fact that the spin space Φ is finite at sufficiently large value of s we have Q < 1. Indeed,

Q = ∞ X m=1 X SK:y∈supp(SK),|supp(SK)|=m w(SK) ≤ ∞ X m=1 exp  −1 2βsm  m|Φ|m = ∞ X m=1 m exp  −1 2βs + ln |Φ|  m  ≤ ∞ X m=1 exp  −1 3βsm  ≤ 1 (24)

at sufficiently large values of βs. Thus, for any fixed β at sufficiently large values of s the expression Q < 1.

Let us show thatP

λ:x∈λ|g(G(λ))| is finite. Indeed, since by (2),

X

B⊂Z1:x∈B

|U (φ(B))|

(12)

converges and is uniformly bounded by C0, there are finite number of interaction elements G = U (φ(B)) (uniformly with respect to configurations φ) for which β|G(λ)| ≥ 1. Now note that if β|G(λ)| < 1 then

g(G(λ)) = exp(−βG(λ)) − 1 < 2β|G(λ)| and X λ:x∈λ |g(G(λ))| = X λ:x∈λ:β|G(λ)|≥1 |g(G(λ))| + X G:x∈λ:β|G(λ)|<1 |g(G(λ))| ≤ constant + 2C0= C3. (25)

Finally, by Eqs. (22)–(25), we have X SD:SD=(SK0,SK,IG0,IG00) w(SD) ≤ w(SK0) Y x∈supp(SK0) (1 + 2C3)2 = w(SK0)(1 + 2C3)2|supp(SK0)| ≤ exp(−βs|supp(SK0)|)(1 + 2C3)2|supp(SK0)| < exp  −1 2βs|supp(SK0)|  . Therefore, the proof of Lemma 4 is completed.

By use of Lemma 4 we can estimate the probability of fixed 2-contour SK0. Indeed, we consider a super-cluster consisting of SK0 and in the first step we fix all 2-contours of order q − 1 and take the summation over all 2-contours of order q, in the second step we fix all 2-contours of order q − 2 and take the summation over all 2-contours of order q − 1, and so on, we repeat this summation process q − 1 times and get the estimation:

X SD:SD=(SK0⊂SD w(SD) < exp  −1 2βs|supp(SK0)|  .

Let A be the event that there is a 2-contour SK0connected to φ(VM) by inter-action elements. We complete the proof of Lemma 3 by proving that at large values of s the absolute probability P (A) of the event A is less then 2 for some 2< 1. Let |supp(SK0)| = l and the distance between supp(SK0) and (VM) is k. Then

P (A) ≤ ∞ X l=1 ∞ X k=0 exp  −1 2βsl  Y g(G(λ))

where the product is taken over all interaction elements between SK0and φ0(VM). Now P (A) ≤ ∞ X l=1 exp  −1 2βsl  Cl 3< ∞ X l=1 exp  −1 3βsl  ≤ exp  −1 4βs  < 2< 1

(13)

if βs is sufficiently large (C3 is a constant defined in (25)).

Lemma 3 is proved. As pointed out the inequality (20) implies the inequalities (19), thus the proof of Lemma 2 is also completed.

Partition functions including only non-long super clusters satisfy the following key lemma which has a geometrically-combinatorial explanation.

Lemma 5.9,10

Ξφ1,φ00,φ2,φ0,(n.l.)= Ξφ1,φ0,φ2,φ00,(n.l.).

Proof. The summations in Ξφ10020,(n.l.)

= Ξφ10200,(n.l.)

are taken over all non-long, non-ordered compatible collections of 2-clusters.

We put a one-to-one correspondence between the terms of these two double partition functions.

Figure 1 shows how this one-to-one correspondence can be carried out. To the term w(D1,100)w(D 1,00 2 )w(D 1,00 3 )w(D 1,00 4 )w(D 2,0 5 )w(D 2,0 6 )w(D 2,0 7 )w(D 2,0 8 )

(the first four factors of this term came from the partition function Ξφ100

and the last four factors of this term came from the partition function Ξφ2,φ0) of the super partition function Ξφ10020

,(n.l.), we correspond the term

w(D11,0)w(D61,0)w(D1,70)w(D1,40)w(D2,500w(D2,200)w(D2,300w(D82,00)

(the first four factors of this term came from the partition function Ξφ1,φ0 and the last four factors of this term came from the partition function Ξφ200

) of the super partition function Ξφ10200

,(n.l.). It can be easily shown that this one-to-one

correspondence is well defined: if some term from Ξφ10200,(n.l.)

corresponding to the term from Ξφ10020

,(n.l.)does not exist (in other words, the corresponding

clusters from Ξφ10

or Ξφ200

are overlapped) then the term from Ξφ10020,(n.l.)

is long super cluster, a contradiction. Therefore, Lemma 5 is proved.

The inequality (12) is a direct consequence of (19) and Lemma 5. The proof of Theorem 2 is completed.

Fig. 1.

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Let P1and P2be two extreme limit Gibbs states corresponding to the boundary conditions φ1 and φ2.6,7

Theorem 3. P1 and P2 are singular or coincide.6,7

Proof of Theorem 1. Due to Theorem 2, P1 and P2 are not singular. Thus, by Theorem 3, P1 and P2coincide. Therefore, Theorem 1 is proved.

Acknowledgments

The author thanks the referees for their useful suggestions.

References

1. R. L. Dobrushin, Theor. Prob. Appl. 18, 201 (1968). 2. R. L. Dobrushin, Func. Anal. Appl. 2, 44 (1968). 3. D. Ruelle, Comm. Math. Phys. 9, 267 (1968). 4. F. Dyson, Comm. Math. Phys. 12, 91 (1969). 5. F. Dyson, Comm. Math. Phys. 21, 269 (1971).

6. Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results (Oxford, Pergamon Press, 1982).

7. H-O. Georgii, Gibbs Measures and Phase Transitions (de Gruyter, 1988).

8. J. Bricmont, K. Kuroda and J. L. Lebowitz, Commun. Math. Phys. 101, 501 (1985). 9. A. Kerimov, J. Stat. Phys. 72, 571 (1993).

10. A. Kerimov, J. Phys. A35, 5365 (2002).

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