Uniqueness of Gibbs states in one-dimensional antiferromagnetic model with long-range interaction

with long-range interaction

Azer Kerimov

Citation: J. Math. Phys. 40, 4956 (1999); doi: 10.1063/1.533009 View online: http://dx.doi.org/10.1063/1.533009

View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i10 Published by the American Institute of Physics.

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Uniqueness of Gibbs states in one-dimensional antiferromagnetic model with long-range interaction

Azer Kerimov

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey 共Received 28 May 1998; accepted for publication 2 April 1999兲

Uniqueness of Gibbs states in the one-dimensional antiferromagnetic model with very long-range interaction is established. © 1999 American Institute of Physics.

关S0022-2488共99兲03309-5兴

I. INTRODUCTION

We study a model on the lattice Z¹ with the Hamiltonian

H共␸共x兲兲⫽_x,y苸Z

兺

U共x⫺y兲␸共x兲␸共y兲⫺␮_x

兺

where the spin variable␸(x) takes the values 0 and 1,␮is a chemical potential. The antiferro- magnetic potential U(x)⬎0 satisfies the following conditions:

共1兲 U(x⫹y)⫹U(x⫺y)⬎2U(x);x,y苸Z¹,x⬎y.

共2兲 The function U(x) can be extended to a twice continuously differentiable function such that U(x)⬃A(x)^⫺␥, U⬘^⬃⫺A␥^{x⫺␥⫺1 and U}⬙^(x)⬃A␥⁽␥⫹1)x^⫺␥⫺2 at x→⬁; where ␥⬎1, and A is a strong positive constant.

The first convexity condition plays a significant role for the structure of the set of all ground states of the model共1兲. The second condition determines the character of the potential’s decrease at infinity and is important in further calculations.

The hypothesis on the uniqueness of the Gibbs states in the model共1兲 was stated by Sinai in 1983共see Ref. 1, Problem 1兲.

It is well known that the condition⌺x苸Z¹,x⬎0xU(x)⬍⬁ automatically implies the uniqueness of the Gibbs states.^2–4We investigate the phase transition problem in the model 共1兲 in the alter- native case, when U(x)⬃Ax^⫺␥, where␥⫽1⫹␣^{, 0}⬍␣⬍1.

The ferromagnetic version of this model关when the potential U(x) is negative兴 was considered by Dyson in his well-known papers.^5,6He proved the existence of two extreme limit Gibbs states P^⫹and P^⫺corresponding to the ground states␸^(x)⫽⫹1 and␸^(x)⫽⫺1 at low temperatures.

A series of papers has been devoted to the investigation of the antiferromagnetic model 共1兲.^1,7–13

The validity of Sinai’s hypothesis for rational values of the density共for almost each value of the external field兲 at low temperatures was proved in Ref. 13.

The main purpose of this paper is to extend the result of Ref. 13 to all values of the external field and to all values of the temperature.

Theorem 1: The model (1) has a unique limit Gibbs state at all values of the temperature

␤^⫺1.

Let us introduce necessary definitions. The set of all periodic configurations we denote by

⌽^per. For every␸苸⌽^per, we define q⫽⌺yx⫽x⫹1⫹p ␸(x), where p is the period of␸. It is obvious that q does not depend on x. Therefore, the density of each periodic configuration is␬⫽q/p. It is more convenient to work with the reciprocal of the density,␩⁽␸^(x))⫽p/q, which represents the aver- age distance between neighboring points at which ␸^(x)⫽1. For every configuration␸苸⌽^per the mean energy h(␸) is defined as follows:

4956

0022-2488/99/40(10)/4956/19/$15.00 © 1999 American Institute of Physics

h共␸共x兲兲⫽1

p_y⫽x⫹1^x

兺

兺

The last expression does not depend on x.

The following definition is useful for describing the zero temperature phase diagram of the model 共1兲.

We fix a positive rational number p/q.

A configuration␸0(x)苸⌽^per with␩⁽␸0(x))⫽p/q is called a special ground state¹if h共␸共x兲兲⫽ inf

␸苸⌽^per,␩共␸兲⫽p/q

h共␸兲.

Hubbard’s criterion (Refs. 1 and 7): Let␸苸⌽^perand r_i(x;␸) denotes the distance between a particle placed at x苸Z¹ and ith particle on the right. If for each x and i

关i␩兴⭐ri共x;␸兲⭐关i␩兴⫹1,

共the square brackets denote the integer part of the enclosed number兲 then ␸ is a special ground state.

The existence of configurations satisfying Hubbard’s criterion 共the special ground states兲 is shown in Ref. 1. The remarkable elegant formula for the special ground states was offered by Aubry. Here we give the construction of the special ground states for each fixed rational value of the density␬^.1

Every rational number p/q has a unique decomposition into a finite continued fraction:

p/q⫽关n0,n₁,...,n_s兴, this means that

n₀⫹ 1

n₁⫹ 1 n₂⫹...⫹ 1

n_s .

The ground state for a configuration with␩⫽关n0,n₁,...,n_s兴 will be constructed by recursion.

共1兲 ␩⫽n0⭓1, n0 is an integer. The periodic configuration with equally distant x at which

␸^(x)⫽1 satisfies Hubbard’s criterion i.e., is a special ground state. In this case ri(x;␸⁾⫽in0, i

⬎0.共2兲␩⫽n0⫹1/n1, where n₀ and n₁ are integers, n₀⭓1, n1⬎1. Then the (n0n₁⫹1) periodic configuration

also satisfies Hubbard’s criterion and is a special ground state.

共3兲 ␩⫽关n0,n₁,...,n_s兴, where n0,n₁,...,n_s are integers, n₀,n₁,...,n_s⭓1. For s⫽0 and s

⫽1 the required configurations are already constructed. Suppose we have already constructed a ground state with s⫽m and ␬⫽关n0,n₁,...,n_m兴. Then the following configuration with s⫽m

⫹1 and␬⫽关n0,n₁,...,n_m⫹1兴 is constructed as

Here,␸⁽ⁿ0,...,n j ), j⫽m⫺1, m,m⫹1, are the blocks from which the ground states for ␩

⫽关n0,...,n_j兴 are obtained by periodic continuations.

The constructed configuration satisfies Hubbard’s criterion and therefore is a special ground state for␩⫽关n0,n₁,...,n_m,n_m⫹1兴.¹

The following explicit expression for the mean energy of the special ground state follows from Hubbard’s criterion:¹

h_␬⫽␬

兺

where m_i⫽关i␩兴, ␲i⫽1⫹mi⫺i␩^.

This formula shows that the function of mean energy as a function of the density ␬ ^is continuous on the set of all rationals and can be extended to a continuous function defined on whole segment 关0, 1兴.

Theorem 2: (Refs. 9 and 1.)共1兲 The function h_␬ is convex.

共2兲 In each rational point the function h_␬ has a left-hand derivative ␮_␬^⫺ and a right-hand derivative␮_␬^{⫹, with}␮_␬^⫹⬎␮_␬^⫺.

共3兲 The Lebesgue measure of the complement of the set 艛_␬(␮_␬^⫺,␮_␬^⫹) in the real line R is zero.

The following theorem gives the full description of the set of all special ground states of the model 共1兲 at rational densities.

Theorem 3: (Ref. 12.) Suppose that the value of the external field␮of the model (1) belongs to the interval (␮_␬^⫺,␮_␬^⫹) for some number␬⫽q/p. Then the special ground state of the model 共1兲 is unique up to translations.

Following Theorem 4 generalizes the main result of Ref. 13 for all values of the temperature and is a special case 共rational densities兲 of Theorem 1.

Theorem 4: Suppose that the value of the external field ␮of the model 共1兲 belongs to the interval (␮_␬^⫺,␮_␬^⫹) for some number␬⫽q/p.

Then the model共1兲 has a unique limit Gibbs state at all values of the temperature␤^⫺1. Suppose that the value of the external field ␮ of the model 共1兲 belongs to the interval (␮_␬^⫺,␮_␬^⫹) for some number␬⫽q/p.

Let us consider an arbitrary configuration␸(x). We say that␸⁽关a,b兴); a,b苸Z¹is a preregu- lar phase, if there exists a special ground state␸␬, such that the restriction of this configuration to 关a,b兴 coincides with␸⁽关a,b兴). We say that␸⁽关c,d兴); c,d苸Z¹ is a regular phase, if there exists a preregular phase␸⁽关a,b兴); a,b苸Z¹, such that c⫺a⬎d0p and b⫺d⬎d0p. Thus, right and left d₀p extensions of a regular phase are ground states.

Let us consider a set A⫽艛i关ai,b_i兴, where␸⁽关ai,b_i兴) is a regular phase and supp PB is the complement of A in Z¹. The connected components of supp PB defined in such a way are called supports of precontours and are denoted by supp PK: supp PK⫽艛i苸Indsupp PK_i.

For each fixed rational density␬the constant d₀ satisfies some technical conditions.¹³In this work we do not need the explicit value of d₀.

Definition 1 (Ref. 13): The pair PK⫽(supp PK,␸⬘(supp PK)) is called a precontour. The set of all precontours is called a preboundary PB of the configuration␸⬘(x). Two precontours PK₁ and PK₂ are said to be connected if dist(supp PK₁,supp PK₂)⬍Nb. The set of precontours ( PK_i;i苸Ind) is called connected if for any two precontours PKcand PK_d;c,d苸Ind there exists a collection ( PK_j

1⫽PKc,..., PK_j

i,..., PK_j

n⫺1, PK_j

n⫽PKd); j_i苸Ind, i⫽1,...,n; such that any two precontours PK_j

i and PK_j

i⫹1, i⫽1,...,n⫺1 are connected. Let 艛in⫽1

PK_i be some maximal con- nected component of the preboundary PB. Suppose that supp PK_i⫽关ai,b_i兴 and bi⬍ai⫹1; i

⫽,...,n⫺1.

The pair K⫽(supp K,␸⬘(supp PK)), where supp K⫽关a1,b_n兴 is called a contour. The set of all contours is called a boundary B of the configuration␸⬘^(x).

In this work we do not need the exact value of the constant N_b.¹²From Ref. 12 it becomes clear that lim_p→⬁N_b⫽⬁. Thus, for irrational values of the density␬ ^Nbis not defined, but as will be seen below, we do not need to define N_b for irrational densities.

Note that supp K⫽(艛in⫽1

supp PK_i)艛(关a1,b_n兴⫺(艛in⫽1

supp PK_i))⫽supp¹K艛supp²K.

The sets supp¹K and supp²K will be, respectively, called the essential and regular parts of the support supp K.

Let the boundary conditions␸^{¯ (x)}⫽关␸^(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed. The set of all configurations␸^{(x); x}苸关⫺V,V兴 we denote via ⌽(V).

It is obvious that for each contour K, such that supp K苸关⫺V⫹(d0⫹1)p,V⫺(d0⫹1)p兴, there exists a configuration ␺K(关⫺V,V兴) such that the boundary of the configuration

␺K(关⫺V,V兴) includes the contour K only:

B共␺K共关⫺V,V兴兲兲⫽K.

Let supp K⫽关a,b兴. It is obvious that the restrictions of the configuration␺K(关⫺V,V兴) to the segments关⫺V,a⫺1兴 and 关b⫹1,V兴 coincide with two ground states␸_␬^{1(x) and}␸_␬^2(x).

A contour K is called an interface contour, if␸␬1

(x)⫽␸␬2

(x).

Note that,␸_␬¹(x) can be obtained by some shifting of the configuration␸_␬^2(x).

An interface contour will be denoted as IK.

Let K be a usual contour 共not an interface contour兲 K,supp K傺关⫺V,V兴 and ␺^K共x兲

⫽␺^([⫺V,V]) if x苸关⫺V,V兴, and ␸^{¯ (x) if x}苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁); IK,supp IK傺 关⫺V,V兴 be an interface contour and ␺IK(x)⫽␺⁽关⫺V,V兴) if x苸关⫺V,V兴, and ␸^{¯ (x) if x}苸 (⫺⬁,V⫺1兴艛关V⫹1,⬁); and ␸^¯_␬^1(x)⫽␸_␬^{1(x), if x}苸关⫺V,V兴, and ␸^{¯ (x) if x}苸(⫺⬁,⫺V

⫺1兴艛关V⫹1,⬁).

Below the configuration␸^¯_␬¹(x) defined for usual contours will be denoted by␸^¯␬(x).

The weights of the usual contour K and interface contour IK will be calculated by the follow- ing formulas:

␥共K兲⫽H共␺K共x兲兲⫺H共␸^¯␬共x兲兲, 共3兲

␥共IK兲⫽H共␺IK共x兲兲⫺H共␸^¯_␬¹共x兲兲. 共4兲 The proof of Theorem 4 is based on the following idea. Let the boundary conditions␸^{¯ (x)}

⫽关␸^(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed. The set of all configurations ␸^{(x); x}苸 关⫺V,V兴 we denote via ⌽(V). Suppose a configuration␸min(x)苸⌽(V) be a configuration with the minimal energy:

H共␸min共x兲兲⫽min_{␸共x兲苸⌽共V兲}H共␸共x兲兲 .

Then the configuration␸min(x) almost coincides with a special ground state of the model共1兲 共Lemma 1 in Sec. II兲. This fact allows us, based on special ground states, to define a common 共for all boundary conditions兲 contour model and after that by using well-known trick¹⁴ 共this trick, which was introduced in Ref. 14 for some special extensions of Pirogov–Sinai theory, is directly applicable to one-dimensional models with long-range interaction兲 to come to noninteracting clusters from interacting contours. Consider an arbitrary segment I, a sufficiently large volume V, two arbitrary boundary conditions ␸^{1(x) and} ␸²(x). It turns out that the dependence of the expression P¹(␸⬘^(I))/P2(␸⬘(I)) on the boundary conditions␸^{1(x) and} ␸²(x) can be estimated through the sum of statistical weights of super clusters connecting the segment I with the boundary and this sum is negligible. Thus, two arbitrary extreme Gibbs states are relatively continuous and hence coincide. In Ref. 13 we developed this method 关the estimation of dependence of the ex- pression P¹(␸⬘^(I))/P2(␸⬘(I)) on the boundary conditions through the sum of statistical weights of super clusters connecting the segment I with the boundary兴 at low temperatures. It turns out that after some modification the method works at all temperatures.

The contents of this paper are as follows. In Sec. II we prove Theorem 4, in Sec. III we complete the proof of Theorem 1.

II. UNIQUENESS OF GIBBS STATES: THE DENSITY␬ISp/q Let us now introduce some necessary facts.

Suppose that the value of the external field ␮ of the model 共1兲 belongs to the interval (␮_␬^⫺,␮_␬^⫹) for some number␬⫽q/p.

Let the boundary conditions␸^1(x)⫽关␸^1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed and

H共␸共x兲兩␸¹共x兲兲⫽⫺␮_x苸Z1,x

兺

兺

U共x⫺y兲␸共x兲␸共y兲

⫹_x,y苸Z1,x⬎y;x苸关⫺V,V兴;y苸关⫺V,V兴

兺

U共x⫺y兲␸共x兲␸¹共y兲

⫹

兺

x,y苸Z¹,x⬎y;x苸关⫺V,V兴,y苸关⫺V,V兴

U共x⫺y兲␸¹共x兲␸共y兲. 共5兲

Lemma 1: Let␸min(x)苸⌽(V) be a configuration with the minimal energy:

H共␸min共x兲兩␸¹共x兲兲⫽min_{␸共x兲苸⌽共V兲}H共␸共x兲兩␸¹共x兲兲 . Then the configuration␸min(x) has the following structure.

The restriction of the configuration␸min(x) on the set关⫺V⫹Nb,V⫺Nb兴 contains at most p⫺1 contours, moreover, all of them are interface contours IKi,, i⫽1,...,m, where m⬍p⫺1 and 兩supp IKi兩⬍3d0p⫹Nb.

Lemma 1 was proved in Ref. 13关see Lemma 12 共Ref. 13兲 and Sec. 5 of Ref. 13兴.

Let H(␸^(x)兩␸^1(x),␸min(x)) denote the relative energy of a configuration␸^(x)关with respect to

␸min(x)]:

H共␸共x兲兩␸¹共x兲,␸min共x兲兲⫽H共␸共x兲兩␸¹共x兲兲⫺H共␸min共x兲兩␸¹共x兲兲.

Consider the Gibbs distribution P¹ on ⌽(V) corresponding to the boundary conditions

␸^1(x)⫽关␸^1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴:

P¹共␸⬘共x兲兲⫽ exp共⫺␤共H共␸⬘共x兲兩␸¹共x兲,␸min共x兲兲兲兲

⌺_{␸共x兲苸⌽共V兲}exp共⫺␤共H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲兲. 共6兲 Let ␸^(x)苸⌽(V) be an arbitrary configuration, the boundary of the ␸(x) includes a finite number of usual contours K_i; i⫽1,...,n, and a finite number of interface contours IKi; i⫽n

⫹1,...,n⫹m. Let Ki⫽Ki; i⫽1,...,n; Ki⫽IKi; i⫽n⫹1,...,n⫹m. The set of all contours of the boundary conditions␸¹(x) will be denoted by K₀.

The statistical weights of contours and interface contours are

w共Ki兲⫽exp共⫺␤␥共Ki兲兲. 共7兲

The following equation is a direct consequence of the formulas共3兲, 共4兲, and 共7兲

exp共⫺␤^H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲⫽Q1ⁿ_i

兿

where the multiplier G(K₀,K₁,...,K_n⫹m) corresponds to the interaction between contours共usual and interface兲, and with the boundary conditions␸^1(x)

G共K0,K₁,...,K_n⫹m兲⫽_{i, j⫽0;i⬍ j}ⁿ

兺

兺

兺

⬍const1⬍Q1⬍const2. The factor Q₁ appears due to the facts that the configuration ␸min(x) not necessarily coincides with a special ground state and is bounded due to Lemma 1.

Now we write down the value of the interaction between the contours K_iand K_j, the value of the interaction between the interface contours IK_iand IK_jand the value of the interaction between contour K_i and interface contour IK_j.

Suppose supp K_l⫽关al,b_l兴; supp IKl⫽关al,b_l兴.

Let

supp IK_i^⫹⫽关bi,a_i⫹1兴 and supp IKi⫺⫽关bi⫺1,a_i兴,

where b₀⫽c, if there exists K苸B(␸⬘(x)), such that supp K⫽关⫺⬁,c兴 and b0⫽⫺⬁ otherwise;

a_m⫹1⫽d, if there exists K苸B(␸⬘(x)), such that supp K⫽关d,⬁兴 and am⫹1⫽⬁ otherwise.

共1兲 The contour Ki苸B(␸⬘(x)) interacts with the contour K_j苸B(␸⬘(x)) through all pairs (x, y ), such that (x, y )苸Int(Ki,K_j) and f⬘^{(x, y ,}␸⁾⫽0 where

Int共Ki,K_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp Ki, y苸supp Kj兴.

The value of the interaction

f⬘共x,y,␸兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺K_i共x兲␺K_i共y兲⫹␸^¯_␬ⁱ共x兲␸^¯_␬ⁱ共y兲

⫺␺K_j共x兲␺K_j共y兲⫹␸^¯_␬^j共x兲␸^¯_␬^j共y兲兲.

共2兲 The interface contour IKi苸B(␸⬘(x)) interacts with the interface contour IK_j 苸B(␸⬘^(x)) 共let aj⬎bi) through all pairs (x,y ), such that (x, y )苸Int(IKi,IK_j) and f⬙^{(x, y )}⫽0, where

Int共IKi,IK_j兲⫽Int¹共IKi,IK_j兲⫹Int²共IKi,IK_j兲⫹Int³共IKi,IK_j兲⫹Int⁴共IKi,IK_j兲, Int¹共IKi,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp IKi and y苸supp IKj兴, Int²共IKi,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp IKi and y苸supp IK⫹j 兴, Int³共IKi,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp IKi⫺ and y苸supp IKj兴, Int⁴共IKi,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp IKi⫺ and y苸supp IK⫹j 兴.

The value of the interaction

f⬙共x,y,␸兲⫽ f1⬙共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺IK_i共x兲␺IK_i共y兲

⫹␸^¯_␬ⁱ共x兲␸^¯_␬ⁱ共y兲⫺␺IK_j共x兲␺IK_j共y兲⫹␸^¯_␬^j共x兲␸^¯_␬^j共y兲兲

if (x,y )苸Int²(IK_i,IK_j),

f⬙共x,y兲⫽ f2⬙共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺IK_i共x兲␺IK_i共y兲⫹␸^¯_␬ⁱ共x兲␸^¯_␬ⁱ共y兲兲 if (x,y )苸Int²(IK_i,IK_j),

f⬙共x,y兲⫽ f3⬙共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺IK_j共x兲␺IK_j共y兲兲⫹␸^¯_␬^j共x兲␸^¯_␬^j共y兲) if (x, y )苸Int³(IK_i,IK_j),

f⬙共x,y兲⫽ f4⬙共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␸^¯␬1,i共x兲␸^¯␬1,i共y兲⫺␸^¯␬2,j共x兲␸^¯␬2,j共y兲兲 if (x,y )苸Int⁴(IK_i,IK_j).

共3兲 The contour Ki苸B(␸⬘(x)) interacts with the interface contour IK_j苸B(␸⬘(x)) through all pairs (x,y ), such that (x,y )苸Int(Ki,IK_j) and f⵮^{(x,y )⫽0, where}

Int共Ki,IK_j兲⫽Int¹共Ki,IK_j兲⫹Int²共Ki,IK_j兲,

Int¹共Ki,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp Ki and y苸supp IKj兴, Int²共Ki,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp Ki and y苸supp IK⫹j 兴 if a_j⬎bi, and

Int²共Ki,IK_j兲⫽关共x,y兲:x,y苸Z¹;x苸supp Ki and y苸supp IK⫺j 兴 if a_i⬎bj.

The value of the interaction

f⵮共x,y兲⫽ f1⵮共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺K_i共x兲␺K_i共y兲

⫹␸^¯_␬ⁱ共x兲␸^¯_␬ⁱ共y兲⫺␺IK_j共x兲␺IK_j共y兲兲⫹␸^¯_␬^j共x兲␸^¯_␬^j共y兲)

if (x,y )苸Int¹(K_i,IK_j),

f⵮共x,y兲⫽ f2⵮共x,y兲⫽U共x⫺y兲共␸⬘共x兲␸⬘共y兲⫺␺K_i共x兲␺K_i共y兲⫹␸^¯_␬ⁱ共x兲␸^¯_␬ⁱ共y兲兲 if (x,y )苸Int²(K_i,IK_j).

For simplicity K_i, i⫽1,...,n⫹m will be denoted by Ki, i苸Ind, where the statistical weights w(K_i) are defined by the formulas共7兲, 共3兲, and 共4兲. Thus, the formula 共8兲 has the form

exp共⫺␤^H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲⫽Q1_i苸Ind

兿

The set of all pairs共x,y兲 in the double sum 共9兲 will be denoted by Y⫽Y(K0,K₁,...,K_n⫹m).

Write共10兲 as follows:

exp共⫺␤^H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲⫽Q1_i苸Ind

兿

兿

From 共11兲 we get

exp共⫺␤^H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲⫽Q1_G

兿

兿

兿

where the summation is taken over all subsets Y⬘ 共including the empty set兲 of the set Y, and g(x, y ,␸⁾⫽exp(⫺␤^f(x,y,␸⁾⁾⫺1.

Consider an arbitrary term of the sum共12兲, which corresponds to the subset Y⬘傺Y. Let the bond (x,y )苸Y⬘. Below, contours and interface contours will be called contours. Consider the set K of all contours such that for each contour K傺K, the set supp K艚(x艛y) contains one point. We call any two contours from K connected. The set of contours K is called Y⬘ connected if for any two contours K_a and K_b there exists a collection (K₁⫽Ka,K₂,..., K_n⫽Kb) such that any two contours K_i and K_i⫹1, i⫽1,...,n⫺1, are connected by some bond (x,y)苸Y⬘^.

The pair D⫽关(Ki,i⫽1,...,s);Y⬘兴, where Y⬘is some set of bonds, is called a cluster provided there exists a configuration ␸(x) such that K_i苸B(␸^{(x)); i}⫽1,...,s; Y⬘傺Y; and the set (Ki, i

⫽1,...,s) is Y⬘ connected. The statistical weight of a cluster D is defined by the formula.

w共D兲⫽_i

兿

兿

⬘ g共x,y,␸兲. 共13兲

Two clusters D₁ and D₂are called compatible provided any two contours K₁ and K₂belong- ing to D₁ and D₂, respectively, are compatible and not connected. A set of clusters is called compatible provided any two clusters of it are compatible.

If D⫽关(Ki,i⫽1,...,s);Y⬘兴, then we say that Ki苸D; i⫽1,...,s.

The following lemma is a direct consequence of the definitions.

Lemma 2: Let the boundary conditions ␸^1(x)⫽关␸^1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed.

If关D1,...,D_m兴 is a compatible set of clusters and 艛im⫽1

supp D_i傺关⫺V,V兴, then there exists a configuration␸(x) which contains this set of clusters. For each configuration␸(x) we have

exp共⫺␤^H共␸共x兲兩␸¹共x兲,␸min共x兲兲兲⫽Q1_Y

兺

⬘^傺Y

兿

where the clusters D_iare completely determined by the set Y⬘. The partition function is

⌶共␸¹共x兲兲⫽Q

兺

where the summation is taken over all nonordered compatible collections of clusters and the factor Q⫽Q(V,␸¹(x)) is uniformly bounded: 0⬍const⬍Q⬍const2.

Lemma 2 shows that we come to noninteracting clusters from interacting contours.

Let P¹and P²be two Gibbs states of the model共1兲 corresponding to the boundary conditions

␸^{1(x) and}␸²(x), respectively.

The following lemma has a key role in the proof of Theorem 4.

Lemma 3: Suppose that the value of the external field ␮ of the model (1) belongs to the interval (␮_␬^⫺,␮_␬^⫹) for some number␬⫽q/p.

Then the measures P¹ and P² are absolutely continuous with respect to each other.

Proof: Let I⫽关a,b兴 be an arbitrary segment and␸⬘(I) be an arbitrary configuration.

In order to prove the lemma we show that there exist two positive constants s and S not depending on I,␸^1(x), ␸^{2(x) and}␸⬘(I), such that

s⭐P¹共␸⬘共I兲兲/P²共␸⬘共I兲兲⭐S. 共14兲 Let P_V¹ and P_V² be Gibbs measures corresponding to the boundary conditions ␸^{1(x), and}

␸^{2(x), x}苸Z¹⫺IV, respectively, where I_V⫽关⫺V,V兴.

Therefore,

lim

V→⬁

P_V¹⫽P¹ and lim

V→⬁

P_V²⫽P²,

where by convergence we mean weak convergence of probability measures.

In order to establish the inequality 共14兲 it will be proved that for each fixed interval I, I傺关⫺M,M兴 there exists a number V0( M ), which depends on M only, such that

s⭐PV

1共␸⬘共I兲兲/PV

2共␸⬘共I兲兲⭐S 共15兲

if V⬎V0. Consider

P_V¹共␸⬘共I兲兲⫽兺_{␸共IV兲:␸共I兲⫽␸⬘共I兲}exp共⫺␤^H共␸共IV兲兩␸¹共x兲,␸min共x兲兲兲O共␸共I兲,V,␸¹兲 兺␸共I_V兲exp共⫺␤^H共␸共IV兲兩␸¹共x兲,␸min共x兲兲兲O共␸共I兲,V,␸¹兲

⫽ ⌶共IV⫺I兩␸¹共x兲,␸⬘共I兲,␸min共x兲兲O共␸共I兲,V,␸¹兲 兺_␸⬙共I兲⌶共IV⫺I兩␸¹共x兲,␸⬙共I兲,␸min共x兲兲O共␸共I兲,V,␸¹兲

where⌶(IV⫺I兩␸^1(x),␸⬘^(I),␸min(x)) denotes the partition function corresponding to the bound- ary conditions ␸^{1(x), x}苸Z¹⫺IV, ␸⬘^{(I), x}苸I and

O共␸共I兲,V,␸¹兲⫽exp(⫺␤_x,y苸Z1;x苸Z

兺

U共x⫺y兲共␸¹共x兲␸共y兲⫺␸¹共x兲␸min共x兲兲).

We can express P_V²(␸⬘(I)) in just the same way.

In order to prove the inequality共15兲 it is enough to establish inequality 共16兲 and inequality 共17兲:

1/2⬍O共␸共I兲,V,␸ⁱ共x兲兲⬍2, i⫽1,2 共16兲 关where the inequalities in 共16兲 are held uniformly with respect to␸^{(I) and}␸ⁱ: for each I there exists V, not depending on␸^{(I) and}␸^{i] and}

1/S⭐⌶共IV⫺I兩␸¹共x兲,␸⬙共I兲,␸min共x兲兲

⌶共IV⫺I兩␸¹共x兲,␸⬘共I兲,␸min共x兲兲

冒

for arbitrary␸⬙^(I).

Indeed, if the inequality共17兲 holds, then

⌶共IV⫺I兩␸¹共x兲,␸⬘共I兲,␸min共x兲兲

兺_{␸⬙共I兲}⌶共IV⫺I兩␸¹共x兲,␸⬙共I兲,␸min共x兲兲

冒

⫽AV

1共␸⬘共I兲兲/AV 2共␸⬘共I兲兲

⫽1

冒 冉

冒

冊

⫽1

冒

Therefore,

1/共1/s兲⭐AV

1共␸⬘共I兲兲/AV

2共␸⬘共I兲兲⭐1/共1/S兲 since the quotient of兺in⫽1a_i/兺in⫽1b_ilies between min(a_i/b_i) and max(a_i/b_i).

Thus, if in addition, the inequality共16兲 holds, then 2^⫺4s⬍PV

1共␸⬘共I兲兲:PV

2共␸⬘共I兲兲⬍2⁴S.

Now we start to prove the inequalities共16兲 and 共17兲.

It can be easily shown that共16兲 is a direct consequence of the condition U(x)⬃Ax^⫺␥, at x

→⬁; where␥⬎1, and A is a strong positive constant.

So, in order to complete the proof of Lemma 3 we must establish the following inequality 关which is just transformed inequality 共17兲兴:

1/S⭐⌶共IV⫺I兩␸¹共x兲,␸⬙共I兲,␸min共x兲兲)⌶共IV⫺I兩␸²共x兲,␸⬘共I兲,␸min共x兲兲

⌶共IV⫺I兩␸²共x兲,␸⬙共I兲,␸min共x兲兲)⌶共IV⫺I兩␸¹共x兲,␸⬘共I兲,␸min共x兲兲⫽⌶^1,⬙⌶^2,⬘

⌶^2,⬙⌶^1,⬘⭐1/s.

共18兲 Consider

⌶^1,⬙⌶^2,⬘⫽⌶共IV⫺I兩␸¹共x兲,␸⬙共I兲,␸min共x兲兲⌶共IV⫺I兩␸²共x兲,␸⬘共I兲,␸min共x兲兲.

The following generalization of the definition of the compatibility allows us to represent

⌶^1,⬙⌶^2,⬘as a single partition function.

A set of clusters is called super compatible provided any of its two parts coming from two partitions sums is compatible. In other words, in super compatibility an intersection of supports of two clusters is allowed.

The following lemma is an analogue of Lemma 2.

Lemma 4: Let boundary conditions ␸^1(x)⫽关␸^1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 and

␸^2(x)⫽关␸^2(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed.

If关D1,...,D_m兴 is a super compatible set of clusters and 艛im⫽1

supp D_i傺关⫺V,V兴, then there exist two configurations ␸^{3(x) and} ␸⁴(x) which contain this set of clusters. For each two con- figurations␸^{3(x) and}␸⁴(x) we have

exp(⫺␤^H共␸³共x兲兩␸¹共x兲,␸min共x兲兲exp(⫺␤^H共␸⁴共x兲兩␸¹共x兲,␸min共x兲兲⫽Q1_G

兺

⬘^傺G,G⬙^傺G

兿

where the clusters D_i are completely determined by the sets G⬘ ^{and G}⬙. The super partition function is

⌶^1,⬙,^2,⬘⫽⌶^1,⬙⌶^2,⬘⫽Q

兺

where the summation is taken over all nonordered super compatible collections of clusters and the factor Q⫽Q(V,␸^1(x),␸²(x)) is uniformly bounded: 0⬍const1⬍Q⬍const2.

Lemma 4 is a direct consequence of the definitions.

An arbitrary connected component of an arbitrary super compatible set of clusters will be called a super clusters. A super cluster SD⫽关(Ki,i⫽1,...,r);G⬘兴 is said to be long if the inter- section of the set (艛im⫽1

supp K_i))艛G⬘ with both I and Z¹⫺IV⫽(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁) is nonempty. In other words, a long super cluster connects the boundary with the segment I.

A set of super clusters is called compatible provided the set of all clusters belonging to these super clusters are super compatible.

It turns out that in our estimates long super clusters are negligible.

Lemma 5: For each fixed interval I, there exists a number V₀(I), which depends on I only, such that if V⬎V0(I)

1/2⌶^1,⬘,2,⬙⬍⌶^{1,⬘,2,⬙,共n.l.兲}⫽

兺

where the summation is taken over all nonlong, nonordered compatible collections of super clusters 关SD1,...,SD_m兴, 艛im⫽1

supp(SD_i)傺IN⫺I corresponding to the boundary conditions

␸^1(x),␸^{2(x), x}苸Z¹⫺IV; ␸⬘^{(x) and}␸⬙^{(x), x}苸I.

Consider a collection of contours K₀,K₁,...,K_n. The value of the interaction of the contour K₀ with the contours K₁,...,K_n we denote by G(K₀兩K1,...,K_n):

G共K0兩K1,...,K_n兲⫽_B苸IG共0兩1,...,n兲

兿

where IG(0兩1,...,n) is the set of all interaction elements intersecting the support of the contour K₀.

Lemma 6:

G共K0兩K1,...,K_n兲⫽_B苸IG共0兩1,...,n兲

兿

⭐const共dist共0兩1,...,n兲兲^⫺␣共兩supp共K0兲兩兲^1⫺␣, 共20兲 where dist(0兩1,...,n) is the distance between the support of K0 and the union of the supports of contours K₁,...,K_n.

In other words, the interaction of K₁,...,K_n on K₀ tends to zero when the distance between them increases, and value of the interaction increases with a rate less than the length of the support of K₀.

The technical Lemma 6 follows from the decreasing conditions of the potential U(x). For the rigorous proof see Ref. 13, Lemma 4.

The following lemma is an analogue of Lemma 5 for clusters共not super clusters兲.

Lemma 7: For each fixed interval I, there exists a number V₀(I), which depends on I only, such that if V⬎V0(I)

1/2⌶^1,⬘⬍⌶^{1,⬘,共n.l.兲}⫽

兺

where the summation is taken over all nonlong, nonordered compatible collections of clusters 关D1,...,D_m兴, 艛im⫽1

supp D_i傺IN⫺I corresponding to the boundary conditions ␸^{1(x), x}苸Z¹

⫺IV; ␸⬘^{(x), x}苸I.

Proof:

⌶^1,⬘⫽⌶^{1,⬘,共n.l.兲}⫹共⌶^1,⬘⫺⌶^{1,⬘,共n.l.兲}兲⫽⌶^{1,⬘,共n.l.兲}⫹⌶^{1,⬘,共l.兲},

where the summation in ⌶^1,⬘,(l.) is taken over all nonordered compatible collections of clusters 关D1,...,D_m兴 containing at least one long cluster, 艛im⫽1supp D_i傺IN⫺I corresponding to the boundary conditions␸^{1(x), x}苸Z¹⫺IV; ␸⬘^{(x), x}苸I.

By dividing both sides of the last equality by⌶^1,⬘, we get

1⫽⌶^{1,⬘,共n.l.兲}/⌶^1,⬘⫹⌶^{1,⬘,共l.兲}/⌶^1,⬘. 共21兲 Now we are going to show that the second term共which is not necessarily positive兲 is negli- gible, that is the absolute value of it is less than 1/2共actually we can show that the absolute value of the second term is less than any fixed positive number at sufficiently large values of V).

The term⌶^1,⬘,(l.)/⌶^1,⬘can be interpreted as a ‘‘probability’’ P共Long兲 of the event that there exists at least one long cluster.

We show that the absolute value of this ‘‘probability’’ is less than 1/2 by the following method. We estimate the density of long clusters: the probability that a given segment belongs to the support of some long cluster. Since some statistical weights of clusters are positive and some negative, we estimate the absolute values of these ‘‘probabilities.’’ We show that for a fixed segment the ‘‘probability’’ that this segment belongs to the support of some long cluster with positive ‘‘probability’’ minus the ‘‘probability’’ that this segment belongs to the support of some long cluster with negative ‘‘probability’’ is less than one. Since the density is less than one, by the law of large numbers a ‘‘typical’’ long cluster has not very long support, and therefore has long bonds. When V tends to infinity, the total length of bonds tends to infinity, and the impact of these bonds tends to zero.

Now we replace a statistical weight w(D_i) of each cluster D_ibelonging to the configuration containing at least one long cluster with its absolute value 共and ‘‘probability’’ of long cluster