interface, surface tension and reentrant pinning transition ... - CiteSeerX

1

INTERFACE, SURFACE TENSION AND REENTRANT PINNING TRANSITION IN THE 2D ISING MODEL C.-E. P stera , Y. Velenikb a D epartement

de Mathematiques, EPF-L CH-1015 Lausanne, Switzerland e-mail: charles.p [email protected] .ch b D epartement de Physique, EPF-L CH-1015 Lausanne, Switzerland e-mail: [email protected] .ch 13 June 1997

Abstract: We report about concentration results for the measure giving the

statistical properties of phase separation lines in the 2D Ising model. The results are sharp and valid for all temperatures below the critical one; they are consequences of a new approach for treating phase separation lines in the 2D Ising model and the following interesting inequality of the surface tension ^(x) of an interface perpendicular to x: for any x, y ^(x) + ^(y) ? ^(x + y) 1 (kxk + kyk ? kx + yk) ; where is the maximum curvature of the Wul shape and kxk the Euclidean norm of x. We apply these new techniques to the study of the pinning transition for the 2D Ising model. One major result is that we can establish the thermodynamical variational problem, which holds at the macroscopic level, for this surface phenomenon, starting from the microscopic description of this model. We can solve explicitely this variational problem and prove that there is a non-perturbative eect, namely reentrant pinning transition. These techniques provide a precise description of typical con gurations at the macroscopic scale and can be used in a variety of dierent situations.

Key Words: Ising model, reentrant pinning transition; interface, surface tension,

positive stiness; correlation length, nite size eects.

Reentrant Pinning Transition

2

1 Introduction Consider a 2D Ising model in some rectangular box with boundary conditions implying the presence of a phase separation line crossing the box from one xed point of one vertical side to another xed point of the other vertical side. We suppose that the model is in the phase coexistence region; the boundary condition is chosen so that above the phase separation line we have the + phase and below it the ? phase. The bottom horizontal side of the box, which we call the wall, is subject to a negative boundary magnetic eld. By varying the temperature or the boundary magnetic eld we can observe the so-called pinning{depinning transition: for values of these parameters for which the + phase wets partially the wall, and under appropriate geometrical conditions, the equilibrium shape of the interface changes from a straight line crossing the box to a broken line touching a macroscopic part of the wall. This macroscopic phenomenon is the pinning{depinning transition. Moreover, it is possible to nd values of the boundary magnetic eld and temperatures 0 < T1 < T2 < Tc such that the interface is not pinned for T < T1 or T > T2 and pinned for T1 < T < T2 ; we can also nd values of the boundary magnetic eld and temperatures 0 < T1 < T2 < T3 < Tc such that for T < T1 or T2 < T < T3 the interface is pinned, while for T1 < T < T2 or T > T3 it is not pinned. These reentrant pinning{depinning transitions are predicted by a macroscopic variational problem for the interface, which is formulated in terms of the surface tension and wall free energies of the model. There is no simple argument based on the microscopic interactions, which can predict these reentrant phase transitions. It is necessary to discuss a macroscopic variational problem. It is important to clearly distinguish the notions of phase separation line and interface. The concept of \interface" is a macroscopic notion; the fundamental thermodynamical function associated with an interface is the surface tension (in [ABCP] similar ideas are developed). The interface gives a non{random description of the boundary between the two phases, which emerges from the properties of the equilibrium measure at a macroscopic scale, when the uctuations of the phase separation line become negligible (compare with the analysis of the Law of Large Numbers for independent random variables by Talagrand in [T]). By contrast, the \phase separation line" gives a random description of the boundary between the two phases at the lattice spacing scale. Let us illustrate this point by considering the so{called boundary condition, which corresponds to a special case of the present paper, where the phase separation line goes from the middle of a vertical side of the box to the middle of the other vertical side. The local structure of the phase separation line is studied in [BLP1] at low temperatures. The de nition of the phase separation line in [BLP1] coincides with the one of Gallavotti in his work [G] about the phase separation in the 2D Ising model; it diers slightly from the one used here, but in no essential way. (Notice that the terminology \interface" is sometimes used for \phase separation line" in [BLP1].) There are three natural scales in the study of the phase separation line. At the scale of the lattice spacing the phase separation line is a random geometrical line, which has a well-de ned intrinsic width, which is nite and depends strongly on the microscopic interaction. Its middle point has uctuations typically of the order of O(L1=2 ), L being the linear size of the box L containing the system. Because of these uctuations the projection


3

of the corresponding limiting Gibbs state, at the middle of the box, when L ! 1, is translation invariant; the magnetization (at the middle of the box) is zero. If we scale the lengths vertically by 1=L1=2 and horizontally by 1=L, then in the limit L ! 1 the phase separation line converges to a Brownian bridge, [Hi] and also [DH]. At that intermediate scale the phase separation is still random, but its properties show some universal features (compare with the Central Limit Theorem). However, at a mesoscopic scale of order O(L), > 1=2, the results of this paper imply that the system has a well{de ned non{random horizontal interface. To describe the system at the scale O(L) we partition the box L into square boxes Ci of linear size O(L); the state of the system in each of these boxes is speci ed by the empirical P ? 1 magnetization jCij t2Ci (t). Then we rescale all lengths by 1=L in order to get a measure for these normalized block-spins in the xed (macroscopic box) Q. When L ! 1 these measures converge to a non{random macroscopic magnetization pro le showing a well-de ned horizontal interface separating the two phases of the model, characterized by a value m of the normalized block-spins, m being the spontaneous magnetization of the model. This coarsed{grained description of the equilibrium state at the thermodynamic limit is in sharp contrast with the above mentioned result implying that the equilibrium state converges to a translation invariant measure at the thermodynamical limit. These two limits are related to properties of the model at two dierent scales, the lattice spacing scale and the macroscopic one. The main results of this paper are concentration estimates for the measure describing the phase separation line, which are consequences of properties of this measure proved in [PV1] and the sharp triangle inequality for the surface tension [I] (section 5). The sharp triangle inequality, Proposition 8.1, has its own interest; we analysed it in appendix. The concentration results of section 5 allow to treat problems involving a nite number of interfaces. To illustrate our technique we have chosen the phenomenon of pinning{depinning transition for one interface, because it is simple and it presents the reentrance phenomenon. We rst study the thermodynamical problem concerning this interface, which leads to a variational problem, which we can solve explicitly since surface tension and wall free energies ([MW], [A]) are known. In the second part of the paper, starting from the microscopic Hamiltonian, we show how the above problem emerges at a suitable large scale. These results can be roughly stated as follows: The typical con gurations of the system can be described by an interface (separating the two phases) which is the solution of the above thermodynamical variational problem. Recently these questions have been studied by Patrick [Pa] in the SOS approximation of the model, using exact calculations. A similar problem is also treated by Patrick and Upton [PU] for the 2D Ising model. However, because they use transfer matrix technique, they cannot study directly the phase separation line; the local observable which they introduce in order to obtain an interpretation in terms of the phase separation line is not natural, as can be seen from the mathematical results of section 7. Moreover, the transfer matrix technique does not seem suitable to treat problems with several interfaces, except for very particular geometries. An important symmetry property of the 2D Ising model is that it is self{dual. Questions about interfaces below the critical temperature can be translated into questions concerning the asymptotic behaviour of the two{point function (or even correlation


4

functions) above the critical temperature. The interface problem considered here has a dual interpretation, which we discussed in subsection 7.2. The dual eects of the pinning transition correspond to nite{size eects for the correlation length. The paper is divided into two parts. In part I we treat the pinning{depinning transition as a macroscopic phenomenon. However, we rst recall in section 2 the de nition of the phase separation line, surface tension and wall free energy starting from the microscopic Hamiltonian of the model; we need the surface tension and the wall free energy in order to formulate the variational problem related to the pinning{depinning transition. We also recall in section 2 the duality transformation. Then we solve the variational problem. The second part of the paper contains the concentration results for the phase separation line. We use in an essential way recent results of [PV1], in particular those of section 5. They are carefully stated in Propositions 2.3 and 5.2 and Lemmas 6.1 and 6.2, so that we can focus our attention on the essential points here. Then we show that the pinning{depinning transition is indeed described at the macroscopic scale by the variational problem of part I, and describe the typical con gurations on that scale. Acknowledgments: We thank M. Troyanov for very useful discussions and suggestions about the geometrical aspect of the sharp triangle inequality.

Part I: Macroscopic theory, reentrance phenomenon 2 De nitions and notations 2.1 Phase separation line We follow [PV1] for the notation and terminology. Throughout the paper O(x) denotes a non{negative function of x 2 IR+, such that there exists a constant C with O(x) Cx. The function O(x) may be dierent at dierent places. Let Q be the square box in IR2 ,

Q := f x = (x(1); x(2)) 2 IR2 : jx(1)j 1=2 ; 0 x(2) 1 g ; (2.1) and L be the subset of ZZ2 (L an even integer) L := f x = (x(1); x(2)) 2 ZZ2 : jx(1)j L=2 ; 0 x(2) L g : (2.2) Notice that L f x = (x(1); x(2)) 2 ZZ2 : x(2) 0 g for each L . The spin variable at x 2 ZZ2 is the random variable (x) = 1; spin con gurations are denoted by 2 Z Z ! 2 f?1; +1g , so that (t)(!) = 1 if !(t) = 1. We always suppose that we have for the box L either the ab boundary condition (ab b.c.) or the ? boundary condition (? b.c.). Let 0 a 1 and 0 b 1 be given; let D be the straight line passing through the points A = (?1=2; a) and B = (1=2; b). The ab b.c. speci es the values of the spins outside L as follows, (x(1)=L; x(2)=L) is below or on D, 8x 62 L ; (x) := ?+11 ifotherwise. (2.3)


5

The ? b.c. speci es the values of the spins outside L as follows,

8x 62 L ; (x) := ?1 :

(2.4)

In L we consider the Ising model de ned by the Hamiltonian

HL = ?

X

h i\L 6=; t;t0

J (t; t0 )(t)(t0) ;

(2.5)

where ht; t0i denotes a pair of nearest neighbours points of the lattice ZZ2 , or the corresponding edge (considered as a unit{length segment) with end{points t; t0 ; the coupling constants J (t; t0 ) are given by t(2) 0 and t0(2) 0, J (t; t0) := 1h ifotherwise, (2.6) with h > 0. Let be the inverse temperature. The Boltzmann factor is expf? HL g and the Gibbs measures in L with ab b.c., respectively ? b.c., are denoted by

h iabL = h iabL( ; h) resp. h i?L = h i?L ( ; h) :

(2.7)

We introduce the dual lattice to ZZ2 ZZ2 := f x = (x(1); x(2)) : x + (1=2; 1=2) 2 ZZ2 g ;

(2.8)

and describe the spin con gurations ! by a set of edges E (!) of the dual lattice. For each edge e of ZZ2 there is a unique edge e of ZZ2 which crosses e, which is 2 written e y e. Let ! 2 f?1; +1gZZ be a spin con guration satisfying the ab b.c.. We set

E (!) := fe L : 9 ht; t0i y e with (t)(!)(t0)(!) = ?1 g : (2.9) We decompose the set E (!) into connected components and use rule A de ned in the picture below in order to get a set of disjoint simple lines called contours.

-

rule A Each con guration ! satisfying the ab b.c. is uniquely speci ed by a family of disjoint contours ( (!); (!)); all contours of = f 1; 2; : : :g are closed1 and is open, with end{points tLl and tLr . We call the phase separation line. Conversely, a family of contours ( 0; 0) is called ab compatible if there exists ! such that ! satis es the ab b.c. and (!) = 0, (!) = 0. In the same way each con guration ! satisfying the ? b.c. is uniquely speci ed by a family of closed contours and we have a notion of ? compatibility. 1 Let A be a set of edges; the boundary A of A is the set of x 2 ZZ2 such that there is an odd number of edges of A adjacent to x. A is closed if A = ; and open if A = 6 ;.


6

For each contour , closed or open, we de ne a set of edges of ZZ2 , co() := f e 2 ZZ2 : 9e 2 ; e y e g : The Boltzmann weight of is

w() :=

Y

h i2co() t;t0

(2.10)

exp(?2 J (t; t0 )) :

(2.11)

Next we de ne three (normalized) partition functions, Z ab(L), Z ab (Lj) and Z ?(L). By de nition

Z ab (L) :=

X

! with ab b:c:

Z ab (Lj) := Z ?(L) :=

w((!))

X

Y

2 (!)

Y

! with ab b:c:: 2 (!) (!)=

X

Y

! with ? b:c: 2 (!)

w( ) ;

(2.12)

w( ) ;

(2.13)

w( ) :

(2.14)

We de ne a weight qL() = qL(; ; h) for each phase separation line of a con guration ! satisfying the ab b.c., 8 < Z ab(L ) w ( ) qL () := : Z ?(L)

j

if is ab compatible in L, (2.15) 0 otherwise. Notice that qL () is not equal to the probability that the phase separation line is , since, in (2.15), we divide by Z ?(L) and not Z ab (L).

2.2 Surface tension and wall free energy We recall the de nition of the surface tension and the dierence of wall free energies. The heuristic arguments for these de nitions can be found e.g. in [P], [FP1] or [FP2]. However, the ultimate justi cation is that these are precisely the quantities which enter into the formulation of the variational problem describing the behaviour of the interface.

2.2.1 Surface tension Consider the model de ned in L, with coupling constants J (t; t0) 1, i.e. h = 1 in (2.6). For each ! compatible with the ab b.c. there is a well-de ned phase separation line (!) with end{points tLl and tLr. Let n = (n(1); n(2)) be the unit vector in IR2 which is perpendicular to the straight line passing through tLl and tLr . By de nition the surface tension ^(n; ) = ^(n) is 1 log Z ab(L) ; ^(n) = ^(n(1); n(2)) := ? Llim !1 ktL ? tL k Z ?( ) l

r

L

(2.16)


7

where ktLl ? tLrk is the Euclidean distance between tLl and tLr. By symmetry of the model we have

^(n(1); n(2)) = ^(?n(1); ?n(2)) = ^(n(2); ?n(1)) = ^(n(2); n(1)) :

(2.17)

Using (2.15) we can write 1 log ^(n) = ? Llim L !1 kt ? tL k l

r

X (!): ! with ab?b:c:

qL ((!)) :

(2.18)

We extend the de nition of the surface tension to IR2 by homogeneity,

^(x) := kxk^(x=kxk) :

(2.19)

Proposition 2.1 The surface tension is a uniformly Lipschitz convex function on IR2 , such that ^(x) = ^(?x). It is identically zero above the critical temperature and strictly positive for all x = 6 0 when the temperature is strictly smaller than the

critical temperature. The main property of ^ is the sharp triangle inequality. For all > c there exists a strictly positive constant = ( ) such that for any x, y 2 IR2 , the norm ^( ) satis es ^(x) + ^(y) ? ^(x + y) (kxk + kyk ? kx + yk) : (2.20) Let x() := (cos ; sin ) and () := ^(x()). Then the best constant is !

d2 () + () > 0 : := inf d2

(2.21)

Remark: The rst part of the proposition follows from [MW] or [LP]. The strict

positivity of follows from the exact solution for (). The second part is proven in appendix. (2.20) was introduced and proven by Ioe in [I]. The statement of (2.20) is dierent in [I], but equivalent to the present one (see proof of Proposition 8.1). The constant is not optimal in [I]. (2.21) is called the positive stiness property. Geometrically, (2.21) means that the curvature of the Wul shape is bounded above by 1=.

2.2.2 Wall free energy The last thermodynamical quantities, which enter into the description of the properties of the interface, are the wall free energies. In the coexistence region the wall free energy depends on the choice of the bulk phase. In fact, only the dierence of wall free energies when the bulk phase is either the + phase or the ? phase has an intrinsic meaning. We de ne the dierence of the contributions of the wall to the free energy when the bulk phase is the + phase, respectively the ? phase, as 00 ( ) Z 1 log ? L ; ^bd = ^bd ( ; h) := ? Llim !1 2L + 1 Z (L)

(2.22)


8

where Z 00 (L) is the partition function with a = b = 0. The de nition of ^bd diers from the analogous quantity used in [PV1] or [PV2], because in these papers the reference bulk phase is the + phase and here it is the ? phase. The quantity ^bd ( ; h) allows to detect the wetting transition through Cahn's criterium (see [FP1] and [FP2]). Since h > 0, 0 < ^bd ( ; h) ^( ). There is partial wetting of the wall if and only if ^bd ( ; h) < ^( ); this occurs if and only if h < hw , where hw has been computed by Abraham [A]. The transition value hw ( ) is the solution of the equation expf2 gfcosh 2 ? cosh 2 hw ( )g = sinh 2 :

(2.23)

^bd ^(1; 0)

hw ( ) 1

h

Figure 1: ^bd as a function of the magnetic eld h, for = 1:4 c .

2.3 Duality A basic property of the 2D Ising model is self-duality. As a consequence of that property many questions about the model below the critical temperature can be translated into dual questions for the dual model above the critical temperature. For example, questions about the surface tension are translated into questions about the correlation length. We refer to [PV1] for a more complete discussion and recall here the main results, which we shall use in section 5. Let EL := f ht; t0i : ht; t0i \ L 6= ; g ; (2.24) be the set of edges in the sum (2.5) giving HL . The dual set of edges is

EL := f e : 9e 2 EL ; e y e g ;

(2.25)

and the dual model is de ned on the dual box L := fx 2 ZZ2 : 9e 2 EL ; x 2 e g (2.26) 2 = f x 2 ZZ : jx(1)j (L + 1)=2 ; ?1=2 x(2) L + 1=2 g : By our choice of L, L IL := fx 2 ZZ2 : x(2) ?1=2g. Throughout the paper we use the following notations: if A ZZ2 , then E (A) is the set of all edges of ZZ2


9

with both end{points in A. The dual Hamiltonian is the free boundary condition (free b.c.) Hamiltonian, that is,

HL = ?

X

h i t;t0

J (t; t0)(t)(t0 ) :

(2.27)

L

In (2.27) the dual coupling constants are related to the coupling constants (2.6) as follows: 0 0 J (t; t ) := h if t(2) = t (2) = ?1=2, (2.28) 1 otherwise. h and the dual temperature are de ned by tanh := expf?2 g ; tanh h := expf?2 hg :

(2.29) (2.30)

The critical inverse temperature c is characterized by tanh c := expf?2 cg ;

(2.31)

so that > c if and only if < c. Notice also that when h is small, then h is large and vice{versa. The Gibbs measure in L with free boundary condition is denoted by h iL = h iL ( ; h). In this paper > c so that < c. For those values of there is a unique Gibbs measure on ZZ2 , which we denote by h i = h i( ). The most important quantity for the 2D Ising model with free b.c. is the covariance function, or two{point function,

h(t)(t0 )i( ) : The decay-rate of the covariance function is de ned for all t; t0 2 ZZ2 as 1 logh(nt)(nt0 )i( ) : (t ? t0 ) = (t ? t0 ; ) := ? nlim 2IN n n!1

(2.32) (2.33)

The following proposition is a typical duality statement; the decay{rate and the surface tension, or the decay{rate of the boundary two{point function and the dierence wall free energy are dual quantities.

Proposition 2.2 For the 2D Ising model the surface tension ^(x; ) and the decay{ rate (x; ) are equal,

^(x; ) = (x; ) 8x : Let > c. Let t; t0 2 IL , t(2) = t0(2) = ?1=2. Then 1 0 0 ? nlim !1 n logh (nt) (nt )iIL ( ; h ) = kt ? t k ^bd ( ; h) :

(2.34) (2.35)

Duality statements are consequences of the fact that for square boxes the low temperature expansion in L with + b.c. (or ? b.c.) and parameters and h is equivalent to the high temperature expansion in L with free b.c. and parameters and h. Let us make this more precise, leaving some details (see [BLP2] and [PV1]).


10

Consider the partition function in L with free b.c., which can be written as X

Y

(t); t2L ht;t0 iL

cosh J (t; t0)(1 + (t)(t0) tanh J (t; t0 )) :

(2.36)

We expand the product; each term of the expansion is labeled by a set of edges ht; t0i: we specify the edges corresponding to factors tanh J (t; t0 ). Then we sum over (t), t 2 ; after summation only terms labeled by sets of edges of the dual lattice ZZ2 with empty boundary give a non{zero contribution. We decompose this set uniquely into a family of connected closed contours using the rule A. Any such family of contours is called compatible. For each (closed) contour we set

w( ) :=

Y

h i2 t;t0

tanh J (t; t0 ) ;

(2.37)

and we introduce a normalized partition function

Z (L) :=

X

Y

:

2 comp: in L

w( ) :

(2.38)

Comparing (2.38) and (2.14) we see that

Z ?(L) = Z (L) :

(2.39)

We now expand the numerator of the two{point function h(t)(t0)iL in a similar way. In this case all non{zero terms of the expansion are labeled by compatible families ( ; ), where all 2 are closed, is open with end{points t; t0. Given an open contour , we introduce a partition function as in (2.13),

Z (Lj) :=

X

Y

:

2

( ;) comp:

w( ) :

(2.40)

The next two formulas are fundamental. For each open contour we de ne the weight of the contour as 8 < Z (L ) qL () = qL(; ; h ) := : w () Z (L)

j

if E (L), (2.41) 0 otherwise. Using this weight we get a random{line representation for the two{point function h(t)(t0)iL . Let be such that = ft; t0 g; we also write : t ! t0 . Then

h(t)(t0 )iL =

X

:t!

t0

qL () :

(2.42)

There are similar representations for

h(t)(t0)iIL ( ; h) := Llim h(t)(t0)iL ( ; h) !1

(2.43)

h(t)(t0)i( ) := Llim h(t)(t0 )i L ( ; h) ; !1

(2.44)

and


11

where L := fx 2 ZZ2 : jx(i)j L; i = 1; 2g. Given : t ! t0 we can de ne weights

qIL (; ; h ) := Llim q (; ; h) ; !1 L

(2.45)

() Z ( L j) : q(; ) := Llim w !1 Z ( )

(2.46)

L

The limits in (2.44) and (2.46) are independent on h . We have the random{line representations X qIL (; ; h ) ; (2.47) h(t)(t0)iIL ( ; h) =

h(t)(t0)i( ) =

:t!t0

X

:t!

t0

q(; ) :

(2.48)

The next proposition is a key{result which is proven in [PV1].

Proposition 2.3 Let < c. Let 1 and 2 be two open contours such that := 1 [ 2 is an open contour and E (L). Then qL (; ; h ) qIL (; ; h) and qL () qL (1 )qL (2) : (2.49) If = ftLl ; tLr g, then qL (; ; h) is equal to (2.15), that is, qL (; ; h ) = qL(; ; h) :

(2.50)

3 The variational problem The interface is a macroscopic non{random object, whose properties are described by a functional involving the surface tension. In Q the interface is a simple recti able curve C with end{points A := (?1=2; a), 0 < a < 1, and B := (1=2; b), 0 < b < 1. We denote by wQ = f x 2 Q : x(2) = 0 g the wall and by jC \ wQj the length of the portion of the interface in contact with the wall wQ. Suppose that [0; w] ! Q, s 7! C (s) = (u(s); v(s)), is a parameterization of the interface. The free energy of the interface C can be written W(C ) :=

Zw 0

h

i

^(u_ (s); v_ (s))ds + jC \ wQj ^bd ? ^(1; 0) ;

(3.1)

because the function ^(x(1); x(2)) is positively homogeneous and ^(x(1); x(2)) = ^(?x(2); x(1)). The interface at equilibrium is the minimum of this functional. Therefore we have to solve the Variational problem: Find the minimum of the functional W among all simple recti able open curves in Q with extremities A = (?1=2; a) and B = (1=2; b). Let D be the straight line from A to B and W be the curve composed of the following three straight line segments: from A to a point w1 on the wall, then along the wall from w1 to w2, and nally from w2 to B . The points w1 and w2 are such that the


12

angles between the rst segment and the wall and between the last segment and the wall are equal, chosen2 in the interval [0; =2], and solutions of cos Y (Y ) ? sin Y 0(Y ) = ^bd ; (3.2) which is known as the Herring-Young equation. (For the case under consideration the existence of Y is an immediate consequence of the Winterbottom construction).

Proposition 3.1 Let Y be the solution of the Herring-Young equation (3.2). 1. If tan Y a + b then the minimum of the variational problem is given by the curve D. 2. If =2 > Y > arctan(a + b) then the minimum of the variational problem is given by D if W(D) < W(W ), by W if W(D) > W(W ) and by both D and W if W(D) = W(W ).

Proof. The proof is an easy consequence of the two following lemmas. Lemma 3.1 states that the minimum is a polygonal line.

Lemma 3.1 Let C be some simple recti able parameterized curve with initial point A and nal point B .

If C does not intersect the wall, then

W(C ) W(D)

(3.3)

with equality if and only if C =D. If C intersects the wall, let t1 be the rst time C touches the wall and t2 the last time C touches the wall. Let Cb be the curve given by three segments from A to C (t1 ), from C (t1 ) to C (t2 ) and from C (t2 ) to B . Then W(C ) W(Cb) : (3.4) Equality holds if and only if C = Cb.

Proof. Since ^ is convex and homogeneous, we have in the rst case by Jensen's

inequality

Zw Zw Zw 1 1 1 W(C ) = w w 0 ^(u_ (s); v_ (s))ds w^( w 0 u_ (s)ds; w 0 v_ (s)ds) = W(D) : (3.5) The inequality is strict if C 6= D as is easily seen using the sharp triangle inequality (2.20). In the second case we apply Jensen's inequality to the part of C between A and C (t1) and between C (t2 ) and B to compare with the corresponding straight segments of Cb. Combining Jensen's inequality and the fact that ^bd ^, we can also compare the part of C between C (t1 ) and C (t2 ) with the corresponding straight segment of Cb. ut

This choice leads to a dierent sign at the right{hand side of the Herring-Young equation (3.2) than in [PV2] formulae (1.5) or (4.60); in these latter references we use ? instead of . 2


13

Lemma 3.2 Let Cb be a polygonal line from A to w^1 2 wQ, then from w^1 to w^2 2 wQ,

and nally from w^2 to B . Let Y be the solution of the Herring-Young equation (3.2). If =2 > Y > arctan(a + b) then W(Cb) W(W ) ;

with equality if and only if Cb = W . If arctan(a + b) Y

W(Cb) > W(D) :

(3.6) (3.7)

Proof. Let 1 2 (0; =2) be the angle of the straight segment of Cb, from A to w^1 , with the wall wQ, and 2 2 (0; =2) be the angle of the straight segment of Cb, from w^2 to B , with the wall wQ. We have W(Cb)

= (1 ) a + ^bd (1 ? a ? b ) + (2 ) b sin 1 tan 1 tan 2 sin 2 = g(1; a) + g(2; b) ;

(3.8)

where we have introduced

g(; x) := () sinx + ^bd(1=2 ? tanx ) :

(3.9)

Let Y be de ned as the solution of the Herring-Young equation (3.2), so that @ g( ; x) = x (sin 0 ( ) ? cos ( ) + ^ ) = 0 : (3.10) Y Y Y Y bd @ Y sin2 Y The second derivative of g(; x) is

@ 2 g(; x) = x( () + 00 ()) ? 2 @ g(; x) : (3.11) @2 sin tan @ Therefore, for 2 (0; =2), we have @ g(; x) = x Z expf? Z 2 dg ( ) + 00 ( ) d : (3.12) @ sin Y

tan Since has positive stiness, i.e. ()+ 00 () > 0, (3.12) implies that Y is an absolute minimum of g(; x) over the interval (0; =2), and that g is strictly monotonous over the intervals (Y ; =2) and (0; Y ). A necessary and sucient condition, that we can construct a simple polygonal line Cb as above, is a + b 1: (3.13) tan tan 1

2

In particular 1 2 [a ; =2] where a := arctan a, and 2 2 [b ; =2] where b := arctan b. Similarly W is a simple curve in Q if and only if

Y 2 [arctan a + b; =2) :

(3.14)


14

h 0:8

i ii iii

0:2

iv Tc T

Figure 2: A sequence of phase transition loci, separating the phase in which the interface is a straight line and the phase in which it is pinned to the wall. The shaded area corresponds to the value of (T; h) so that ^bd ( ; h) < ^((1; 0); ). The four curves correspond to: i) a = 0:1, b = 0:1; ii) a = 0:1, b = 0:2; iii) a = 0:1, b = 0:4; iv) a = 0:4, b = 0:4. Observe that the system in case i) exhibits reentrance (see also Fig. 3).

From the preceding results we have W(Cb) g (1 ; a) + g (2 ; b) ;

with

Y a 2 = Y b

1

=

if Y 2 [a ; =2], otherwise, if Y 2 [b ; =2], otherwise.

(3.15) (3.16)

If (3.14) holds, then (3.15) implies W(Cb) W(W ). If (3.14) does not hold, then the two segments from A to the wall, and from B to the wall intersect at some point P . c be the simple polygonal curve going from A to P , then from P to B . We Let W have (this follows from Lemma 3.1 and ^(1; 0) ^bd ) c) : g(Y ; a) + g(Y ; b) W(W

(3.17)

Applying again Lemma 3.1 we get c ) > W(D) : W(W

(3.18)

ut

4 Reentrance and pinning transition The results of the preceding section show that, when the parameters a and b are wellchosen, the system under consideration can undergo a phase transition from a phase in which the interface is pinned to the wall on a macroscopic distance to a phase


15

h

h

0:82

0:82

0:80

0:80

0:78

1 T 0:78 1 T Figure 3: This gure shows part of the phase transition line for a = 0:1; b = 0:1 (left), and a = 0:1; b = 0:12 (right). For values of the parameters T and h below these curves the interface is pinned, while it is a straight line above these curves. Increasing the temperature along the dashed lines, we see that the system exhibits reentrance; this corresponds to the two situations discussed in the introduction.

in which it does not touch the wall. It is interesting to consider the corresponding phase diagram. Figure 2 shows a set of phase transition loci, depending on the parameters a and b, in the T -h plane (T = 1=k being the temperature). The shaded area corresponds to the set of parameters

f(T; h) : ^bd( ; h) < ^((1; 0); )g :

(4.1)

In other words, the boundary of that region is the wetting transition line: If we set a = b = 0, then for values of the temperature and boundary magnetic eld inside this set, the phase separation line is pinned to the wall microscopically (partial wetting), while for values of the parameters outside this set it takes o and uctuates far from the wall (complete wetting). Notice that in the macroscopic limit, the interface lies always along the wall in this case. The four curves i) to iv) in Figure 2 represent the phase transition loci for various values of the parameters a > 0 and b > 0. For any value of the parameters and h above the curve, the system's interface is the straight line, while, for any value of these parameters below the curve, it is pinned. Clearly, since a and b are strictly positive, the phase transition must occur inside the shaded region , since when ^bd( ; h) = ^((1; 0); ), Jensen's inequality implies that the interface is always a straight line. The phenomenon of reentrance described in the introduction can be seen in Figures 2 and 3. Suppose a = b = 0:1 and h is slightly above 0:8 (this corresponds to the dashed line of the rst picture in Fig. 3). At very low temperature, the interface does not touch the wall; if we increase the temperature, then there is a rst transition and the interface becomes tied to the wall; if we increase further the temperature, then a second transition takes place and the interface is again the straight line; nally, at T = Tc, the system becomes disordered. In fact for slightly dierent values of a and b, there can even be one more transition, as shown in the second picture of Figure 3.

Part II: Interface, concentration of measure

In this part we prove new concentration results for the phase separation line and give an expression for the probability of a coarse{grained description of the phase


16

separation line in terms of the surface tension. This allows us to make the connection with the macroscopic variational problem discussed in part I. Then we discuss the typical phase separation lines at a macroscopic scale.

5 Concentration properties By duality, properties of interfaces at temperatures below Tc are related to properties of the random{line representation of the two{point function of the Ising model at temperatures above Tc. The results of this section are given for the two{point function and are valid for all temperatures above the critical temperature Tc. Our concentration results are based on the sharp triangle inequality, which allows us to improve Propositions 6.1 and 6.2 of [PV1]. The random{line representation for the two{point function h(t)(t0)iL is the formula X qL () : (5.1) h(t)(t0 )iL = :t!t0

On the set of all open contours , such that = ft; t0 g, qL () de nes a measure whose total mass is h(t)(t0)iL . The same is true for the similar representations of h(t)(t0)i( ) or h(t)(t0)iIL . It is therefore important to have good upper and lower bounds for these quantities. We recall some basic results. Proposition 5.1 is proven in [MW] and the last part in [PV1]; Proposition 5.2 is proven in [PV1]. We set := fx 2 IL : x(2) = ?1=2 g and L := \ L.

Proposition 5.1 Let < c. 1. There exists K such that

^(y ? x)g : h(x)(y)i K expkf? x ? yk1=2

(5.2)

2. Let ^(1; 0) = ^bd and x; y 2 . Then there exists K 0 such that

^bd (y ? x)g : h(x)(y)iIL K 0 expf? kx ? yk3=2

(5.3)

3. Let ^(1; 0) > ^bd and x; y 2 . Then there exists K 00 > 0 such that

h(x)(y)iIL K 00 expf?^bd kx ? ykg :

(5.4)

Proposition 5.2 Let < c and 0 < h < 1. 1. Let x, y be two dierent points of ZZ2 . Then

h(x)(y)i =

X

:x!y

q() expf?^(y ? x)g :

(5.5)


17

2. Let x, y be two dierent points of L . Then

h(x)(y)iL h(x)(y)iIL =

X

:x!y

qIL () expf?^bd ky ? x)kg : (5.6)

3. Let x, y be two dierent points of L. Then X

:x!y

qL (; ; h) h(x)(y)i( ) :

(5.7)

\E ( )=;

4. Let x, y, z be three dierent points of L. Then X

qL ()

X

qL ()

X

qL () :

(5.8)

Remark: Statement 4. of Proposition 5.2 can be written as X qL () h(x)(y)iL h(y)(z)iL :

(5.9)

:x!z y3

:x!y

:y!z

:x!z y3

There are variants of 4.. Let : x ! z and y 2 ; let 1 be the part of from x to y and 2 be the part of from y to z. Then X

or

:x!z ;y3 1 \E ( )=;

qL (; ; h) h(x)(y)iL ( ; 1) h(y)(z)iL ( ; h) :

(5.10)

qL (; ; h) h(x)(y)iL ( ; 1) h(y)(z)iL ( ; 1) :

(5.11)

X :x!z ;y3 1 \E ( )=; 2 \E ( )=;

We prove Propositions 5.3 to 5.5, which state concentration properties of the random lines contributing to the two-point function. Given x, y 2 ZZ2 and > 0 we set

S (x; y; ) := ft 2 ZZ2 : kt ? xk + kt ? yk ky ? xk + g ; (5.12) and @ S (x; y; ) is the set of z 2 ZZ2nS (x; y; ) such that there exists an edge hz; z0 i with z0 2 S (x; y; ). Proposition 5.3 Let x, y 2 L and > 0. Let S1 := S (x; y; ). Then ( is de ned in (2.20) and (2.21)) X

: x7!y 6E (S1 ) \E (S1 )\E (L )=;

+

X

z1 ;z2 2L

qL () j@ S1 j expf?g expf?^(y ? x)g

expf?^(z1 ? x)g expf?^bd (z2 ? z1 )g expf?^(y ? z2 )g :

(5.13)


18

Proof. Let s 7! (s) be a parameterization of the open contour from x to y. Let s1 be the rst time (if any) such that (s1) 2 L and s2 the last time such that (s2) 2 L . X X X qL () (5.14) qL () t2@ S1 :x!y ;3t \E (L )=;

: x7!y 6E (S1 ) \E (S1 )\E (L )=;

+

X

X

z1 ;z2 2 L

:x!y (s1 )=z1 ;(s2 )=z2

qL () :

Using Proposition 5.2, the remark following it and the sharp triangle inequality, we get X X X qL () h(x)(t)iL ( ; 1) h(t)(y)iL ( ; 1) (5.15) t2@ S1

t2@ S1 :x!y ;3t \E (L )=;

X t2@ S1

expf?(^ (t ? x) + ^(y ? t))g

X

expf?(^ (y ? x) + )g t2@ S1 j@ S1 j expf?g expf?^(y ? x)g ; and

X

X

z1 ;z2 2L

:x!y (s1 )=z1 ;(s2 )=z2

qL ()

X z1 ;z2 2L

expf?^(z1 ? x)g

expf?^bd (z2 ? z1 )g expf?^(y ? z2)g :

Remarks. 1. If h 1, then the statement (5.13) simpli es, X qL () j@ S1 j expf?g expf?^(y ? x)g : :x7!y 6E (S1 )

(5.16)

ut

(5.17)

2. At the thermodynamical limit we can improve (5.17) using Proposition 5.1. X q() K1 j@ S1 jkx ? yk1=2 expf?gh(x)(y)i : (5.18) :x7!y 6E (S1 )

The replacement of expf?^(y ? x)g by h(x)(y)i is signi cant since h(x)(y)i is the total mass of the measure de ned by q () on the set of all with = fx; yg. 3. If 1 < h < hc , then the sharp triangle inequality applied two times to (5.13) gives X qL () expf?^(y ? x)g j@ S1 j expf?g (5.19) : x7!y 6E (S1 ) \E (S1 )\E (L )=;

Reentrant Pinning Transition +

19

X z1 ;z2 2L

expf?(kz1 ? xk + kz2 ? z1 k + ky ? z2 k ? kx ? yk)g :

Proposition 5.4 Let x, y 2 L with x(2) = y(2) = ?1=2, and > 0. Let S2 := S (x; y; ). If ^(1; 0) = ^bd , that is h hc , then X qL () O(kx ? yk + )j@ S2 j expf?g expf?^(y ? x)g : (5.20) :x7!y 6E (S2 )

Proof. If ^(1; 0) = ^bd and u, v 2 IL with u(2) = v(2) = ?1=2, then ^bd ku ? vk = ^(u ? v) : (5.21) Let : x 7! y, with 3 t, t 2 @ S2 . We consider as a parameterized curve, s 7! (s), from x to y. We set t = (s); we denote by s1 the last time before s such that (s1) 2 L; we denote by s2 the rst time after s such that (s2) 2 L. We have

X

qL ()

:x!y

6E (S2 )

X X t2@ S2 :x!y

X

qL ()

3t

X

X

t2@ S2 u;v2L

(5.22)

:x!y ;t2 (s1 )=u;(s2 )=v

qL () :

Using (5.21), Proposition 5.2, GKS inequalities and the sharp triangle inequality, we get X X X qL h(x)(u)iIL h(u)(t)ih(t)(v)ih(v)(y)iIL u;v :x!y

3u;t;v

u;v

X u;v

X u;v

expf?^(u ? x) ? ^(t ? u)g expf?^(v ? t) ? ^(y ? v)g expf?^(t ? x)g expf?(kx ? uk + ku ? tk ? kx ? tk)g

expf?^(y ? t)g expf?(ky ? vk + kv ? tk ? ky ? tk)g O(kx ? yk + ) expf?(^ (t ? x) + ^(y ? t))g : Then the proof is as in (5.15).

ut

In the case of partial wetting, i.e. ^(1; 0) > ^bd (1; 0), the previous proposition can be improved to re ect the fact that the contours stick to the wall, even microscopically.

Proposition 5.5 Suppose h > hc (partial wetting for the dual model). Let x, y 2 L with x(2) = y(2) = ?1=2, x(1) < y(1), and let i > 0, i = 1; 2. Let S3 := ft 2 L : x(1) ? 1 t(1) y(1) + 1 ; ?1=2 t(2) 2 g : (5.23) Then, there exists a constant C ( ) > 0 such that X

:x!y 6E (S3 )

qL () j@ S3 j expf?21 ^bd g + j2j expf?C2g expf?^bd ky ? xkg : (5.24)


20

Proof. Let @ S3 := ft 2 S3 : t(2) = 2 or t(1) = x(1) ? 1 or t(1) = y(1) + 1 g; we can write

X :x!y 6E (S3 )

qL ()

X

X

qL ()

t2@ S3 :x!y 3t

X

X

t2@ S3 :x!y t(2) 0, 0 < a < 1, 0 < b < 1 and W = W ( ; h) be the

minimum of the functional W. Then there exists C > 0 and L0 = L0 ( ; h) such that, for all L L0 , Z ab(L) expf?CW Lg Z ?(L) : (6.1)

L Remark. The dual statement of Proposition 6.1 is exp f? W Lg L L h(tl )(tr )iL LC :

Proof. We can write, using Proposition 2.3, Z ab(L) = Z ?(L)

X

: =ftLl ;tLr g

(6.2)

qL () :

(6.3)

Let C be the simple recti able curve in Q which realizes the minimum of the variational problem (or one of the minima in case of degeneracy). Let K1 > 0, which will be chosen large enough below. We rst consider the case C = D. Let uLl and uLr be the points of L with uLl(1) = tLl(1) + [K1 log L], uLr(1) = tLr (1) ? [K1 log L], which are closest to the straight line from tLl to tLr . We need the following result

Lemma 6.1 Let B be a rectangular box in L, and x, y two points on its boundary. Let d > 0 and u, v be two points in B such that u and v are closest to the straight line from x to y and kx ? uk = ky ? vk = d. If the distance of B to L is larger than C 0 log L, with C 0 > 2, then X qL () expf?O(log L)g :x!y E (B)

X :u!v E (B)

q () :

(6.4)

Proof. Following the proof of Proposition 6.1 of [PV1], from (6.38) to (6.42), we get

X :x!y E (B)

qL () expf?O(log L)g

X :u!v E (B)

qL () :

(6.5)

Since the distance of B to is larger than C 0 log L and C 0 > 2, then it follows from point 4. of Lemma 5.3 in [PV1] that there exists a constant C~ > 0 such that for L large enough and all E (B) ~ () : qL () qIL () Cq (6.6)

Reentrant Pinning Transition tLl

22

uLl

S1 uLr

B

tLr

L

Figure 4: The ellipse S1 ; the box B is the whole box minus the bottom strip.

ut

This proves the lemma.

Let C 0 > 2. We introduce B := ft 2 L : t(2) > [C 0 log L]g. Let S1 be the elliptical set of Proposition 5.3 with x = uLl, y = uLr, = O(K1 log L) so that E (S1) E (B) (see Fig. 4). From Lemma 6.1, applied to the box B with x = tLl , y = tLr , u = uLl and v = vlL, and the second remark following Proposition 5.3, we get Z ab (L) X q () (6.7) L Z ?(L) :tL !tL r l E (B)

expf?O(log L)g expf?O(log L)g

X L :uL l !ur E (S1 )

q ()

X

:uLl !uLr

q() ?

X L :uL l !ur 6E (S1 )

q()

expf?O(log L)gh(uLl)(uLr)i 1 ? K1 j@ S1 j kuLl ? uLrk1=2 expf?g L L expf?^(tl ? tr )g ;

LC for some positive constant C , by taking K1 large enough. We now consider the case C = W . Since h > 0, the angle Y satis es 0 < Y < =2. Denote by w1L and w2L the two points on L which are closest to the corners of the polygonal line W scaled by L. We de ne three rectangular boxes (see Fig. 5) B1 = ft 2 L : t(1) w1L(1) ; t(2) > [C 0 log L]g ; (6.8) L L B2 = ft 2 L : w1 (1) t(1) w2 (1)g ; (6.9) B3 = ft 2 L : w2L(1) t(1) ; t(2) > [C 0 log L]g : (6.10) Moreover let r1L, resp. r2L, be the point of L closest to the straight line through tLl and w1L, resp. tLr and w2L, such that r1L(2) = [C 0 log L] + 1=2, resp. r2L(2) = [C 0 log L] + 1=2. Let 1 , resp. 2 , be a shortest open contour from r1L to w1L, resp. from r2L to w2L. We de ne A as the set A of open contours = 1 [ 1 [ 2 [ 2 [ 3 such that


23

B1 tLl

B2

B3

uLl

S1 uLr

tLr

S3 vlL

vrL S2 r1L r2L L w1L w10 L w20 Lw2L Figure 5: The three boxes Bi , i = 1; : : : ; 3 and their elliptical subsets; the two bold segments represent the two shortest contours i , i = 1; 2.

i E (Bi), i = 1; : : : ; 3; 1 : tLl ! r1L; 2 : w1L ! w2L; 3 : r2L ! tLr.

We can write

Z ab (L) Z ?(L)

X q ()

2A

L

expf?O(log L)g

Y

X

i i E Bi ): i as above (

qL (i) :

(6.11)

We apply Lemma 6.1 to the sum over 1 with B = B1, x = tLl, y = r1L, d = K1 log L; we denote by uLl and vlL the corresponding points u and v. We apply the same lemma to the sum over 3 with B = B2, x = r2L, y = tLr, d = K1 log L; we denote by uLr and vrL the corresponding points u and v. The sum over 2 is taken care of by the following

Lemma 6.2 Let w10 L := w1L + ([K1 log L]; 0), w20 L := w2L ? ([K1 log L]; 0). Then X X qIL () : (6.12) qL () expf?O(log L)g :w1L !w2L E (B2 )

:w10 L !w20 L E (B2 )

Proof. Same proof of that of Proposition 6.2 in [PV1].

ut

We nally introduce three elliptical sets. S1 is constructed as in Proposition 5.3 with x = uLl, y = vlL and = O(K1 log L) such that E (S1) E (B1 ); S3 is


24

constructed as in Proposition 5.3 with x = vrL, y = uLr and = O(K1 log L) such that E (S3 ) E (B3); S2 is constructed as in Proposition 5.4 with x = w10 L, y = w20 L and = [K1 log L] (and therefore E (S2) E (B2 )). Applying Propositions 5.3 and 5.4 as before gives the conclusion,

Z ab (L) expf?LW g Z ?(L) LC for some positive constant C .

(6.13)

ut

We prove using the above proposition, that the surface tension of a (very) coarsegrained version of the phase separation line cannot be too large compared to W . Let be the open contour. We construct a polygonal line approximation P := P () of . Let s 7! (s) be a unit-speed parameterization of . If (s)(2) > ?1=2 for all s, then let P be the straight line from tLl to tLr. Otherwise, let s1 be the rst time such that (s)(2) = ?1=2 and s2 the last time such that (s)(2) = ?1=2; we write wîL = (si), i = 1; 2. We also introduce [P ] := f! : P ((!)) = Pg. By construction, if s < s1 or s > s2 then (s)(2) 6= ?1=2. We can therefore apply Proposition 5.2 to estimate the probability of the event [P ],

PLab [ [P ] ] expf?W(P )g :

(6.14)

Proposition 6.2 Let > c, h > 0, 0 < a < 1, 0 < b < 1. Then there exists L0 = L0 ( ; h) such that, for all L L0 and T > 0, PLab[ fW(P ()) W L + T g ] expf?T + O(log L)g : (6.15) Proof. Let I (T ) := f E (L) : = ftLl ; tLrg ; W(P ()) WL + T g :

(6.16)

Then, from Propositions 6.1, 2.3 and (6.14), ? L) X PL[ fW(P ()) W L + T g ] = ZZab( q () (L) 2I (T ) L expfW L + O(log L)g

expfW L + O(log L)g

expfW L + O(log L)g

(6.17) X 2I (T )

qL ()

X

X

P: : P )W L+T P ()=P

W(

X

P: W(P )W L+T

qL ()

expf?W(P )g

expf?T + O(log L)g :

(The number of dierent coarse{grained polygonal lines is bounded by O(L2).) ut


25

7 Typical con gurations The main result of the paper is a statement about concentration properties of the probability of the phase separation line. An immediate consequence of Theorem 7.1 is that at a suitable scale, when L ! 1, the phase separation line de nes a non{ random object, the interface, which is characterized as the solution of the variational problem discussed in section 3, that is, the interface in Q is either the straight line D or the broken line W . We obtain an essentially optimal description of the location of the interface up to the scale of normal uctuations of the phase separation line.

7.1 Main result The weight of a separation line in L, going from tLl to tLr, is given by qL (; ; h) = qL (; ; h). These weights de ne a measure on the the set of the phase separation lines, such that the total mass is X

E (L ): =ftLl ;tLr g

Consequently

qL (; ; h ) = h(tLl)(tLr)iL ( ; h) : PLab[] = h(tLq)L((t)L )i : l

r

L

(7.1)

(7.2)

Let D and W be the curves in Q introduced in section 3. We set

IiL := f x 2 L : kx ? wiLk (ML log L)1=2 g ; i = 1; 2 ;

(7.3)

and

L := M log L : (7.4) We de ne two sets of contours. The set TD contains all E (L) such that a1 : = ftLl; tLrg; a2 : is inside S (tLl ; tLr; L ).

The set TW contains all E (L), considered as parameterized curves s 7! (s), such that

b1 : b2 : b3 : b4 : b5 :

= ftLl; tLrg, (0) := tLl ; 9s1 such that (s1) 2 I1L and for all s < s1 , (s) \ L = ;; 1 := f(s) : s s1g is inside S (tLl ; (s1); L); 9s2 such that (s2) 2 I2L and for all s2 < s, (s) \ L = ;; 3 := f(s) : s2 sg is inside S ((s2 ); tLr; L);


26

b6 : 2 := f(s) : s1 s s2 g is inside fx 2 L : x(2) L ; (s1)(1) ? L x(1) (s2)(1) + Lg :

Theorem 7.1 Let > c, h > 0, 0 < a < 1, 0 < b < 1. There exist M > 0 and L0 = L0 (h; ; M ) such that, for all L L0, the following statements are true. 1. Suppose that the solution of the variational problem in Q is the curve D. Then

PLab[TD ] 1 ? L?O(M ) :

(7.5)

2. Suppose that the solution of the variational problem in Q is the curve W . Then

PLab[TW ] 1 ? L?O(M ) :

(7.6)

3. Suppose that the solution of the variational problem in Q is the either the curve D or the curve W . Then

PLab[TD [ TW ] 1 ? L?O(M ) :

(7.7)

Comment: The results of Theorem 7.1 are optimal in the following sense: At a ner

scale we do not expect the phase separation line to converge to some non{random set, but rather to some random process. It is known that uctuations of a phase separation line of length O(L), which is not in contact with the wall, are O(L1=2) (see [Hi] and [DH]). On the other hand, if the phase separation line is attracted by the wall on a length O(L), then we expect that its excursions away from the wall have a size typically bounded by O(log L).

Proof. 1. Suppose that the minimum of the variational problem is given by D, W(D) = W.

Let W be the minimum of the functional over all simple curves in Q, with end{ points A and B , and which touch the wall wQ. By hypothesis there exists > 0 with W = W + . We set S1 := S (tLl ; tLr ; L); for L large enough S1 \ L = ; since a > 0 and b > 0. We apply Proposition 6.1. We have X PLab [f 62 TD g] = h(tL )1(tL)i qL() (7.8) l r L 62TD X LC expfWLg qL () : 62TD

We apply Proposition 5.3.

X q ()

+

X z1 ;z2 2L

62TD

L

j@ S1 j expf?g expf?^(tLr ? tLl )g

expf?^(z1 ? tLl)g expf?^bd (z2 ? z1 )g expf?^(tLr ? z2 )g :

(7.9)


27

We can bound above the last sum by O(L2) expf?LW g. Therefore C +1 PLab[f 62 TD g] O(LLM ) + O(LC +2) expf?Lg : (7.10) This proves the rst statement. 2. Suppose that the minimum of the variational problem is given by W , W(W ) = W. Then there exists > 0 such that W(D) = W + . We estimate PLab [f 62 TW g] in several steps. Notice that condition b1 is always satis ed. 1: The probability that condition b2 is satis ed, but not b3 , can be estimated by Proposition 5.3; it is smaller than O(LC +1)=LM . 2: The probability that condition b4 is satis ed, but not b5 , is estimated in the same way; it is smaller than O(LC +1)=LM . 3: The probability that conditions b2 and b4 are satis ed, but not b6 , can be estimated by Proposition 5.5; it is smaller than L?O(M ) . 4: We estimate the probability that condition b2 is not satis ed. The case with condition b5 is similar. If does not intersect L , then this probability is smaller than O(LC ) expf?Lg, since W(D) = W + . Suppose that there exist s1 and s2, with (si) 2 L, (s) \ L = ; for all s < s1 and (s) \ L = ; for all s2 < s. Let pLi := (si), i = 1; 2. Under these conditions, b2 is not satis ed if and only if pL1 62 I1L. Let C (pL1 ; pL2 ) be the polygonal curve from tLl to pL1 , then from pL1 to pL2 and nally from pL2 to tLr . Then the probability of this event is bounded above by X

X

L pL 1 2L : p2 2L L 2 6 I pL 1 1

expf?W(C (pL1 ; pL2 ))g

(7.11)

O(L2) minfexpf?W(C (pL1 ; pL2 ))g j pL1 2 LnI1L ; pL2 2 Lg : Suppose that C denotes the polygonal line giving the minimum; scaled by 1=L we get a polygonal line in Q, denoted by C , from A to some point P1, then from P1 to P2 and nally from P2 to B . Let be the angle between the straight line from A to P1 with the wall. We have W(C ) = LW(C ) L(g ( ; a) + g (Y ; b)) : (7.12) By hypothesis j ? Y j L11=2 O((M log L)1=2 ) : (7.13) Therefore (use a Taylor expansion of g around Y and the monotonicity of g(; x) on [0; Y ], respectively [Y ; =2]) there exists a positive constant such that M log L W(C ) g (Y ; a) + g (Y ; b) + (7.14) L = W + M Llog L :

We conclude that the probability, that condition b2 is not satis ed, is bounded above by O(LC +2)=LM . If M is large enough, the second statement of the theorem is true. 3. The proof of the third statement of the theorem is similar. ut


28

7.2 Finite size eects for the correlation length We show that we get nite size eects for the two{point function when h > hc , where hc ( ) := (hw ( )). In this paper we always assume that 0 < h < 1, so that we also have 0 < h < 1. Notice that 0 < hw ( ) < 1 for any 0 < < c; consequently 1 < hc ( ) < 1 for any c < < 1. By duality we can interpret Proposition 6.1 and Theorem 7.1 at temperatures above Tc. The fact that tLl and tlr are points on the boundary of L is not important for the dual model above the critical temperature, as one can check easily from the proofs of sections 6 and 7. Let < c and 0L be the subset of ZZ2 obtained by translating L by (0; ?L=2), that is, 0L := f t 2 ZZ2 : t + (0; L=2) 2 L g : (7.15) We consider the Ising model with free b.c. on 0L with coupling constants 0 0 J (t; t ) := h > 0 if t(2) = t (2) = ?1=2 ? L=2, (7.16) 1 otherwise. The corresponding Gibbs state is denoted by h i0L or by h i0L ( ; h). When L ! 1 the states h i0L ( ; h) converge to the unique in nite Gibbs state h i( ) of the model with coupling constant 1, independently of the value of h , since < c. The (horizontal) correlation length ( ) is therefore independent of h and is given by the formula 1 log h (t)(t + (L; 0)) i( ) : ( ) := ? Llim (7.17) !1 L Theorem 7.1, as well as Proposition 6.1, show that in general we do not get the same result if we take the thermodynamical limit and the limit in (7.17) together. Indeed, we can nd h and tL such that the distance of tL to the boundary of the box 0L is O(L) and 1 log h (t )(t + (L; 0)) i 0 ( ; h ) 6= ( ) ; ? Llim (7.18) L L L !1 L because in the random{line representation of the two{point correlation function the random lines are concentrated near a part of the boundary of the box 0L. Borrowing the terminology of [SML] about the long{range order, we can say that there is no equivalence in general between the \short" correlation length ( ) and a \long" correlation length like in (7.18). Proposition 2.2 states that this equivalence holds when h = h = 1, the correlation length ( ) being equal to the surface tension of an horizontal interface of the dual model. The reason for the validity of Proposition 2.2 can be formulated in physical terms: the dual model is in the complete wetting regime. It is interesting to observe that although the only quantities involved are 2-point functions and the system is ferromagnetic, the system exhibits reentrance which shows that one should not always expect monotonicity in such cases.


29

8 Appendix: sharp triangle inequality The main property of the surface tension, which we use in this paper, is the sharp triangle inequality (STI). We recall some basic facts about the Wul shape and prove that STI is equivalent to the property that the curvature of the Wul shape is bounded above. This slightly extends the result of Ioe [I]. In particular we do not suppose that the surface tension is dierentiable. Our approach is geometrical.

8.1 Convex body and support function Let ^ : IR2 ! IR, x 7! ^(x), be a positively homogeneous convex function, which is strictly positive at x 6= 0. In this section hy; xi denotes the Euclidean scalar product of y 2 IR2 and x 2 IR2. The conjugate function ^ ,

^ (y) := sup fhy; xi ? ^(x)g ; x

(8.19)

is the indicator function of a convex set W IR2 de ned by ^, y 2 W ^ (y ) = 01 ifotherwise. (8.20) In Statistical Mechanics, when ^ is the surface tension, W is called the Wul shape and (8.23) the Wul construction. Because ^ is strictly positive at x 6= 0, the interior of W is non{empty (W is a convex body) and contains 0. The function ^ is the support function of W , ^(x) = sup hy; xi : (8.21) y 2W

It is a norm if and only if W = ?W . Given x, we de ne the half{space H (x),

H (x) := fy : hy; xi ^(x)g : We have the important relation,

W = fx 2 IR2 : hx; yi ^(y) ; 8y 6= 0g =

(8.22) \ y6=0

H (y) :

(8.23)

A pair of points (y; x) 2 IR2 IR2 is in duality (in the sense of Convexity Theory) if and only if

hy; xi = ^(x) + ^ (y) = ^(x) :

(8.24)

If (y; x) are in duality and x 6= 0, then y 2 @W , the boundary of W . Moreover, in such a case (y; x) are in duality for any positive scalar . In the following x^ is always a unit vector in IR2; there is at least one x 2 @W , which is in duality with x^ for any x^. The geometrical meaning of x^ is the following: there is a support plane for W at x normal to x^. By convention the pair (x ; x^) is in duality, so that hx ; xî = ^(^x). We may have y1 6= y2, such that (y1; x^) and (y2; x^) are in duality.


Lemma 8.1 1. Suppose that y1 and y2 are two dierent points of F (^x) := fy 2 IR2 : (y; x^) are in dualityg : Then y = y1 + (1 ? )y2 2 F (^x), for all 2 [0; 1]. 2. The set F (^x) is equal to the set of subdierentials of ^ at x^, @ ^(^x) := fy 2 IR2 : ^(z + x^) ^(^x) + hy; zi 8z 2 IR2 g : Proof. 1. follows from (8.24); 2. is proven e.g. in [PV2] section 4.1.

30

(8.25)

(8.26)

ut

The set F (^x) of Lemma 8.1 is a facet of W with (outward) normal x^ (0 belongs to interior of W ). Therefore, existence of a facet is equivalent to non{uniqueness of the subdierentials of ^ at x^ or to non dierentiability of ^. For a given y 2 @W we may have two dierent vectors x^1 and x^2 , such that (y; x^1) and (y; x^2) are in duality. This situation corresponds to the existence of a corner of W at y.

Lemma 8.2 There is a corner of W at y if and only if there exists a segment [^x1 ; x^2 ] := fx : x = x^1 + t(^x2 ? x^1 ) ; t 2 [0; 1]g, with x^1 6= x^2 , on which ^ is ane. Proof. Suppose that there is a corner at y. Then hy; x^1 + t(^x2 ? x^1 )i = (1 ? t)^ (^x1 ) + t^(^x2 ) ysup hy; x^1 + t(^x2 ? x^1)i 2W = ^((1 ? t)^x1 + tx^2 ) :

(8.27)

Since ^ is convex we have equality in (8.27). Suppose that ^ is ane on [^x1; x^2 ]. Let x1=2 := 1=2(^x1 + x^2 ). If y 2 @ ^(x1=2 ), then y 2 @ ^(^xk ), k = 1; 2. Indeed, for all z

^(z) ? ^(x1=2 ) ? hy; z ? x1=2 i 0 :

(8.28)

But, if ^ is ane on [^x1 ; x^2 ], then 2 1X 2 k=1f ^(^xk ) ? ^(x1=2 ) ? hy ; x^k ? x1=2 i g = 0 :

Therefore

^(^xk ) = ^(x1=2 ) + hy; x^k ? x1=2 i : From this it follows that y 2 @ ^(^xk ), ^(z) ^(x1=2 ) + hy; z ? x1=2 i = ^(xk ) + hy; z ? x^k i 8z ; which implies in our case that (y; x^k ) are in duality.

(8.29) (8.30) (8.31)

ut


31

8.2 Curvature Let W1 and W2 be two convex bodies; we say that @W1 is tangent to @W2 at x if W1 and W2 have a common support plane at x . We recall the notion of curvature of W at x . Let U be an open neighbourhood of x . Let Ti(x ; U ) be the family of discs D with the following properties 1. @ D is tangent to @W at x; 2. W \ U D \ U . We allow the degenerate cases where the disc is a single point or a half{plane. Consequently Ti(x ; U ) 6= ;. We denote by (D) the radius of the disc D and set

(x ; U ) := supf(D) : D 2 Ti (x ; U )g :

(8.32)

Clearly (x ; U1) (x; U2 ) if U1 U2 . The lower radius of curvature at x is de ned as (x ) := supf(x ; U ) : U open neighb. of x g : (8.33) Similarly, we introduce Ts (x; U ) 6= ; as the family of discs D with the following properties 1. @ D is tangent to @W at x; 2. W \ U D \ U . We set

(x ; U ) := inf f(D) : D 2 Ts(x ; U )g : The upper radius of curvature at x is de ned as (x) := inf f(x ; U ) : U open neighb. of x g :

(8.34) (8.35)

Given x ; y 2 @W , x 6= y, let C (x ; y ) be the circle of radius y , which is tangent to @W at x and goes through y; we suppose that C (x ; y ) intersects the interior of W . If y 2 U , then

(x ; U ) y (x ; U ) :

(8.36)

If (x ) := (x ) = (x), then the radius of curvature at x is (x ) and the curvature at x is (x ) := 1=(x ) (see Chapter 1 of [S]). From (8.36) we get (x ) = (x) () ylim !x y = (x ) :

(8.37)

Indeed, if limy!x y = , then for every " > 0 there exists a neighbourhood U such that for all y 2 U , jy ? j ". Therefore ? " (x ; U ) (x ; U ) + ". If x is a corner, then (x ; U ) = 0 for any open neighbourhood U 3 x and (x ; U ) is as small as we wish, provided U is small enough. Therefore (x ) = 0. However,


32

we may have (x ) = 0 when x is not a corner, as the following example shows. We de ne a convex body by its boundary,

f z (1)(t) := cos(t)j cos(t)j:6 ; z (2)(t) := sin(t)j sin(t)j:6 ; t 2 [0; 2] g :

(8.38)

There is a unique support line at every point of the boundary. At the four points (t = k=2, k = 0; : : : 3) it is elementary to verify that (x ) = 0.

Lemma 8.3 Let W be a convex compact body such that its lower radius of curvature

is bounded below uniformly by K0 > 0. Then, given < K0 , there is a circle

C (x ; ) W of radius , which is tangent to @W at x for any x .

Proof. Since the lower radius of curvature is positive, there is no corner. Consequently, for any y 2 @W there is unique y^ in duality with y. The hypothesis also implies that at every x 2 @W there is a disk D(x) with the properties: the radius (D(x )) of D(x ) is non-zero, D(x ) W and @ D(x ) is tangent to @W

at x. Since W is convex, the convex envelope of all these discs is a subset of W . Therefore, since W is compact, we can nd > 0, such that (D(x)) for any x . Let x 2 @W and y^ be given. Let D(x; y^) H (^x) \ H (^y) be the largest disc, which is tangent to @H (^x) at x . If x^ = y^, then the radius r(x ; y^) of D(x ; y^) is in nite, otherwise it is nite. Since ^( ) is continuous, r(x ; y^) is a continuous function of y^ at any y^ 6= x^. We set r(x ) := inf r(x; y^) : (8.39) y^ Let fy^ng be a minimizing sequence, such that limn r(x; y^n) = r(x) and limn y^n =: y^. There are two cases: y^ = x^ and y^ 6= x^. If y^ = x^, then r(x ) K0 . Suppose the converse, r(x) < K0 . Then, for any n such that r(x; y^n) < K0, we can nd a disc Dn and a neighbourhood Un of x , such that Dn is tangent to @W at x and

W \ Un Dn \ Un D(x; y^n) \ Un :

(8.40)

Let zn be the point of contact of D(x; y^n) with @H (^yn). Since W is convex, @W intersects @ D(x ; y^n) at some point tn belonging to the circle arc of @ D(x ; y^n) from x to zn . Since r(x ; y^n) < K0 and x^ = limn y^n, we also have limn zn = x and thus limn tn = x . But this contradicts (8.36). Thus r(x ) K0; for any < K0 there is a circle C (x ; ) of radius , which is tangent to @W at x 2 @W ; C (x ; ) W by (8.23). If y^ 6= x^, then r(x; y^) = r(x) and the disc D(x; y^) W by (8.23). Let r := inf x r(x); we claim that r K0 . Suppose the converse, r < K0 . Let fzn g be a minimizing sequence such that r(zn ) < K0 , limn r(zn ) = r and limn zn =: z . For every n there exists y^n such that r(zn ; y^n) = r(zn ) and D(zn ; y^n) is the largest disc in W , which is tangent to @W at zn . Since r < K0 , there exists y^ 6= z^, so that

r = r(z ; y^) :

(8.41)


33

Indeed, if D(z) W is a disc tangent to @W at z , then by convexity the convex envelope of D(z) and D(zn ; y^n) is a subset of W . If (D(z )) > r, then the discs D(zn ; y^n) are not the largest discs in W which are tangent to @W at zn , when n is suciently large. The existence of D(z ; y^) and the convexity of W imply the existence of an open set U , such that

@W \ U = @ D(z ; y^) \ U :

(8.42)

Indeed, D(z ; y^) is tangent to @W at z and also at some y in duality with y^; moreover, at any point x 2 @W there exists a disc of radius r contained in W , tangent to @W at x. But this contradicts r < K0. ut

8.3 Sharp Triangle Inequality

Proposition 8.1 Let W be a convex compact body and ^ be its support function. Then the following statements are equivalent. 1. The lower radius of curvature of @W is uniformly bounded below by K0 > 0. 2. There exists a positive constant K1 such that for any x^ and y^

hx ? y; xî K1 kx^ ? y^k2 :

(8.43)

3. There exists a positive constant K2 such that for any x, y 2 IR2

^(x) + ^(y) ? ^(x + y) K2(kxk + kyk ? kx + yk) :

(8.44)

Remarks: 1. Suppose that the curvature is bounded above everywhere by . Then

1. holds with K0 = 1=. 1. implies 2. with K1 = 1=2; this follows by modifying slightly the proof given below: if hx ? y; xî 4K1, then there exists v^ such that hx ? y; xî = 2K1kx^ ? v^k2 and hx^; yî hx^; vî. 2. implies 3. with K2 = 1=. 2. The validity of the sharp triangle inequality implies absence of corner for W , since it prevents ^ to be ane on segments [^x1 ; x^2] with x^1 6= x^2 . However, the example before Lemma 8.3 shows that the converse is not true.

Proof. We prove 1 ) 2. Let x ; y 2 @W , x 6= y and 0 < 2K1 < K0. The circle C (x ; 2K1) of radius 2K1 , center d, which is tangent to @W at x , is a subset of W (Lemma 8.3). If then for any y^ Suppose that

hx ? y; xî 2K1 ;

(8.45)

hx ? y; xî K1 =2kx^ ? y^k2 :

(8.46)

hx ? y; xî 2K1 :

(8.47)


34

We can nd z 2 C (x ; 2K1) such that, if v^ := (z ? d )=kz ? d k and ' is the angle between the unit vectors x^ and v^, then hx^; vî 0 and

hx ? y; xî = 2K1(1 ? cos ') = K1kx^ ? v^k2 :

(8.48)

We claim that

hx^; yî hx^; vî : (8.49) Suppose the converse. Let C (y; 2K1) W be the circle of radius 2K1, which is

tangent to @W at y. By hypothesis the line perpendicular to v^ at z and the support line at y perpendicular to y^ intersects at an interior point of H (^x); this implies that C (y; 2K1) 6 H (^x), which is in contradiction with W H (^x). Therefore and

kx^ ? y^k kx^ ? v^k

(8.50)

hx ? y; xî = K1 kx^ ? v^k2 K1kx^ ? y^k2 :

(8.51)

We prove 2 ) 1. Suppose that

hx ? y; xî K1 kx^ ? y^k2 :

(8.52)

Let C (x ; y ) be the circle of radius y , which is tangent to @W at x and goes through y; let c be its center and u^ := (y ? c)=ky ? c k. Assume furthermore that hx^; uî 0. Let v := x^ + u^ and v^ = v=kvk. Then y kx^ ? u^k2 = hx ? y; xî (8.53) 2 and hx ? y; vî = 0. Since @W is convex, there exists z \between" x and y such that z^ = v^ and kx^ ? z^k kx^ ? y^k : (8.54) On the other hand, kx^ ? u^k kx^ ? v^k + kv^ ? u^k (8.55) and kx^ ? v^k = kv^ ? u^k = kx^ ? z^k : (8.56) If x^ = y^, then y = 1; otherwise, using (8.53) to (8.56), 2y kx^ ? v^k2 2y kx^ ? u^k2 (8.57) = hx ? y; xî K1kx^ ? y^k2 K1kx^ ? v^k2 : Since this holds for any y in a neighbourhood of x , we have (x) K1 =2. We prove 2 ) 3. We set +y : (8.58) z := x + y ; z := (x + y) ; z^ := kxx + yk


35

We have, using (8.21) and (8.24),

^(x) + ^(y) ? ^(z) = hx ; xi + hy; yi ? hz ; zi = hx ? z ; xi + hy ? z ; yi = kxk hx ? z ; xî + kyk hy ? z ; yî :

(8.59)

By elementary trigonometry

kxk + kyk ? kzk = 21 kxk kx^ ? z^k2 + kyk ky^ ? z^k2 ;

(8.60)

so that (8.43) implies (8.44). We prove 3 ) 2. Let (x ; x^) and (y; y^) be given; we set z := x^ + y^. Using (8.60), kx^ ? z^k = ky^ ? z^k and kx^ ? y^k kx^ ? z^k + kz^ ? y^k we have

hx ? y; xî = hx ; xî + hy; yî ? hy; x^ + yî ^(^x) + ^(^y) ? ^(z) K2(kx^k + ky^k ? kzk) = K2kx^ ? z^k2 K42 kx^ ? y^k2 :

(8.61)

ut


36

References [A]

Abraham D.B.: Solvable model with a roughening transition for a planar Ising ferromagnet. Phys. Rev. Lett. 44, 1165{1168 (1980) [ABCP] Alberti G., Bellettini G., Cassandro M., Presutti E.: Surface tension in Ising systems with Kac potentials. J. Stat. Phys. 82, 743{796 (1996) [BLP1] Bricmont J., Lebowitz J.L., P ster C.-E.: On the local structure of the phase separation line in the two-dimensional Ising system. J. Stat. Phys. 26, 313-332 (1981) [BLP2] Bricmont J., Lebowitz J.L., P ster C.-E.: On the surface tension of lattice systems. Annals of the New York Academy of Sciences 337, 214{223 (1980) [DH] Dobrushin, R.L., Hryniv O.: Fluctuations of the phase boundary in the 2D Ising ferromagnet. Preprint ESI 355 (1996), to appear in Commun. Math. Phys. [FP1] Frohlich J., P ster C.-E.: Semi{in nite Ising model I. Thermodynamic functions and phase diagram in absence of magnetic eld. Commun. Math. Phys. 109, 493{523 (1987) [FP2] Frohlich J., P ster C.-E.: Semi{in nite Ising model II. The wetting and layering transitions. Commun. Math. Phys. 112, 51{74 (1987) [G] Gallavotti G.: The phase separation line in the two{dimensional Ising model. Commun. Math. Phys. 27, 103{136 (1972) [Hi] Higuchi Y.: On some limit theorems related to the phase separation line in the two{dimensional Ising model. Z. Wahrsch. verw. Geb. 50, 287-315 (1979) [I] Ioe D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74, 411{432 (1994) [LP] Lebowitz J.L., P ster C.-E.: Surface tension and phase coexistence. Phys. Rev. Lett. 46, 1031{1033 (1981) [MW] McCoy B.M., Wu T.T.: The Two{dimensional Ising Model. Harvard University Press, Cambridge, Massachusetts (1973) [Pa] Patrick A.: The in uence of boundary conditions on Solid-On-Solid models. Preprint DIAS-STP-96-02 (1996) [P] P ster C.-E.: Interface and surface tension in Ising model. In Scaling and self{similarity in physics. ed. J. Frohlich, Birkhauser, Basel, pp. 139{161 (1983) [PU] Patrick A., Upton P.J.: Surface phase transitions in two dimensions: metastability and re{entrance. Preprint DIAS-STP-96-08 (1996)

Reentrant Pinning Transition [PV1] [PV2] [S] [SML] [T]

37

P ster C.-E., Velenik Y.: Large deviations and continuum limit in the 2D Ising model. To appear in Prob. Th. Rel. Fields. P ster C.-E., Velenik Y.: Mathematical theory of the wetting phenomenon in the 2D Ising model. Helv. Phys. Acta 69, 949{973 (1996) Spivak M.: A comprehensive introduction to dierential geometry, vol 2, second edition, Publish or Perish, Berkeley (1979) Schultz T.D., Mattis D.C., Lieb E.H.: Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36, 856{871 (1964) Talagrand M.: A new look at independence. Ann. Prob. 24, 1-34 (1996)