Nonperturbative analysis of a model of random surfaces

Nuclear Physics B251[FS13] (1985) 553-564 '~; North-Holland Publishing Company

NONPERTURBATIVE ANALYSIS OF A MODEL OF RANDOM SURFACES D.B. ABRAHAM Department of Theoretical Chemistry. Oxford Universio'. Oxford OXI 3TG, UK

J.T. CHAYES* and L. CHAYES* Department of Mathematics and Physics. Hart:ard Unit,er~itv, Cambridge, MA 02138, USA Rcceived 13 August 1984

The truncated pair function for a particular system of random surfaces on Z '1 is analyzed for all temperatures strictly below the critical temperature. Throughout this regime, it is shown that (i) the pair function has Ornstein-Zernike behavior, (ii) the mass. or inverse correlation length, is analytic. (iii) the number of random surfaces has the expected asymptotic scaling, and (iv) the surfaces do not undergo a breathing transition. The results are established by using a random-surfacc Schwingcr-Dyson equation which should be applicable in the nonperturbativc analysis of other models of random paths and surfaces.

I. Introduction In this n o t e , we c h a r a c t e r i z e the n o n c r i t i c a l b e h a v i o r o f a m o d e l o f r a n d o m s u r f a c e p a i r c o r r e l a t i o n s i n t r o d u c e d a n d s t u d i e d in [1,2]. T h e p a i r f u n c t i o n s are of the form e -" alsl ,

(1.1)

w h e r e / 3 - go-2 c o r r e s p o n d s to the inverse t e m p e r a t u r e or s q u a r e o f the i n v e r s e c o u p l i n g , ~ is a p r e p r e s c r i b e d set of lattice s u r f a c e s a n d

ISI

d e n o t e s the a r e a o f the

s u r f a c e S ~ S. T h e class o f surfaces, ~, w h i c h we s t u d y are s e l f - a v o i d i n g " t u b e s " which obey a solid-on-solid (SOS) constraint along a preferred direction. The pair correlation

so d e f i n e d is e x p e c t e d to be a g o o d a p p r o x i m a t i o n

to the g l u e b a l l

* J.T.C. and L.C. hold National Science Foundation Mathematical Sciences Postdoctoral Research Fellowships. Research supported in part by the National Science Foundation under grant no. PHY-8203669. 553

554

D.B. Abraham et al. / Random surfaces

propagator in the strong coupling phase of Ising-type gauge theories, and by duality in d = 3, to the truncated pair function in the low-temperature phase of uniaxial ferromagnets. Mrdels of the form (1.1) have been considered by other authors [3-7], in the context of both walks and surfaces. Such models are expected to elucidate certain features of the stochastic geometry of interacting paths and surfaces which occur in graphical expansions of euclidean quantum field theories [8] and statistical mechanical systems [9]. For example, simple random walks, self-avoiding walks [6, 7] and branched polymers [6] may be written in the form (1.1). A planar random surface model of this form has been studied by Durhuus, FrOhlich and Jrnsson [3-5]. In [1,2] we established certain properties of the pair function (1.1) under the assumption that various correlation lengths of the system were distinct (a condition which was verified for large fl). These properties included Ornstein-Zernike (OZ) decay, analyticity of the mass, and power-law corrections to exponential growth of a quantity which counts the number of surfaces contributing to the pair function. It is also possible (see sect. 4) to prove absence of breathing of the surfaces under the same assumption. The purpose of this note is to show that the correlation lengths are distinct for all noncritical temperatures, and thereby establish these results up to the critical point. This essentially completes the analysis of the noncritical behavior of these surfaces. Previous results along these lines have been obtained only for the simplest systems, such as "stiff" walks in d = 2 and simple random walks, for which it is possible to derive closed-form expressions for the generating function (1.1). (It should be noted, however, that the arguments developed by Brydges and Spencer [7] may be used to prove OZ decay up to the critical point for self-repelling walks above their upper critical dimension [10].) The techniques we use are an extension of those developed in [2], in which a random surface Ornstein-Zernike (or renewal) equation was used to control the asymptotic behavior of the pair function. The new ingredient in our analysis is the incorporation of vertex factors into the renewal equation, which produces a random-surface Schwinger-Dyson equation. The number of vertex factors is controlled by thermodynamic properties of the surfaces, derived in [2]. Given some (rather nontriviai) modifications, this approach should be useful in establishing OZ behavior nonperturbatively in other models of the form (1.1), for both walks and surfaces. The organization of this paper is as follows. In sect. 2, we define the class of surfaces under consideration and, for completeness, review the relevant results from [2]. In sect. 3, the Schwinger-Dyson equation is derived and used to extend the results of [2] up to the critical point. In sect. 4, we consider the phenomenon of breathing, introduced in [1,2] to characterize the asymptotic scaling of the width of typical surfaces contributing to the pair function. It is shown that whenever the correlation lengths are distinct (and thus for all noncritical temperatures), the surfaces contributing to the pair function

D.B. Abraham et al. / Random surfaces

555

do not breathe. Since it is expected that the surfaces do breathe at the critical point, this indicates that the breathing of surfaces derives the destabilization of this system.

2. Definitions and basic properties In this section, we review the definitions and relevant properties of the randomsurface pair correlation analyzed in [1,2]. The reader is referred to [2] for more detailed descriptions, and for proofs of some of the results given below. For conceptual simplicity, attention is restricted to the three-dimensional cubic lattice. Let r ° be the elementary ring composed of four edges of the dual lattice ( l + ~)3 which lie in the plane x = ~ and surround the x-axis. Let rtt~_,.,m denote the translation of r ° by the lattice vector (L, a, b). We define S~t.:~,b) to be the class of surfaces, S, with boundary 0S = r ° u rt° .,~,1, such that the intersection of S with any of the planes x = 1. . . . . L is a single closed ring. We refer to these surfaces as SOS tubes. Our principal object of study is the pair correlation (or generating) function

Q,/.:..b)(fl) =

Z

e

(2.1)

-/~lsl

It will be convenient to consider an enlarged class of pair functions, specified by different boundary conditions. To this end, let (rt'lg = 0. 1.... ) denote all self-avoiding rings in the plane x = ½, such that (say) the upper leftmost plaquette of each r ~ is pierced by the x-axis. (This choice prevents overcounting translates of identical ~v rings.) The surfaces $~L:~,h) are those which obey the SOS restriction and have r"Ur¢~.... bl as their boundary. The corresponding correlation function will be denoted by Q~L:.,m- Finally, the master functions are defined by

OZ."= ~ Q~'t[:,.b,,

Qt. =--O~.~-

(2.2)

1

(2.3)

a,h

P R O P O S I T I O N 2.1

The mass, or inverse correlation length, M(,8) = lim 1.~;

['

- ~ logQ~'[(,8) ,

exists and is independent of tt and v. Furthermore, there is a critical temperature/~, 0 /~, and M(fl) = - oc for/3 //3, M ( B ) is concave, increasing and continuous in B. Finally, for each finite L, Q / . ~ e M~/S~I.

(2.4)

D.B. Abraham et al. / Random surfaces

556

More generally, there are fl-independent constants b ~" such that ~u

t. ~< b~'~e •

M([3)l.

(2.5)

Proof.

The existence and properties of M(fl) for O l_ have been established in [2] (see, in particular, propositions 2.1, 2.2, 2.6 and corollaries 2 and 3 to theorem 5.3). To demonstrate equality of the limits (2.3) independent of boundary conditions, observe that by explicit construction (2.6a) while L+ 2 Q00

~

~

~u e

- tqArea(r~+ A r e a ( r ~ ) + Per( r ~' ) + P e r ( r "

) -. 21

[]

(2.6b)

Another quantity of interest is the direct correlation function, C(t.: ~. b)(fl), defined by summing over the subclass of SOS tubes which have as their boundary r ° u r(°l . , o , b ) ' but which do not contain a translate of r ° in any of the intermediate planes, x = 1. . . . . L. It was shown ([2], proposition 2.3) that the corresponding master function, Ct., has a well-defined mass, which we denote by Mc(/3). The results derived in [2] were proved under the assumption that Mc(B ) > M(fl), a condition which was verified for/3 large enough. In sect. 3, we shall prove that Me(/3 ) > M(/3) for all 13 > ft. In order to do this, we shall need certain thermodynamic properties of the surfaces, which are given below. These properties may be derived by rewriting the pair functions (2.1) and (2.2) in the form

Q,L;0.0, = E r k ( L ) e

t,kL,

(2.7a)

k

I:1 ~ = Y" T ~ ( L ) e - '~k1,

(2.7b)

k

where F~(L) and P~'~(L) are the number of surfaces of the required type with exactly kL plaquettes. Clearly, for any finite L, there are only certain values of k for which F~'"(L) is nonzero. However, for all rational k > 4, there is always a sequence of lengths, ~ ' " ( k ) , tending to infinity, such that F ~ ( L ) ~ 0 for all L ~ ~'~(k). For such k, it was shown in [2] that F~(L) grows exponentially in L. The relevant properties of the growth constant are summarized in the following. PROPOSITION

2.2

For each rational k > 4,

(2.8)

~'(k)LE~'"(k)

exists and is independent of ~t and v. Furthermore, for rational k > 4, ~'(k) is

D.B. Abraham et al. / Random surfact:~

557

concave in k. In particular, this implies that ~'(k) extends to a concave, continuous function on all real k >/4. Finally, there are k-independent constants d ~ such that, for all finite L, T'~'"( L ) ~
(2.9)

In particular d °°= 1, so that T'~(L)- r'ff~(L)~< e ~~k~l. Proof. The existence and properties of ~'(k) are a consequence of propositions 5.1 and 5.2 of [2]. That the limit (2.8) is independent of boundary conditions follows from geometric estimates along the lines of eq. (2.6). [] It can be shown (theorem 5.3 of [2]) that, for fl >1fl, M(fl) and ~'(k) are related by a Legendre transform, i.e. - M(/3) = sup [~'(k ) - / 3 k ],

(2.10)

so that ~'(k) is identified as the surface entropy. In particular, this means that the inverse temperature, /3, and the cooering factor, k, are conjugate variables for the system. Moreover, combining the relation (2.10) with standard variational arguments, it is easy to prove the following elementary, but useful proposition. PROPOSITION 2.3

For each/3 >/~, there is a finite k(/3) which maximizes the r.h.s, of eq. (2.10). Remark. If ~'(k) contains a linear region, then the maximizer of proposition 2.3 is not unique. However, even in this case, for every/3 >/3 the largest value of k which maximizes the r.h.s, of eq. (2.10) is finite. We shall call this value k(/3). Indeed, it will follow from corollary 3 to theorem 3.3. that, for this model, k(/3) is always unique. 3. Separation of the masses

The principal result of this section is that the masses, M(fl) and Mc(fl), are strictly separated whenever fl > ft. As corollaries, we obtain OZ decay of Q~t.:~.m, analyticity of the mass M(fl), and power-law corrections to the exponential growth of l'k(L) for all noncritical temperatures. Suppose fl >/~, so that by proposition 2.3, k(fl) < oo. Then there are only a finite number of rings, r ° . . . . . r", with Per(r ~') ~
D.B. Abraham et uL / Random surfaces

55g P R O P O S I T I O N 3.1

Let fl >/3. Then for all ~t, u = 0, 1 . . . . . n(fl), MB(fl ) - tlim . ~ [ - L1 l°gB~'~ ]

(3.1)

exist and is independent of ~ and u. Moreover MB(fl ) > M ( f l ) .

(3.2)

Proof. Existence of the limit independent of b o u n d a r y conditions follows from s t a n d a r d subadditivity and geometric estimates. Thus we may restrict attention to B z. for the p r o o f of (3.2). Let X be the size of the smallest ring strictly larger than k(fl). Evidently,

Bt.(fl)~< E I - ~ ( L ) e "~1 k~,~

~XL

sup X~
el¢,k, ,,~k]~.+ ~

F'~(L)e a~/.

(3.3)

h~2,k

T h e second inequality follows from the observation that there are at most XL values of k ~ [ h, 2 ~, ) such that f ' k ( L ) 4: 0, and from using the bound l-'k( L ) ~< e ~tk }z o f p r o p o s i t i o n 2.2. Since, by propositions 2.2 and 2.3, ~ ( k ) - f l k is concave and m a x i m i z e d by k(fl)< ?~, it follows that the second term in (3.3) is exponentially smaller than the first, and that

-MB(fl)<~sup[~(k)-Bk]=~(h)-flX<-M(fl ).

[] (3.4)

T h e strategy of our proof is to control the asymptotic behavior of Q t, by using the larger mass of the direct correlation functions. In previous work [1,2], such a strategy was implemented by relating the single direct correlation function, C i,. to Q t_ in an equation of the convolution form (i.e., an O Z or renewal equation). This is derived as follows: observe that the contribution to Q L from surfaces which have no nodes (translates of r °) in the first N steps may be expressed as a product of C,v and Q t, u- S u m m i n g over N yields the Ornstein-Zernike equation. Here we have m a n y direct correlation functions, B 1 , ~, ~, = 0, 1 . . . . n(fl). Thus, in order to write a renewal equation for Q / , we must incorporate vertex factors into the convolution form. Indeed, the contribution to Q t. from surfaces which do not contain any of the rings r ° . . . . . r" until the N t h step is given by "¢"'# ~" ~ 0 wn{}, N - 1 V~,Cb~ " ~ 1 . ,0 N " T h e vertex term V ~'= e - t~p~r{,,~ in simply the weight of the intermediate ring. The ratio of the above sum to B x Q t _ ,, should be regarded as the vertex factor, V~,.

D.B. Abraham et al. / Random surfaces

559

LEMMA 3.2

Suppose /3>/3. Then the Laplace transform Q(z)=F, IQtZ L is analytic for Izl < e M~), has a simple pole at z = e M~a~ and has a meromorphic extension in the region Izl < e u"~a). Proof. Using the previous observation, we have I,

Q L = ~ L -t- E

~ ~°~-lV"~°-N,

(3.5)

N = I ~=0

with BoO" = oo~ = 8%. This is the Schwinger-Dyson equation for Q ~. More generally, we may write

t Qt. = ~

B x Q , . _ N,

(3.6)

N= 0 where Q L and B t, are (n + 1) × (n + 1 ) matrices with the elements (Q t,) ~'' = V"Q ~,~ l and ( B t ) "~= V~B~_I. By convention, we have set Q0 = ! and B0 = 0, so that eq. (3.6) is only valid for L >/- 1. The Laplace transform of eq. (3.6) is given by (3.7)

= 1 +

Observe that as Izl ~0,B'(z)~0, so that ~ - B A ( z ) has an inverse for z sufficiently small. Using Cramer's rule, we may express (Q^(z)) °° (and hence Q(z)) as an explicit function of the B(z). Although Q ( z ) is defined a priori only for IzL < e M~a~, by proposition 3.1 the expression for Q ( z ) in terms of the B(z) defines a meromorphic extension of Q ( z ) in the larger region [z I < e u.~t~. We now argue that Q ( z ) has a simple pole at z = e Mtt~. Consider the function l(x), restricted to x real. (Observe that Q0 = 1, so that Q l(0)= 1.) An elementary continuity argument shows that Q ~(x)$0 as x q'e M. Since Q ( z ) has a meromorphic extension in this region, we need only rule out the possibility of a multiple zero. This is done by using the a priori bound (2.4) of proposition 2.1, which implies that for x < eM - l ( x ) >/1 - x e

TM

(3.8)

a condition which would be violated if Q - l ( x ) had a multiple zero. [] Remark. At this point, it is conceptually straightforward to repeat the analysis of ref. [2], replacing the Ornstein-Zernike equation of [2] with the Schwinger-Dyson equation (3.6). However, the matrix form of the Schwinger-Dyson equation introduces unnecessary technical complications. Alternatively, one may derive a Schwinger-Dyson equation for the single direct correlation function C. This is the approach followed below.

560

D.B. Abraham et aL / Random surfaces

T H E O R E M 3.3

For all /3 >/~, Mc(/3 ) > M(/3).

Proof. Let /3>/3. Clearly Ms(/3)>/Mc(/3 ). If Mt~(/3)=Mc(/3 ), we are done. Otherwise, we derive a Schwinger-Dyson equation of the form (3.6) for the function C~!, obtained by summing all nodeless SOS surfaces with boundary r l U r(l/..... ~. Since MB(/3)> Mc(/3), we may repeat the analysis of lemma 3.2 to show that C l l ( z ) has a simple pole at z = e u,~/~) and a meromorphic extension in the region Izl < e u'(¢~). That the same is true for C ( z ) -= C°°(z) follows from upper and lower bounds analogous to eq. (2.6). Now Q ( z ) and C(z) are related by the Ornstein-Zernike equation discussed earlier. In the notation of lemma 3.2, this equation may be expressed as ( t ~ ' ( z ) ) ' ~ = 1 - 1 / ( Q ^ ( z ) ) '"'.

(3.9)

By lemma 3.2, eq. (3.9) shows that ((~^(z)) °°--, 1 as z ~ e M~. However, by the previous paragraph, e M~I/~ is a singularity of ((~'(z)) °°. Thus M(/3)--/: Me(/3), which means that M c ( f l ) > M(/3). [] Using theorems 2.4, 4.1, 5.5 and 5.6 of [2] this implies the following.

COROLLARIES

For all /3 >/3: 1. The master function Q L has pure exponential decay in the sense that there are constants •t(/3) and ~2(/3) such that

(3.10) 2. The pair function Q(t.:,,h~(fl) has the asymptotic behavior

[ (1)1

Q(L:.,m(fl) = (const) 1 e M~¢)"e I~"~+b2'/""l 1 + 0

o

(3.11)

In particular, this means that Q(t.;~,m has the Ornstein-Zernike form, and that the hitting distribution of endpoints in the final plane is normal. 3. The mass M(fl) is real analytic. 4. The surface counting function F k ( L ) has the asymptotic behavior

[ (1)]

I'k(L ) = (const) L 3 ~ e ~'k'L 1 + 0 ~ -

.

(3.12)

Remark. The generalization to higher dimensions gives the power-law correction L (d 1)/2 for Qtt.:o.b), and the factor L ,//2 for I'~(L).

D.B. Abraham et al. / Random surfaces

561

4. Absence of breathing In this section, we study quantities which measure the width of typical tubes contributing to the pair function. In particular, if r~ is the cross section of a tube of length 2L + 1 at its midplane, we would like to determine whether the perimeter of r~ scales with L as L --, oo. In a sense to be made precise below, if Per(rc) gets large as L ~ oo, the tubes are said to breathe; otherwise, they are said to be stable. Partial results on breathing for SOS tubes have been obtained in [1,2], where it was shown that the tubes are stable whenever fl is larger than the critical point of a related lower-dimensional model. In this section, we shall show that whenever M c ( f l ) > M(fl) (and thus for all fl >/~), the tubes are stable. Definition. Let r~ denote the central ring of a tube S ~ 5t21•+~;~.br We define 5 r ( h ) = PrObL{ Per(r~) > h )

I__L__ -- O21-1 E • a.b

_E

e-OlSlx{Per(r¢(S))>h},

(4.1)

SE~I21..I;a.h)

where X is the characteristic function of the surface event. If fl>~/3 and . q ( h ) l i m i n f r _ ~ . ~ t (h) is bounded below uniformly in h, the tubes are said to breathe. If ~q(h) tends to zero with h, the tubes are said to be stable. Remarks. 1. This differs slightly from the definition of breathing used in [1, 2], in that the latter applied only to tubes constrained to encircle the x-axis. 2. One could of course examine the above questions in the ensemble of tubes with the second endpoint fixed at (L,0,0). At least for this system, the results are identical. 3. It was pointed out to us by Michael Aizenman that if we consider a restricted breathing criterion, characterized by the parameter Probt.{Per(rc)= 4), then exponential decay of Q 1. alone implies absence of breathing. Indeed, Probt.{Per(r~) = 4} = e - 4 ° Q ~ , / O 2 j . + t which tends to a nonzero constant as L ~ oo whenever Qt. has pure exponential decay. The strategy of the general proof is the following. We first argue that a finite tube does not breathe. Next, we use separation of the masses to prove that, with large probability, a region about the midplane may be replaced by a finite tube. The width of a finite tube is characterized by

1 Y'~ e t~lSlx{qr~S, P e r ( r ( S ) ) > h } , ~L'"'b(h)- Q~ZL+I:,.t,) s~5,:,.,:,.h,

(4.2)

which is simply the probability that there is any ring along the tube with perimeter larger than h.

562

D.B. Abraham et al. / Random surfaces

PROPOSITION 4.1 Suppose f l > / ~ . Then for all finite (L,a,b), given any e > 0, there is an ho(L,a,b,e) suchthat 81.... h(h) < e i f h > h o. Proof. This follows from the fact that whenever fl > fi, Qt2L+l:~.bl is a convergent sum, and thus its tail can be made arbitrarily small. [] In order to replace a region about the midpoint with a finite tube, we begin by showing that, with large Probability, a node occurs " n e a r " the center. Consider the tubes contributing to Q2L~1" Take A < L, and let ffc ( A ) denote the event that a node occurs in the region ( L + ½) - A ~< x ~< L + ½, i.e. the region of length A to the left of the central plane. Similarly, let °3L(A) be the event that a node occurs in the corresponding region to the right of the central plane.

LEMMA 4.2 Suppose M c ( f l ) >

M(fl). Then

ProbL(gL_(A))=Probt.(~Tc(A))>~l_(const)e-IM,qlJ, MtB,IA

(4.3)

Proof. The probability that the event 9X_(A) does not occur may be written explicitly as 1

L-A

E Q2L+I N=0

2L. E

1

QN-1VOCR - N IvOQ2L}I

R,

(4.4)

R=I.,} 1

with the convention O - 1 = 1/V°. Now we use the exponential decay of Q2L+I given in corollary 1 to theorem 3.3 to bound the denominator of (4.4). The numerator is bounded by the a priori bound (eq. (2.4)) on Q, and by the analogous estimates for C. This gives an upper bound on (4.4) of the form (const)e IM~(,~) M(B)]A, which is the desired result. [] Next we consider the event .6K(A) = ~}L_(A) n N + ( A ) that a node occurs on both sides of the midplane. This event essentially allows us to replace the tube of length 2 L + 1 with one of length less than 2A + 1. However, since we do not know the (y, z) coordinates of the nodes, we have no guarantee that as L ---, ~ this shorter tube will remain of finite extent in the (y, z) directions. In order to examine this, let us enumerate all positions of nodes which give us the event ~ (A). These may be labelled by the points r.=(x_,y_,z_), with (L+½)-A<~x ~ L + ~ , y _ ~ Z + ½, z _ ~ / ' + ½. We define .°3L_(A, r_) to be the event that a node does occur in the required region, and that r_ is the point of occurrence of the leftmost such node. The events ~.~ _(A, r_) form a disjoint partition of the event ".'3"l_(A). Similarly, we define ~:'X+(A, r+) to be the event that a node occurs in the required region to the right of the midplane, and that r ~ = (x+, y + , z + ) is the position of the rightmost such node, L+ ~ <~x, <~(L+ ~ ) + A . y ~ Z + 7, 1 z ~ / ' + ~. Finally, we denote by -

D.B. Abraham et al. / Random surfaces

563

~ ( A , r_, r+) = 9L_(A, r_)n~.TC+(A, r+) the event that ~ ( A ) occurs, and that r. and r , are the outermost nodes. The following lemma shows that if the event ?)~(A) does occur, then with large probability, the azimuthal separation of the outermost nodes does not exceed the order of A. L E M M A 4.3

Suppose Mc(fl ) > g(fl). Then

P r o b t . ( l y , _ y _ l + ] z . _ z i < ~ l A l ~ . ( A , r ,r~))>~l_(const)e

~A

(4.5)

Proof. The probability that, conditioned on ~ ( A , r_, r+), the above event does not occur may be written as 1 "~ 2(x,

E x_)+l

lY--):

E

Q , 2 , x . - ~ _ ) , l : t y , - y i.iz. ~ i)-

(4.6)

!lz+-z_l

This quantity may be bounded above by an analysis along the lines of that given in appendix A of [2]. In particular, one may define an angle-dependent mass M(a, b) for the pair function Q~t.:at..bL). It can be shown (proposition A.1 of [2]) that M(a, D) is jointly convex. The crucial estimate (proposition A.2 of [2]) is that M(a, b) has a lower bound of the form M(a,b)>/~,, + ~.2(la[ + Ibl).

(4.7)

In this case, (4.7) may be proved by using the normal distribution of endpoints established in corollary 2 to theorem 3.3. Given (4.7), the quantity (4.6) is bounded above by an estimate of the form (const)e which is the desired result. [] ] I ~ 2A ,

THEOREM

4.4

Whenever fl > fl,

lim ~(h)=O,

(4.8)

h~c~

i.e. the tubes are stable. Proof. Let E(h) denote the event (Per(rc) > h ). Choose A, 1 << A < L. We use the event DL(A) and its complement, 9L(A) ¢, to decompose ~c;(h). By lemma 4.2, we have

~qt.(h) = ProbL(•(h ) N ~.3~(A)) + Probt.(E(h ) (~ ~C(A) c) ~< Probt.( 'E (h) (h c3"C(A)) + (const) e - t M'ta) - 'w'B))A

(4.9)

564

D.B. Abraham et al. / Random surfaces

Using the disjoint partition of N.(A), the first term may be rewritten as

E

Probt.(?'~ (A, r_, r+)) P r o b t . ( E ( h ) l ? Y c ( A , r , r ))

r~,r

(.~,

.Y_)~2A + 1

~ < ( 1 - ( c o n s t ) e - ~ A)

E

Prob1.(gL(A, r_, r, ))

r, ,r_

(x+-- ~_)~2A +I Iv . - v - I + l z _ - z I~tA

(4.10)

× P r o b L ( ~ ( h ) l N . ( A . r_, r ) ) . The inequality follows from lemma 4.3. Now define @6"A( h ) =

ProbL(~,-~(h)lN(A,r

sup

,r+) ) .

(4.11)

(x. x..)~<2A+ 1 lY--Y I ~ I : . - z - I ~ < ~ t A

Note that ¢%A(h) is independent of L, and that by proposition 4.1, llmj,_~: ,t~(h) = 0. Finally, bounding Probt.(?3t(A)) above by one, we have

~(h) ~ (1- (const)e I~2A)~[~A(h)'-b(consI)e-[M((BJ-M'B']A Taking h ~ 00, and noting that A is arbitrary, we obtain the desired result.

(4.12) []

Two of us (J.T.C. and L.C.) would like to thank M. Aizenman, D. Brydges, M. Campanino and J. FrOhlich for useful discussions on related problems. J.T.C. and L.C. would also like to express their gratitude to the National Science Foundation for its support and the opportunities this has provided. References [1] D.B. Abraham, J.T. Chayes and L. Chayes0 Statistical mechanics of lattice tubes, Phys. Rev. I)30 (1984) 841 [2] I).B. Abraham, J.T. Chayes and L. Chayes, Random surface correlation functions, Commun. Math. Phys.. in press [3] B. Durhuus, J. FrOhlich and T. Jbnsson, Nucl. Phys. B225[FSg] (1983) 185 [4] B. Durhuus, J. Fr6hlich and T. Jonsson, Phys. Lett. 137B (1984) 93 [5] B. Durhuus, J. Fr6hlich and T. Jonsson, Nucl. Phys. B240[FS12] (1984) 453 [6] A. Bovier, G. Felder and J. Fr~hlich, Nucl. Phys. B230[FSI0] (1984) 119 [7] D. Brydges and T. Spencer, Self-avoiding walk in five or more dimensions, preprint [8] J. Fr6hlich, Quantum field theoo' in terms of random walks and random surfaces (Cargese, 1983) [9] M. Fisher. Boltzmann Medalist Address (1983) [10] D. Brydges, private communication