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Collapse of random surfaces in the connected plaquettes model
Volume 112A, number 3,4
PHYSICS LETTERS
21 October 1985
C O L L A P S E OF R A N D O M SURFACES IN T H E C O N N E C T E D P L A Q U E T T E S M O D E L Hal T A S A K I Department of Physics, Faculty of Science, University of Tokyo. Hongo. Bunl~vo-ku, Tokyo 113. Japan
and Takashi H A R A Institute of Physics, College of Arts and Sciences. University of Tol~vo, Komaba, Meguro-ku. Tol~vo 153, Japan
Received 10 December 1984; accepted in revised form 15 August 1985
Critical phenomena in a simple stochastic geometric model of random surfaces are studied. We find that, at the critical point, the characteristic area remains finite while the susceptibilitydiverges. Therefore. random surfaces collapse into branched polymers in the scaling limit.
Recently, there has been a considerable interest in mathematical theories of random surfaces [ 1 - 8 ] . They are the natural extensions of the problems of random walks, and are believed to shed light on such physical subjects as continuum limit of lattice gauge theories, critical phenomena observed in various surfaces and interfaces, and stochastic (or quantal) dynamics of string-like objects [9]. In the present note, we study the "connected plaquettes model" (CPM) which is a simple stochastic geometric system of random surfaces. We show that the model is in general characterized by finite characteristic area, and thus describes a surface system. But at the critical point, we find that the characteristic area remains finite while the susceptibility diverges. Therefore, a (possible) continuum limit of the model only describes a scalar field theory or a system of branched polymers [1]. Mathematical details of the proofs and discussions on the other related models will be reported elsewhere [10]. Definitions. Let Z d be a d-dimensional hyper-cubic lattice (d/> 2), and P denote the set of all plaquettes (1 × 1) squares) in Zd. By a cluster C, we mean an arbitrary finite subset of P which is connected in the sense that all the plaquettes attaching to a common
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
bond are connected (fig. 1). Note that each plaquette in Z d appears at most once in C, and hence the present model has self-avoiding property. A cluster C is said to span a collection of loops 22 = L1U...ULn (fig. 1) if there exists a connected surface S (a two-dimensional PL manifold immersed in Zd) supported on C which satisfies aS = 22 (~S is the boundary of S). We denote by e 0 the set of all clusters containing the origin, and by CA? the set of all clusters spanning a collection of loops 22. Our connected plaquettes model(CPM) is described by the (truncated) expectation values for a collection of loops 22, defined by
Fig. 1. A cluster which spans a loop L. 115
Volume 112A, number 3,4
G(13;/2) = C ~ ~ e-t~lCI,
(1)
where 13is a real parameter and ICI denotes the number of plaquettes in C. Note that the CPM can be naturally interpreted as describing a problem of forming random patterns from small two-dimensional objects (i.e., plaquettes), where the parameter (-13) corresponds to the chemical potential of the system. Another interpretation of CPM is to regard it as an effective theory for more complicated statistical mechanical systems such as self-avoiding random surfaces and lattice gauge theories. Actually, using the random surface representation [2], one can show [10] that abelian lattice gauge theories asymptotically reduce to the present model in the limit (g2~)13~oo. Moreover eq. (1) provides rigorous upper bounds for one- and two-loop expectation values in abellan lattice gauge theories [2,10,11 ]. Finally, we stress that the CPM is one of the simplest stochastic geometric systems of random surfaces, and thus is worth studying even from a purely mathematical point of view. We also note that since the CPM has the self-avoiding and non-planar characters, its properties (especially critical phenomena) are different from those of the planar random surface (PRS) extensively studied in refs. [1,6]. In the following, we investigate critical phenomena and the accompanying scaring limit of the CPM, which are expected to reflect universal properties independent of the specific details of the definitions.
General properties, existence of a critical point. Since the CPM is a surface model, it is expected that there are two physical quantities of basic importance. The characteristic length ~ and the characteristic area o are defined by ~(13)-1 = _ llm sup In G(13;0P0 U Opl)/l, l-~oo o(13)-1 = - rim sup in G(13;L(l,/))/l 2 ,
21 October 1985
PHYSICS LETTERS
(2)
denote the number of elements in O~?(n). Then we have the following elementary but important lemma. Lemma 1. The asymptotic behaviour of NA2(n) is given by n N~?(n) ~ (const) (power correction) Cd,
(3)
where cd is a constant depending only on the dimensionatity d, and satisfying 2 ( d - 1) ~
Proof. Eq. (3) for N0(n) follows from the subadditivity inequality N°(n + m)/(n + m) >>-(const) {NO(n)/n} {No(m)/m} , in a way similar to that in ref. [1]. As for general NZ?(n), the two inequalities
N't2(n + m) >-(const)N-CJ(n)NO(m)/m, N22(n) <.NO(n) establish eq. (3) with the same Ccl asNo(n). The bounds for cd follow from elementary estimates for the entropy of clusters. Noting that the expectation value (1) can be rewritten as ¢o
G(13;.~)= ~
N~?(n)e -~n
n=0
the lemma implies the following. Proposition 2. The CPM has a critical point 13c= lnc d > 0. For/3 > 13c,the model is well-defined, and the characteristic length and area satisfy the following inequalities: (13-13c)~<~(/3)-1 ~<13, (13-13c)-..<0(13)-1 ~<13.
(4)
For 13< 13c,the model is ill-defined. Thus we find that, for noncritical values of 13,the CPM has finite and nonzero characteristic length and area, and thus is actually describing a system of random surfaces.
I---~oo
respectively, Here L(I, l) denotes an l × l square loop, P0 is a plaquette containing the origin, and Pl is its translate by a distance I. In the field theoretic notation [1,2,6], m = ~-1 is called the glue-ball mass, and r = o -1 is called the string tension. For a positive integer n, let C ~ ( n ) be a subset of C-(2characterized by the condition If[ = n, and N~(n) 116
Critical behaviour of susceptibility and characteristic length. Let us define the susceptibility (or mean size of the clusters) by X(13)= ~
G(13;0p0L)0p).
pGP
Using the tree graph inequality [10]
G(/~; apoUopluap2)~< e 2t3 ~
p~P
nected self-avoiding surface S with aS = L(I, l) and SCC. Thus we can bound the loop expectation value G(fl; L(I,/)) as
G(/~; ~p0U~P)
X G0~; aplU0P) G(fl; bp2tA~p),
=
and the standard techniques [12], we can prove [10] the following proposition which establishes the existence of a critical phenomenon at/3 c. Proposition 3. X(~) -~ on as/3 ~ 13c. Moreover if we assume X ~ (fl-/3c)-~,we have 7 >~ 1/2. Note that the mean-field theory [11 3] (i.e., d on limit) predicts the equality 3' = 1/2. In ref. [10], we show that the analysis based on the higher order tree graph inequality and the arguments similar to those in ref. [12] strongly suggests that 3' = 1/2 holds exactly for d > d u = 8. As for the characteristic length ~, we still have no rigorous results like proposition 3. But from the above results on the behaviour of X, we expect that ~(/3) also diverges as (~-~c) -v with v I> 1/4. Again the exponent v is expected to take its mean-field value 1/4 for d > d u. Moreover, the standard scaling arguments [ 1 ] imply the relation v = I/D, where D is the Hausdorff dimension of clusters in the scaling limit (i.e. a typical cluster with ICI = n has length of order n 1/D). Since the self-avoided property requires the inequality D ~< d, we see that the exponent v(= 1/1) >1l/d) is strictly larger than 1/4 for d < 4. Therefore,we conclude that the critical phenomena in the CPM are inevitably not mean-field like in d = 2, 3 dimensions.
Critical behaviour of characteristic area. Observe that in order for a random surface model to have a scaling limit (continuum limit) with finite (rescaled) characteristic length and area, the following scaling relation must be valid: o(/3) "~ ~03)2 -+ on
21 October 1985
PHYSICS LETTERS
Volume 112A, number 3,4
as/3 ~ ~c"
(5)
To investigate whether eq. (5) is satisfied in the CPM, we study the behaviour of the characteristic area o, when/~ approaches/~c. Let L be a connected self.avoiding loop (i.e. each bond is used at most once). From an arbitrary connected surface S with aS = L, one can always construct a self-avoiding surface S' (i.e. each plaquette is used at most once) satisfying aS' = L and S'CS, by eliminating from S the plaquettes used more than twice. Therefore to each CEG L(/,/) there corresponds at least one con-
c~Lq,0
<
Z; S;aS=L(/,/) C~S
(6)
where, in the last equation, the first summation runs over connected self-avoiding surfaces, and the second summation runs over all clusters including S. Let NS(n) be the number of clusters in e 0(n) including S. Introducing a probability that an arbitrarily chosen cluster in C0(n) includes S, i.e. Prob(S, n) = NS(n)/NO(n), the second summation in (6) can be rewritten as on
C~S
e-#•l = ~ /VS(n)e-~n n=0
= ~
Prob(S,n)N0(n) e - ~ •
(7)
n=0
Here we present a rough (and nonrigorous) estimate for Prob(S, n) which leads to one of our main results. In ref. [10], we will descr~e a more tedious entropy estimate which justifies the present one. For a j~laquette p in Z d, consider the quantity Prob(p, n) = NP(n)/ N0(n) which denotes the probability that an arbitrary cluster contains p. From a given cluster C~p, we remove the plaquette p, and attach a new plaquette p' to one of the edges of the maximum plaquette in C\p (i.e. a plaquette with maximum coordinate number with respect to some lexiographic ordering [1 ] ), in a way that p' then becomes the maximum plaquette in (C\p)Up'. If we neglect the possibility that Ckp becomes disconnected, this procedure yields 2 ( d - 1) different clusters not including p. Thus we arrive at the estimate Prob(p, n) ~
(8)
p~S
Now combining eqs. (6)-(8) with the fact [1 ] that the number of connected self-avoiding surfaces with ISI -- A is bounded by ( 2 d - 3 ) A , we have G(~; L(I,/)) -~
Volume 112A, number 3,4
PHYSICS LETTERS
__>
--~
21 October 1985
---->
SCALINGLIMIT o
I-]
Fig. 2. Schematic view of the collapse of random surfaces into branched polymers, when the critical point is approached. The scales are adjusted to fix the rescaled characteristic length. Thus we conclude that the scaling relation (5) is unfortunately violated in the CPM. In the scaling limit where the (rescaled) characteristic length is adjusted to be finite, the (rescaled) characteristic area inevitably vanishes. Therefore the CPM, in such a limit, only describes a system of branched polymers or a scalar field theory (fig. 2). A similar "collapse of random surfaces" has been observed in various simple models of random surfaces [1,5,7]. It is a challenging subject to overcome this apparently universal difficulty, and to construct well defined measures for random surfaces. See refs. [ 6 - 8 ] for recent researches in this direction.
[4]
[5]
[6] We wish to thank Professor M. Suzuki for stimulating discussions and a careful reading of the manuscript, Professor M. Kato, Dr. T. Hattori, K.-I. Kondo, and H. Watanabe for fruitful discussions, and Professor J. Fr6hlich for instructive comments on the random surface representation. We are also grateful to the referee for instructive comments on the manuscript. The present work is supported in part by Research Institute for Fundamental Physics.
References [ 1] B. Durhuus, J. Fr/Slichand T. J6nsson, Nucl. Phys. B225 [FS9] (1983) 185;Phys. Lett. 137B (1984) 93; Nucl. Phys. B240 [FS12] (1984) 453. [2] J. Fr6hlich, in: Proc. Cargese summer school 1983, eds. G. 't Hooft et al. (Plenum, New York, 1985); K.H. Miitter, Nucl. Phys. B251 [FS13] (1985) 735; H. Tasaki and T. Hara, unpublished. [3] R. Schrader, preprint ITP-SB-84-44; D.B. Abraham, J.T. Chayes and L. Chayes, Phys. Rev. 118
[7]
[8]
[9]
[10]
[11] [12]
[13]
D30 (1984) 841; Commun. Math. Phys. 96 (1984) 439; Nucl. Phys. B251 [FS13] (1985) 553; A. Maritan and A. SteUa, Phys. Rev. Lett. 53 (1984) 123. M. Aizenman, J.T. Chayes, L. Chayes, J. Fr6hlich and L. Russo, Commun. Math. Phys. 92 (1983) 19; J.T. Chayes and L. Chayes, Nucl. Phys. B235 [FS11 ] (1984) 19; M. Ham and M. Aizenman, preprint. D.J. Gross, Phys. Lett. 138B (1984) 185; T. Eguchi and T. Filk, preprint UT-425 (1984); T. Eguchi, R. Nakayama and S.K. Yang, Nucl. Phys. B251 [FS13] (1985)401; F. David, Nucl. Phys. B257 [FS141 (1985) 45; J. Ambj~n, B. Durhuus and J. Fr6hlich, NORDITA preprint 85•2. B. Durhuus, J. Fr6hlich and T. J6nson, Reflection positivity and tree inequalities in a theory of planar random surfaces, preprint. T. Hara, J. Kobayashi and H. Tasaki, Universality classes of random walks, surfaces, and volumes, talk at RIMS (June, 1985) and a paper in preparation. H. Tasaki and T. Hara, A hierarchical model for planar random surfaces, talk at RIMS (June, 1985) and a paper in preparation. T. Funaki, Nagoya Math. J. 89 (1983) 129; T. Hara and H. Tasaki, Proc. Stochastic processes and their applications (1985). H. Tasaki and T. Hara, Connected plaquettes model - a stochastic geometric approach to lattice gauge theories and pattern formation problems, in preparation. H. Tasaki, J. Stat. Phys. 38 (1985) 873. M. Aizenman, Commun. Math. Phys. 86 (1982) 1; M. Aizenman and C.M. Newman, J. Star. Phys. 36 (1984) 107; J. Fr6hlich, Nucl. Phys. B200 [FS4] (1982) 281; A. Bovier, G. Felder and J. Fr6hlich, Nucl. Phys. B230 [FS10] (1984) 119. J.-M. Drouffe, G. Parisi and N. Sourlas, Nucl. Phys. B161 (1979) 397.
PHYSICS LETTERS
21 October 1985
C O L L A P S E OF R A N D O M SURFACES IN T H E C O N N E C T E D P L A Q U E T T E S M O D E L Hal T A S A K I Department of Physics, Faculty of Science, University of Tokyo. Hongo. Bunl~vo-ku, Tokyo 113. Japan
and Takashi H A R A Institute of Physics, College of Arts and Sciences. University of Tol~vo, Komaba, Meguro-ku. Tol~vo 153, Japan
Received 10 December 1984; accepted in revised form 15 August 1985
Critical phenomena in a simple stochastic geometric model of random surfaces are studied. We find that, at the critical point, the characteristic area remains finite while the susceptibilitydiverges. Therefore. random surfaces collapse into branched polymers in the scaling limit.
Recently, there has been a considerable interest in mathematical theories of random surfaces [ 1 - 8 ] . They are the natural extensions of the problems of random walks, and are believed to shed light on such physical subjects as continuum limit of lattice gauge theories, critical phenomena observed in various surfaces and interfaces, and stochastic (or quantal) dynamics of string-like objects [9]. In the present note, we study the "connected plaquettes model" (CPM) which is a simple stochastic geometric system of random surfaces. We show that the model is in general characterized by finite characteristic area, and thus describes a surface system. But at the critical point, we find that the characteristic area remains finite while the susceptibility diverges. Therefore, a (possible) continuum limit of the model only describes a scalar field theory or a system of branched polymers [1]. Mathematical details of the proofs and discussions on the other related models will be reported elsewhere [10]. Definitions. Let Z d be a d-dimensional hyper-cubic lattice (d/> 2), and P denote the set of all plaquettes (1 × 1) squares) in Zd. By a cluster C, we mean an arbitrary finite subset of P which is connected in the sense that all the plaquettes attaching to a common
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
bond are connected (fig. 1). Note that each plaquette in Z d appears at most once in C, and hence the present model has self-avoiding property. A cluster C is said to span a collection of loops 22 = L1U...ULn (fig. 1) if there exists a connected surface S (a two-dimensional PL manifold immersed in Zd) supported on C which satisfies aS = 22 (~S is the boundary of S). We denote by e 0 the set of all clusters containing the origin, and by CA? the set of all clusters spanning a collection of loops 22. Our connected plaquettes model(CPM) is described by the (truncated) expectation values for a collection of loops 22, defined by
Fig. 1. A cluster which spans a loop L. 115
Volume 112A, number 3,4
G(13;/2) = C ~ ~ e-t~lCI,
(1)
where 13is a real parameter and ICI denotes the number of plaquettes in C. Note that the CPM can be naturally interpreted as describing a problem of forming random patterns from small two-dimensional objects (i.e., plaquettes), where the parameter (-13) corresponds to the chemical potential of the system. Another interpretation of CPM is to regard it as an effective theory for more complicated statistical mechanical systems such as self-avoiding random surfaces and lattice gauge theories. Actually, using the random surface representation [2], one can show [10] that abelian lattice gauge theories asymptotically reduce to the present model in the limit (g2~)13~oo. Moreover eq. (1) provides rigorous upper bounds for one- and two-loop expectation values in abellan lattice gauge theories [2,10,11 ]. Finally, we stress that the CPM is one of the simplest stochastic geometric systems of random surfaces, and thus is worth studying even from a purely mathematical point of view. We also note that since the CPM has the self-avoiding and non-planar characters, its properties (especially critical phenomena) are different from those of the planar random surface (PRS) extensively studied in refs. [1,6]. In the following, we investigate critical phenomena and the accompanying scaring limit of the CPM, which are expected to reflect universal properties independent of the specific details of the definitions.
General properties, existence of a critical point. Since the CPM is a surface model, it is expected that there are two physical quantities of basic importance. The characteristic length ~ and the characteristic area o are defined by ~(13)-1 = _ llm sup In G(13;0P0 U Opl)/l, l-~oo o(13)-1 = - rim sup in G(13;L(l,/))/l 2 ,
21 October 1985
PHYSICS LETTERS
(2)
denote the number of elements in O~?(n). Then we have the following elementary but important lemma. Lemma 1. The asymptotic behaviour of NA2(n) is given by n N~?(n) ~ (const) (power correction) Cd,
(3)
where cd is a constant depending only on the dimensionatity d, and satisfying 2 ( d - 1) ~
Proof. Eq. (3) for N0(n) follows from the subadditivity inequality N°(n + m)/(n + m) >>-(const) {NO(n)/n} {No(m)/m} , in a way similar to that in ref. [1]. As for general NZ?(n), the two inequalities
N't2(n + m) >-(const)N-CJ(n)NO(m)/m, N22(n) <.NO(n) establish eq. (3) with the same Ccl asNo(n). The bounds for cd follow from elementary estimates for the entropy of clusters. Noting that the expectation value (1) can be rewritten as ¢o
G(13;.~)= ~
N~?(n)e -~n
n=0
the lemma implies the following. Proposition 2. The CPM has a critical point 13c= lnc d > 0. For/3 > 13c,the model is well-defined, and the characteristic length and area satisfy the following inequalities: (13-13c)~<~(/3)-1 ~<13, (13-13c)-..<0(13)-1 ~<13.
(4)
For 13< 13c,the model is ill-defined. Thus we find that, for noncritical values of 13,the CPM has finite and nonzero characteristic length and area, and thus is actually describing a system of random surfaces.
I---~oo
respectively, Here L(I, l) denotes an l × l square loop, P0 is a plaquette containing the origin, and Pl is its translate by a distance I. In the field theoretic notation [1,2,6], m = ~-1 is called the glue-ball mass, and r = o -1 is called the string tension. For a positive integer n, let C ~ ( n ) be a subset of C-(2characterized by the condition If[ = n, and N~(n) 116
Critical behaviour of susceptibility and characteristic length. Let us define the susceptibility (or mean size of the clusters) by X(13)= ~
G(13;0p0L)0p).
pGP
Using the tree graph inequality [10]
G(/~; apoUopluap2)~< e 2t3 ~
p~P
nected self-avoiding surface S with aS = L(I, l) and SCC. Thus we can bound the loop expectation value G(fl; L(I,/)) as
G(/~; ~p0U~P)
X G0~; aplU0P) G(fl; bp2tA~p),
=
and the standard techniques [12], we can prove [10] the following proposition which establishes the existence of a critical phenomenon at/3 c. Proposition 3. X(~) -~ on as/3 ~ 13c. Moreover if we assume X ~ (fl-/3c)-~,we have 7 >~ 1/2. Note that the mean-field theory [11 3] (i.e., d on limit) predicts the equality 3' = 1/2. In ref. [10], we show that the analysis based on the higher order tree graph inequality and the arguments similar to those in ref. [12] strongly suggests that 3' = 1/2 holds exactly for d > d u = 8. As for the characteristic length ~, we still have no rigorous results like proposition 3. But from the above results on the behaviour of X, we expect that ~(/3) also diverges as (~-~c) -v with v I> 1/4. Again the exponent v is expected to take its mean-field value 1/4 for d > d u. Moreover, the standard scaling arguments [ 1 ] imply the relation v = I/D, where D is the Hausdorff dimension of clusters in the scaling limit (i.e. a typical cluster with ICI = n has length of order n 1/D). Since the self-avoided property requires the inequality D ~< d, we see that the exponent v(= 1/1) >1l/d) is strictly larger than 1/4 for d < 4. Therefore,we conclude that the critical phenomena in the CPM are inevitably not mean-field like in d = 2, 3 dimensions.
Critical behaviour of characteristic area. Observe that in order for a random surface model to have a scaling limit (continuum limit) with finite (rescaled) characteristic length and area, the following scaling relation must be valid: o(/3) "~ ~03)2 -+ on
21 October 1985
PHYSICS LETTERS
Volume 112A, number 3,4
as/3 ~ ~c"
(5)
To investigate whether eq. (5) is satisfied in the CPM, we study the behaviour of the characteristic area o, when/~ approaches/~c. Let L be a connected self.avoiding loop (i.e. each bond is used at most once). From an arbitrary connected surface S with aS = L, one can always construct a self-avoiding surface S' (i.e. each plaquette is used at most once) satisfying aS' = L and S'CS, by eliminating from S the plaquettes used more than twice. Therefore to each CEG L(/,/) there corresponds at least one con-
c~Lq,0
<
Z; S;aS=L(/,/) C~S
(6)
where, in the last equation, the first summation runs over connected self-avoiding surfaces, and the second summation runs over all clusters including S. Let NS(n) be the number of clusters in e 0(n) including S. Introducing a probability that an arbitrarily chosen cluster in C0(n) includes S, i.e. Prob(S, n) = NS(n)/NO(n), the second summation in (6) can be rewritten as on
C~S
e-#•l = ~ /VS(n)e-~n n=0
= ~
Prob(S,n)N0(n) e - ~ •
(7)
n=0
Here we present a rough (and nonrigorous) estimate for Prob(S, n) which leads to one of our main results. In ref. [10], we will descr~e a more tedious entropy estimate which justifies the present one. For a j~laquette p in Z d, consider the quantity Prob(p, n) = NP(n)/ N0(n) which denotes the probability that an arbitrary cluster contains p. From a given cluster C~p, we remove the plaquette p, and attach a new plaquette p' to one of the edges of the maximum plaquette in C\p (i.e. a plaquette with maximum coordinate number with respect to some lexiographic ordering [1 ] ), in a way that p' then becomes the maximum plaquette in (C\p)Up'. If we neglect the possibility that Ckp becomes disconnected, this procedure yields 2 ( d - 1) different clusters not including p. Thus we arrive at the estimate Prob(p, n) ~
(8)
p~S
Now combining eqs. (6)-(8) with the fact [1 ] that the number of connected self-avoiding surfaces with ISI -- A is bounded by ( 2 d - 3 ) A , we have G(~; L(I,/)) -~
Volume 112A, number 3,4
PHYSICS LETTERS
__>
--~
21 October 1985
---->
SCALINGLIMIT o
I-]
Fig. 2. Schematic view of the collapse of random surfaces into branched polymers, when the critical point is approached. The scales are adjusted to fix the rescaled characteristic length. Thus we conclude that the scaling relation (5) is unfortunately violated in the CPM. In the scaling limit where the (rescaled) characteristic length is adjusted to be finite, the (rescaled) characteristic area inevitably vanishes. Therefore the CPM, in such a limit, only describes a system of branched polymers or a scalar field theory (fig. 2). A similar "collapse of random surfaces" has been observed in various simple models of random surfaces [1,5,7]. It is a challenging subject to overcome this apparently universal difficulty, and to construct well defined measures for random surfaces. See refs. [ 6 - 8 ] for recent researches in this direction.
[4]
[5]
[6] We wish to thank Professor M. Suzuki for stimulating discussions and a careful reading of the manuscript, Professor M. Kato, Dr. T. Hattori, K.-I. Kondo, and H. Watanabe for fruitful discussions, and Professor J. Fr6hlich for instructive comments on the random surface representation. We are also grateful to the referee for instructive comments on the manuscript. The present work is supported in part by Research Institute for Fundamental Physics.
References [ 1] B. Durhuus, J. Fr/Slichand T. J6nsson, Nucl. Phys. B225 [FS9] (1983) 185;Phys. Lett. 137B (1984) 93; Nucl. Phys. B240 [FS12] (1984) 453. [2] J. Fr6hlich, in: Proc. Cargese summer school 1983, eds. G. 't Hooft et al. (Plenum, New York, 1985); K.H. Miitter, Nucl. Phys. B251 [FS13] (1985) 735; H. Tasaki and T. Hara, unpublished. [3] R. Schrader, preprint ITP-SB-84-44; D.B. Abraham, J.T. Chayes and L. Chayes, Phys. Rev. 118
[7]
[8]
[9]
[10]
[11] [12]
[13]
D30 (1984) 841; Commun. Math. Phys. 96 (1984) 439; Nucl. Phys. B251 [FS13] (1985) 553; A. Maritan and A. SteUa, Phys. Rev. Lett. 53 (1984) 123. M. Aizenman, J.T. Chayes, L. Chayes, J. Fr6hlich and L. Russo, Commun. Math. Phys. 92 (1983) 19; J.T. Chayes and L. Chayes, Nucl. Phys. B235 [FS11 ] (1984) 19; M. Ham and M. Aizenman, preprint. D.J. Gross, Phys. Lett. 138B (1984) 185; T. Eguchi and T. Filk, preprint UT-425 (1984); T. Eguchi, R. Nakayama and S.K. Yang, Nucl. Phys. B251 [FS13] (1985)401; F. David, Nucl. Phys. B257 [FS141 (1985) 45; J. Ambj~n, B. Durhuus and J. Fr6hlich, NORDITA preprint 85•2. B. Durhuus, J. Fr6hlich and T. J6nson, Reflection positivity and tree inequalities in a theory of planar random surfaces, preprint. T. Hara, J. Kobayashi and H. Tasaki, Universality classes of random walks, surfaces, and volumes, talk at RIMS (June, 1985) and a paper in preparation. H. Tasaki and T. Hara, A hierarchical model for planar random surfaces, talk at RIMS (June, 1985) and a paper in preparation. T. Funaki, Nagoya Math. J. 89 (1983) 129; T. Hara and H. Tasaki, Proc. Stochastic processes and their applications (1985). H. Tasaki and T. Hara, Connected plaquettes model - a stochastic geometric approach to lattice gauge theories and pattern formation problems, in preparation. H. Tasaki, J. Stat. Phys. 38 (1985) 873. M. Aizenman, Commun. Math. Phys. 86 (1982) 1; M. Aizenman and C.M. Newman, J. Star. Phys. 36 (1984) 107; J. Fr6hlich, Nucl. Phys. B200 [FS4] (1982) 281; A. Bovier, G. Felder and J. Fr6hlich, Nucl. Phys. B230 [FS10] (1984) 119. J.-M. Drouffe, G. Parisi and N. Sourlas, Nucl. Phys. B161 (1979) 397.