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Monte Carlo method for random surfaces
Nuclear Physics B251[FSI31 (1985) 665-675 9 North-Holland Publishing Company
M O N T E CARLO M E T I t O D FOR RANDOM SURFACES B. BERG
IL hlstitut f~r Theoretische Physik der Unieersitgtt llamburg, Luruper Chaussee 149, D-2000 llamburg 50 A. BILLOIRE
Service de Physique Thborique, Orme des Merisiers. CEN-SACI~t Y, 91191 Gif-sur- Yeette, Cedex, France D. FOERSTER
Max-Planck hlstitut, Fohringer Ring 6. D8000 Mfinchen 40, West Germany Received 1 June 1984
Previously two of the authors proposed a Monte Carlo method for sampling statistical ensembles of random walks and surfaces with a Boltzmann probabilistic weight. In the present paper we work out the details for several models of random surfaces, defined on d-dimensional hypercubic lattices.
1. Introduction
Random surfaces have attracted a lot of interest [1], because of their relation to relativistic string theory, non-abelian gauge theories and problems in statistical mechanics. Only few analytic results exist and numerical studies may be very useful. In ref. [2] a local stochastic process was proposed, which allows one to generate statistical ensembles of walks and surfaces with a Boltzmann probabilistic weight. If applicable the procedure allows Monte Carlo (MC) simulations of rather large walks and surfaces, because the needed amount of computer memory is only proportional to the length of the walk, respectively the area of the surface. The MC procedure of ref. [2] (MCP2) has a wide range of possible applications. For instance one may carry out a stochastic inversion of large matrices, which are relevant for fermions in lattice QCD. Kuti [3] advocated later the von Neumann-Ulam method for similar purposes. A test is the stochastic computation of the average of y-traces (Tr)L over walks of fixed length L (for thd precise definition of Tl:t. see ref. [4]). In table 1 we compare the exact answer [4] with results as obtained by the MCP 2 [5] and by the von Neumann-Ulam method [6]. The MCP2 is more efficient because
666
B. Berg et al. / Random surfaces TABLE 1 S t o c h a s t i c c a l c u l a t i o n of -f-traces L 4 6 8 10 12 14 16 18
20 22 24
(Tr)~.~ t -8 - 18.53 - 22.66 -25.16 -25.78 - 24.74 - 22.18 - 18.20 - 12.91 -6.36 + 1.41
( T O t . (BBF), = I T E R -8 - 18.52 - 22.70 -25.36 -26.12 - 25.2 - 21.8 - 22 - 16 -5 29
+0 __. 0.01, 4- 0.04, 4- 0.12 4- 0.24 4- 0.6 4- 0.6 4- 4 4- 8 4- 8 + 30
3.25 106 2.91 106 2.55 106 3.03 106 2.71 106 2.45 106 2.23 106 2.04 106 3.24 106 3.01 106
(TOt . (KUTI) - 7.98 4-_0.09 - 18.55 + 0.53 - 22.9 + 1.0 -25+3 -27_+_3 -25 + 14
E x a c t r e s u l t s a r e t a k e n f r o m ref. [4]. ( T r ) L (BIIF) c o r r e s p o n d s to rcf. [51 a n d = I T E R gives tile n u m b e r o f M C P 2 i t e r a t i o n s at single links. ( T r ) L ( K U T I ) h a s b e e n t a k e n f r o m ref. [6] a n d relies o n a b o u t 106 p a t h s .
it allows one to fix the initial and final points of sampled random walks. For another related investigation see ref. [7]. Another relevant field for applications are problems in polymer physics. The MCP 2 has turned out to be useful in a numerical study of self-avoiding random walks in four (and three) dimensions [8]. In good agreement with renormalization group results [9] logarithmic scaling violations of mean field theory were found in four dimensions. Statistical ensembles of polymer gases are studied in ref. [10]. In the present paper we are interested in random surfaces (RS). In the next section we fix the notation and introduce three models of RS. The first two models are tile exact analogs of the two models of free random walks for which the MCP 2 was tested in ref. [2]. T h e third model are self-avoiding random surfaces. For these models we describe in all details the MCP 2 in sect. 3. The (tedious) work of writing tile computer program along these lines has already been carried out in ref. [11], where first numerical results are also reported. Our way of simulating RS differs considerably from other methods for which the interested reader is referred to the literature [12], in spirit related is ref. [13]. The explicit use of a lattice prevents, however, sampling very large surfaces.
2. Models of random surfaces Let us consider a d-dimensional hypercubic lattice. We will introduce three models of random surfaces (RS). By a surface we mean a finite collection of plaquettes, where each plaquette may appear several times, together with a prescription for gluing the plaquettes together.
B. Berg et al. / Random surfaces
667
3
Fig. 1. F.xampleof a spike.
Each plaquette of the surface contains four lattice links. Via each of its links a surface plaquette is either glued together with precisely one other surface plaquette (containing the same link), or the surface link is a boundary link. The boundary (if any) is not allowed to move. We are interested in calculating expectation values with respect to the partition function Z = }-~.e -/~as .
(2.1)
s
The sum is over all connected surfaces with fixed boundary and fixed Euler characteristic. A s is the area ( = number of plaquettes with multiplicity) of surface S. If the b o u n d a r y is empty, the smallest surface of Euler characteristic 2 is made of a doubly occupied plaquette. The smallest surfaces of Euler characteristic 0 (orientable or not) and 1 are made of a four times occupied plaquette suitably connected. So far we have defined our first model. We call spikes the configurations where the surface folds back upon itself, see fig. 1 for an example. A precise definition is that a spike is generated if one (or more) of the four neighbours of some surface plaquette is the same lattice plaquette. Such spikes are allowed in our first model. In our second model spikes are forbidden. According to the random walk language in ref. [2] we call surfaces of the first model bosonic random surfaces (BRS)* and those of the second model fermionic random surface (FRS). There are no really good reasons for this notation. The original motivation was, that random paths with spikes do not contribute in the partition function of a theory of Wilson fermions. " They are called elsewhere (planar) random surfaces. See for instance Durhuus et al. [1].
668
B. Berg et al. / Random surfaces
Our tlfird model are self-avoiding random surfaces: no link of the lattice is contained in more than two plaquettes of the surface. Boundary links are contained in precisely one plaquette of the surface.
3. Monte Carlo procedure for random surfaces In accordance with detailed balance (for a review see [14]) we establish in this section a system of transition probabilities for local shifts. In dimensions d >/4 these shifts allow one to generate all surfaces within each of the models introduced previously. In the limit of infinite statistics the Monte Carlo procedure (MCP2) generates random surfaces (RS) with the weights of the modified partition function Z r'~c = ~ A s e - / ~ ' q .
(3.1)
S
Finally we will optimize the transition probabilities. Our procedure for modifying the surface is the following: (i) We randomly select a plaquette RP of the surface. (ii) We shift the p!aquette RP by one lattice unit in a direction orthogonal to the plane spanned by RP (RP plane). These shifts are restricted in a model-dependent way and precise probabilities will be established later. (iii) The shift takes RP along 4 plaquettes. 0 ~< ~r ~<4 of such plaquettes are occupied by neighbour plaquettes glued to RP (before the shift). These ~r neighbour plaquettes are annihilated and 4 - ~r new plaquettes are created. The surface area changes thus by AA = 4 - 2 N = 0 , +2, +4. RP, old and new plaquettes are then glued together in an obvious way. Note that the above-described modification of the surface does not affect the topology of the surface. 3.1. L O C A L T O P O L O G Y
A somewhat tedious step is to list completely all different local "topological" cases for such a procedure. By means of fig. 2 we fix our notation in a way also convenient for a computer implementation of the MCPv RP denotes the randomly chosen plaquette. Let 0~, ( # = 1 . . . . . d) be the unit vectors along the axis of our hypercubic lattice. The vectors f~ = +0~,,
(i = 1 . . . . . 4 ) ,
(3.2)
are defined to point in the direction of the (glued) neighbour plaquette NPi (i = 1 . . . . . 4). In this way/t~ is uniquely defined for each neighbour plaquette NPv The case of a boundary link is treated like a neighbour plaquette in the RP plane, with f~ pointing away from RP.
B. Berg et a L / Ramlomsurfaces
669
N~
N~
N~ lit
~p
P I.!(0
Fig. 2. Notation for a randomly chosen plaquette.
In the following p ( a A), zlA = + 4, + 2,0 is the probability for an allowed shift (in a given direction), which changes the area by the amount AA. This shift probability (SP) only depends on AA. Further there will be 12 different "zero-shift probabilities" (ZSP): pO. These are probabilities for leaving the surface unchanged. Now we list the complete classification of all local topological cases. Afterwards the probabilities are determined and optimized. (i) All 5 plaquettes (RP, NP,.) are in the same plane: for each i = 1. . . . . 4 we have /t, = p. or p., = v. RP can be shifted in any of the 2 d - 4 directions orthogonal to the RP plane with equal SP P(+4). The probability for leaving the surface unchanged (ZSP) is thus P0~ = 1 - ( 2 d - 4 ) P ( + 4 ) . (ii) One of the neighbour plaquettes NPi is out of the RP plane: #,,, :#/.t and #,, :# v,#,, = / t or #,,=v for i = 2,3,4. Here and in the following % . . . . . 74 is a suitable chosen cyclic permutation of 1. . . . . 4. RP can be shifted in the direction/.t=, with probability P ( + 2 ) and in ( 2 d - 5 ) other directions with equal probability P( + 4). SP: P ( + 4 ) , P ( + 2 ) ;
ZSP: P ( ~
(iii) Two of the neighbour plaquettes NPi are out of the RP plane: #,,l ~: # and #,,, ~= v, and further: (a)
/.t~,,4:#,
bt,,2:#v;
It,,=/.t
or
#,,=v
fori=3,4,
(b)
~,,jv~tt,
/.t,,34:v;
#~,,=/t
or
#,,,=v
fori=2,4.
B. Bcrg ct a L / Ra,ulom surfaces
670
(iii.1) f~., ~f~2 if (a) holds, or f~, g: f~., if (b) holds. SP: P ( + 4 ) , P ( + 2 ) ; (iii.2) ./~, "--L2 if (a)
ZSP: P0ii.l) o = 1-2P(+2)-(2d-6)P(+4).
holds, or f., =.L,
SP: P ( + a ) , P ( 0 ) ;
if (b) holds.
zsP: Pt~i~.2)~_- 1 -e(o)-(2d-5)P(+4)
(iv) Three of the neighbour plaquettes NP~ are out of the RP plane: /t~., 4:/t and /G,, 4: v for i = 1,2,3; p.~, = bt or ~,,, = i:. (iv.l) All L, (i = 1,2,3) are different.
SP: / ' ( + 4 ) , P ( + 2 ) ;
(i~.2)
1-3P(+2)-(2d-7)P(+4).
L=L2,L orL ,L2=L orL,=L,~L,.
sP: P ( + 4 ) , e ( + 2), /'(0) ;
(~v.3)
ZSP: Pc~ =
zsp: F'~i~
1-p(0)-P(+2)-(2d-6)P(+4).
L=L=L.
SP: P ( + 4 ) , P ( - 2 ) ;
ZSP: P~,~.3)o _ 1 - P(-2)-(2d-
5)P(+4).
(v) All neighbour plaquettes NP, are out of the RP plane: /t~ ~ p. and /~, 4: v for i = 1,2,3,4. (v.1) All J] are different (can only happen in d>~ 5).
Sp: P ( + a ) , P ( + 2 ) ;
(v.2) SP:
L,=L 4:L,,L and
P(+4),P(+2),P(O);
(v.3)
or
L,=L ,L:L andL2*L.
ZSP: P ( ~
zsp:/'t~3~~ __ 1 - 2 P ( 0 ) - ( 2 d -
6)P(+4)
L =L =L,§
SP=P(+4),P(+2),P(-2);
(v.5)
L#=L
Pt~
L=L, 4:L=L, or L=L .L=L. sp: e ( +4), /'(o) ;
(v.4)
ZSP:
ZSP: p t O . a ) = l _ e ( _ Z ) _ p ( + 2 ) _ ( 2 d _ 6 ) P ( + 4 ) .
L,=L=L=L. SP: ? ( + 4 ) , P ( - 4 ) ;
ZSP: ?(~
1-e(-4)-(2d-5)e(+4).
We remark that d = 5 is the lowest dimension where the P ( + 4 ) shift of case (v.l) can be realized.
B. Berg et al. / Random surfaces
671
3.2. TRANSITION PROBABILITIES
The transition probabilities P(z~A) are chosen such that a detailed balance [14] is fulfilled:
A'p(+aA) AP(-aA)
= e(A-.A')
P(A'~A)
A' e -/~A'- A) = ~---e A' -pa'4 .
A
(3.3)
There AA = A ' - A . The extra factor A'/A in front of P(+AA)/P(-AA) comes from the random choice of the plaquette on an already existing surface and amounts to actually simulating the partition function (3.1) (not (2.1)). The speed of going through surface space is increased by setting as many ZSP as possible equal to zero. Therefore it is easy to see that the MCP 2 is optimized with et,,
(3.4)
= P ,5> = 0 .
Eqs. (3.3) and (3.4) uniquely determine the transition probabilities P(A A). With c: = e -2p,
(3.5)
we obtain 1 l+c 2 P(0) = ~- 1 + ( 2 d - 5)c 2' l+c
2
(3.6a)
1
P(-2)=
l+c
1+(2d-5)c
P(-4) =
1 1 + ( 2 d - 5)c 2 '
2'
P(+2)=cP(-2),
P ( + 4 ) = c2p(-4).
(3.6b)
(3.6c)
In practical applications the "bottle neck" of the procedure is P ( - 4 ) [11]. We are still left with two problems: (i) Within each model considered we have to show that all surfaces can be generated by our procedure. (ii) For each model we have to specify the shifts which are not allowed. Point (i) is obviously true for our first model (BRS). For FRS the only case which seems to cause trouble is depicted in fig. 3. There are three different ways of gluing the plaquettcs 1, 2, 3 and 4 together: (1,2),(3,4),(a);
(1,3),(2,4),(b);
(1,4),(2,3),(c).
(3.7)
Plaquettes (i, j ) are glued together. At the first look it is not obvious how to reach
672
B. Berg et al. / Random surfaces
i
Fig. 3. Three random surfaces. situation (c). In d >I 4 dimensions this is possible as follows: consider fig. 4, shif plaquette 2 of this figure in the direction (+)~4, then shift plaquette 1 two times ir direction + e 2 , finally shift plaquette 2 back in direction (T-)~ 4. In d--- 3 dimension: this does not work, and not all FRS can be reached with our present rules (Additional rules may, of course, be added.) A shift is rejected whenever it would modify the nature of the surface (e.g generates a spike on a fermionic surface) or has no inverse (according to our rule:
Fig. 4. A special random surface.
B. Berg et al. / Random surfaces
673
/",,,7
C Fig. 5. The simple way to generate a spike.
for surface deformation). Therefore our MCP2 has the character of a mixture between heat bath and Metropolis algorithm [14]. For FRS, one rejects shifts like moving in fig. 5 the plaquette 1 in direction 3. Tiffs would lead to a spike. One subtle point is that spikes may be generated in a way which has no analog in the case of a random walk. Consider again case (3.7) (c). W e have to reject a shift of plaquette 1 in +~2 direction, because a spike would be generated along the dotted link in fig. 3. The situation which would lead to problems with detailed balance is depicted in fig. 6. Plaquettes 2 and 3 are both connected to plaquette 1 but are not connected together whereas they are close by. A shift of plaquette 1 in direction 3 has no inverse and is thus rejected. This rule also applies to bosonic surfaces. For SRS, we have to introduce a "logical" lattice of link variables additionally to reject all shifts, which would touch links already occupied by the surface.
2
( Fig. 6. A problem with detailed balance.
674
B. Berg et aL / Random surfaces
BRS are slightly more subtle. Transitions which resolve a spike are only allowed if tile inverse transition would again create the spike. For instance shifting in fig. 1 one of the two plaquettes drawn in the (1,2) plane upward has to be rejected. Otherwise, as already noted in [2], detailed balance would be violated. In s u m m a r y : BRS and FRS can be simulated with a local upgrading procedure. O n l y i n f o r m a t i o n about nearest a n d next-nearest neighbours of a r a n d o m l y chosen plaquette is required. Introducing a logical lattice allows also to simulate SRS. Different models could be considered by defining new probabilities.
4. Conclusions W e explicitly worked out the ideas of ref. [2] for a non-trivial case. The M C P 2 allows M C simulations of very large r a n d o m surfaces. A major disadvantage is that p r o g r a m m i n g is very tedious. In ref. [11] the c o m p u t e r implementation has been carried out and the first numerical results obtained: (i). BRS d o not allow an efficient simulation because of a very low acceptance rate for p r o p o s e d shifts. (ii) F R S allow, however, precise numerical results and for SRS the efficiency is expected to increase even further. (iii) F u r t h e r applications of the method will be the subject of future work [15]. W e would like to thank E. Marinari, B. Pearson, J. B. Zuber and P. Windey for their stimulating interest.
References [1] D. Focrstcr, Phys. Lctt. 77B (1978) 597; G. Padsi, Phys. Len. 81B (1979) 357; A.M. Polyakov, Phys. Letl. 82B (1979) 247; J. Frrhlich, Carg~se lectures 1979 (Plenum, New York); D. J. Wallace and R.K.P. Zia, Phys. Roy. Len. 43 (1979) 808; M.J. Lov.'e and D.J. Wallace, Phys. Lctt. 93B (1980) 433; D. Wcingarten, Phys. Lctt. 90B (1980) 280; J.M. Drouffe, G. Parisi and N. Sourlas, Nucl. Phys. BI61 (1980) 397; D. Foerster, Nucl. Phys. BI70tFSI] (1980) 107; M. L~seher, K. Symanzik and P. Weisz, Nucl. Phys. B173 (1980) 365; A.A. Migdal, Nucl. Phys. B189 (1981) 253; A.M. Polyakov, Phys. Lett. 103B (!981) 207; F. David, Phys. Left. 120B (1981) 193; thesis, Paris (1982); T. Eguclti and H. Kawai, Phys. Lctt. I10B (1982) 143; A.B. Zamolodchikov, Phys. Lett. II7B (1982) 87; T. Eguchi and II. Kawai, Phys Lett. lI4B (1982) 247; D. Durhuus, J. Frrhlich and T. Jonsson, Nucl. Phys. B225[FSg](1983) 185; B240[FSI2] (1984) 453; J. Frrhlich, Carg~se lectures 1983 (Plenum, New York) to appear; D.J. Gross, Phys. Lett. 138B (1984) 185; A. Maritan and A. Stella, preprint (1984);
B. Berg et al. / Random surfaces
[2] [31 [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15]
675
T. Eguchi and T. Filk, Phys. Lett. 149B (1984) 381; B. Duplantier, Phys. Lett. 141B (1984) 239 B. Berg and D. Foerster, Phys. Lett. 106B (1981) 323 J. Kuti, Phys. Rev. Lett. 49 (1982) 183 B. Berg, A. Billoire and D. Foerster, Left. Math. Phys. 6 (1982) 293 B. Berg, A. Billoire and D. Foerster, June 1981 (unpublished) J. Kuti and J. Polonyi, Proc. 6th Johns Hopkins workshop, Florence, June 1982, p. 15 ff. M. Karowski, R. Schrader and H. J. Thun, G6ttingen preprint (Feb. 84) C. Arag~o de Carvalfio, S. Caraci01o and J. Fr6hlich, Nucl. Ptiys. B215[FS5] (1983) 209; C. Arag~o de Carvalho and S. Caraciolo, J. de Phys. 44 (1983) 323 A.I. Laskin and D.E. Khmelnitski, JETP (Soy. Phys.) 29 (1969) 1123; E. BrO_in, J. le Guillou and J. Zinn-Justin, Phys. Rev. D8 (1978) 2418 M. Karowski, H.J. Than, W. Helfrich and F. Rys, J. of Phys. A16 (1983) 4073 B. Berg and A. Billoire, Phys. Lett. 297B (1984) 297 T. Eguchi and M. Fukugita, Phys. Lett. llTB (1982) 223; H. Kawai and Y. Okamoto, Phys. Lett. 130B (1983) 415; A. Billoire, D.J. Gross and E. Marinari, Phys. Lett. 139B (1984) 75 T. Sterling and J. Greensite, Phys. Lctt. 121B (1983) 345 K. Binder, in Phase transitions and critical phenomena, vol. 5B, ed. C. Domb and M.S. Green (Academic Press, New York, 1976) B. Baumann and B. Berg, in preparation
M O N T E CARLO M E T I t O D FOR RANDOM SURFACES B. BERG
IL hlstitut f~r Theoretische Physik der Unieersitgtt llamburg, Luruper Chaussee 149, D-2000 llamburg 50 A. BILLOIRE
Service de Physique Thborique, Orme des Merisiers. CEN-SACI~t Y, 91191 Gif-sur- Yeette, Cedex, France D. FOERSTER
Max-Planck hlstitut, Fohringer Ring 6. D8000 Mfinchen 40, West Germany Received 1 June 1984
Previously two of the authors proposed a Monte Carlo method for sampling statistical ensembles of random walks and surfaces with a Boltzmann probabilistic weight. In the present paper we work out the details for several models of random surfaces, defined on d-dimensional hypercubic lattices.
1. Introduction
Random surfaces have attracted a lot of interest [1], because of their relation to relativistic string theory, non-abelian gauge theories and problems in statistical mechanics. Only few analytic results exist and numerical studies may be very useful. In ref. [2] a local stochastic process was proposed, which allows one to generate statistical ensembles of walks and surfaces with a Boltzmann probabilistic weight. If applicable the procedure allows Monte Carlo (MC) simulations of rather large walks and surfaces, because the needed amount of computer memory is only proportional to the length of the walk, respectively the area of the surface. The MC procedure of ref. [2] (MCP2) has a wide range of possible applications. For instance one may carry out a stochastic inversion of large matrices, which are relevant for fermions in lattice QCD. Kuti [3] advocated later the von Neumann-Ulam method for similar purposes. A test is the stochastic computation of the average of y-traces (Tr)L over walks of fixed length L (for thd precise definition of Tl:t. see ref. [4]). In table 1 we compare the exact answer [4] with results as obtained by the MCP 2 [5] and by the von Neumann-Ulam method [6]. The MCP2 is more efficient because
666
B. Berg et al. / Random surfaces TABLE 1 S t o c h a s t i c c a l c u l a t i o n of -f-traces L 4 6 8 10 12 14 16 18
20 22 24
(Tr)~.~ t -8 - 18.53 - 22.66 -25.16 -25.78 - 24.74 - 22.18 - 18.20 - 12.91 -6.36 + 1.41
( T O t . (BBF), = I T E R -8 - 18.52 - 22.70 -25.36 -26.12 - 25.2 - 21.8 - 22 - 16 -5 29
+0 __. 0.01, 4- 0.04, 4- 0.12 4- 0.24 4- 0.6 4- 0.6 4- 4 4- 8 4- 8 + 30
3.25 106 2.91 106 2.55 106 3.03 106 2.71 106 2.45 106 2.23 106 2.04 106 3.24 106 3.01 106
(TOt . (KUTI) - 7.98 4-_0.09 - 18.55 + 0.53 - 22.9 + 1.0 -25+3 -27_+_3 -25 + 14
E x a c t r e s u l t s a r e t a k e n f r o m ref. [4]. ( T r ) L (BIIF) c o r r e s p o n d s to rcf. [51 a n d = I T E R gives tile n u m b e r o f M C P 2 i t e r a t i o n s at single links. ( T r ) L ( K U T I ) h a s b e e n t a k e n f r o m ref. [6] a n d relies o n a b o u t 106 p a t h s .
it allows one to fix the initial and final points of sampled random walks. For another related investigation see ref. [7]. Another relevant field for applications are problems in polymer physics. The MCP 2 has turned out to be useful in a numerical study of self-avoiding random walks in four (and three) dimensions [8]. In good agreement with renormalization group results [9] logarithmic scaling violations of mean field theory were found in four dimensions. Statistical ensembles of polymer gases are studied in ref. [10]. In the present paper we are interested in random surfaces (RS). In the next section we fix the notation and introduce three models of RS. The first two models are tile exact analogs of the two models of free random walks for which the MCP 2 was tested in ref. [2]. T h e third model are self-avoiding random surfaces. For these models we describe in all details the MCP 2 in sect. 3. The (tedious) work of writing tile computer program along these lines has already been carried out in ref. [11], where first numerical results are also reported. Our way of simulating RS differs considerably from other methods for which the interested reader is referred to the literature [12], in spirit related is ref. [13]. The explicit use of a lattice prevents, however, sampling very large surfaces.
2. Models of random surfaces Let us consider a d-dimensional hypercubic lattice. We will introduce three models of random surfaces (RS). By a surface we mean a finite collection of plaquettes, where each plaquette may appear several times, together with a prescription for gluing the plaquettes together.
B. Berg et al. / Random surfaces
667
3
Fig. 1. F.xampleof a spike.
Each plaquette of the surface contains four lattice links. Via each of its links a surface plaquette is either glued together with precisely one other surface plaquette (containing the same link), or the surface link is a boundary link. The boundary (if any) is not allowed to move. We are interested in calculating expectation values with respect to the partition function Z = }-~.e -/~as .
(2.1)
s
The sum is over all connected surfaces with fixed boundary and fixed Euler characteristic. A s is the area ( = number of plaquettes with multiplicity) of surface S. If the b o u n d a r y is empty, the smallest surface of Euler characteristic 2 is made of a doubly occupied plaquette. The smallest surfaces of Euler characteristic 0 (orientable or not) and 1 are made of a four times occupied plaquette suitably connected. So far we have defined our first model. We call spikes the configurations where the surface folds back upon itself, see fig. 1 for an example. A precise definition is that a spike is generated if one (or more) of the four neighbours of some surface plaquette is the same lattice plaquette. Such spikes are allowed in our first model. In our second model spikes are forbidden. According to the random walk language in ref. [2] we call surfaces of the first model bosonic random surfaces (BRS)* and those of the second model fermionic random surface (FRS). There are no really good reasons for this notation. The original motivation was, that random paths with spikes do not contribute in the partition function of a theory of Wilson fermions. " They are called elsewhere (planar) random surfaces. See for instance Durhuus et al. [1].
668
B. Berg et al. / Random surfaces
Our tlfird model are self-avoiding random surfaces: no link of the lattice is contained in more than two plaquettes of the surface. Boundary links are contained in precisely one plaquette of the surface.
3. Monte Carlo procedure for random surfaces In accordance with detailed balance (for a review see [14]) we establish in this section a system of transition probabilities for local shifts. In dimensions d >/4 these shifts allow one to generate all surfaces within each of the models introduced previously. In the limit of infinite statistics the Monte Carlo procedure (MCP2) generates random surfaces (RS) with the weights of the modified partition function Z r'~c = ~ A s e - / ~ ' q .
(3.1)
S
Finally we will optimize the transition probabilities. Our procedure for modifying the surface is the following: (i) We randomly select a plaquette RP of the surface. (ii) We shift the p!aquette RP by one lattice unit in a direction orthogonal to the plane spanned by RP (RP plane). These shifts are restricted in a model-dependent way and precise probabilities will be established later. (iii) The shift takes RP along 4 plaquettes. 0 ~< ~r ~<4 of such plaquettes are occupied by neighbour plaquettes glued to RP (before the shift). These ~r neighbour plaquettes are annihilated and 4 - ~r new plaquettes are created. The surface area changes thus by AA = 4 - 2 N = 0 , +2, +4. RP, old and new plaquettes are then glued together in an obvious way. Note that the above-described modification of the surface does not affect the topology of the surface. 3.1. L O C A L T O P O L O G Y
A somewhat tedious step is to list completely all different local "topological" cases for such a procedure. By means of fig. 2 we fix our notation in a way also convenient for a computer implementation of the MCPv RP denotes the randomly chosen plaquette. Let 0~, ( # = 1 . . . . . d) be the unit vectors along the axis of our hypercubic lattice. The vectors f~ = +0~,,
(i = 1 . . . . . 4 ) ,
(3.2)
are defined to point in the direction of the (glued) neighbour plaquette NPi (i = 1 . . . . . 4). In this way/t~ is uniquely defined for each neighbour plaquette NPv The case of a boundary link is treated like a neighbour plaquette in the RP plane, with f~ pointing away from RP.
B. Berg et a L / Ramlomsurfaces
669
N~
N~
N~ lit
~p
P I.!(0
Fig. 2. Notation for a randomly chosen plaquette.
In the following p ( a A), zlA = + 4, + 2,0 is the probability for an allowed shift (in a given direction), which changes the area by the amount AA. This shift probability (SP) only depends on AA. Further there will be 12 different "zero-shift probabilities" (ZSP): pO. These are probabilities for leaving the surface unchanged. Now we list the complete classification of all local topological cases. Afterwards the probabilities are determined and optimized. (i) All 5 plaquettes (RP, NP,.) are in the same plane: for each i = 1. . . . . 4 we have /t, = p. or p., = v. RP can be shifted in any of the 2 d - 4 directions orthogonal to the RP plane with equal SP P(+4). The probability for leaving the surface unchanged (ZSP) is thus P0~ = 1 - ( 2 d - 4 ) P ( + 4 ) . (ii) One of the neighbour plaquettes NPi is out of the RP plane: #,,, :#/.t and #,, :# v,#,, = / t or #,,=v for i = 2,3,4. Here and in the following % . . . . . 74 is a suitable chosen cyclic permutation of 1. . . . . 4. RP can be shifted in the direction/.t=, with probability P ( + 2 ) and in ( 2 d - 5 ) other directions with equal probability P( + 4). SP: P ( + 4 ) , P ( + 2 ) ;
ZSP: P ( ~
(iii) Two of the neighbour plaquettes NPi are out of the RP plane: #,,l ~: # and #,,, ~= v, and further: (a)
/.t~,,4:#,
bt,,2:#v;
It,,=/.t
or
#,,=v
fori=3,4,
(b)
~,,jv~tt,
/.t,,34:v;
#~,,=/t
or
#,,,=v
fori=2,4.
B. Bcrg ct a L / Ra,ulom surfaces
670
(iii.1) f~., ~f~2 if (a) holds, or f~, g: f~., if (b) holds. SP: P ( + 4 ) , P ( + 2 ) ; (iii.2) ./~, "--L2 if (a)
ZSP: P0ii.l) o = 1-2P(+2)-(2d-6)P(+4).
holds, or f., =.L,
SP: P ( + a ) , P ( 0 ) ;
if (b) holds.
zsP: Pt~i~.2)~_- 1 -e(o)-(2d-5)P(+4)
(iv) Three of the neighbour plaquettes NP~ are out of the RP plane: /t~., 4:/t and /G,, 4: v for i = 1,2,3; p.~, = bt or ~,,, = i:. (iv.l) All L, (i = 1,2,3) are different.
SP: / ' ( + 4 ) , P ( + 2 ) ;
(i~.2)
1-3P(+2)-(2d-7)P(+4).
L=L2,L orL ,L2=L orL,=L,~L,.
sP: P ( + 4 ) , e ( + 2), /'(0) ;
(~v.3)
ZSP: Pc~ =
zsp: F'~i~
1-p(0)-P(+2)-(2d-6)P(+4).
L=L=L.
SP: P ( + 4 ) , P ( - 2 ) ;
ZSP: P~,~.3)o _ 1 - P(-2)-(2d-
5)P(+4).
(v) All neighbour plaquettes NP, are out of the RP plane: /t~ ~ p. and /~, 4: v for i = 1,2,3,4. (v.1) All J] are different (can only happen in d>~ 5).
Sp: P ( + a ) , P ( + 2 ) ;
(v.2) SP:
L,=L 4:L,,L and
P(+4),P(+2),P(O);
(v.3)
or
L,=L ,L:L andL2*L.
ZSP: P ( ~
zsp:/'t~3~~ __ 1 - 2 P ( 0 ) - ( 2 d -
6)P(+4)
L =L =L,§
SP=P(+4),P(+2),P(-2);
(v.5)
L#=L
Pt~
L=L, 4:L=L, or L=L .L=L. sp: e ( +4), /'(o) ;
(v.4)
ZSP:
ZSP: p t O . a ) = l _ e ( _ Z ) _ p ( + 2 ) _ ( 2 d _ 6 ) P ( + 4 ) .
L,=L=L=L. SP: ? ( + 4 ) , P ( - 4 ) ;
ZSP: ?(~
1-e(-4)-(2d-5)e(+4).
We remark that d = 5 is the lowest dimension where the P ( + 4 ) shift of case (v.l) can be realized.
B. Berg et al. / Random surfaces
671
3.2. TRANSITION PROBABILITIES
The transition probabilities P(z~A) are chosen such that a detailed balance [14] is fulfilled:
A'p(+aA) AP(-aA)
= e(A-.A')
P(A'~A)
A' e -/~A'- A) = ~---e A' -pa'4 .
A
(3.3)
There AA = A ' - A . The extra factor A'/A in front of P(+AA)/P(-AA) comes from the random choice of the plaquette on an already existing surface and amounts to actually simulating the partition function (3.1) (not (2.1)). The speed of going through surface space is increased by setting as many ZSP as possible equal to zero. Therefore it is easy to see that the MCP 2 is optimized with et,,
(3.4)
= P ,5> = 0 .
Eqs. (3.3) and (3.4) uniquely determine the transition probabilities P(A A). With c: = e -2p,
(3.5)
we obtain 1 l+c 2 P(0) = ~- 1 + ( 2 d - 5)c 2' l+c
2
(3.6a)
1
P(-2)=
l+c
1+(2d-5)c
P(-4) =
1 1 + ( 2 d - 5)c 2 '
2'
P(+2)=cP(-2),
P ( + 4 ) = c2p(-4).
(3.6b)
(3.6c)
In practical applications the "bottle neck" of the procedure is P ( - 4 ) [11]. We are still left with two problems: (i) Within each model considered we have to show that all surfaces can be generated by our procedure. (ii) For each model we have to specify the shifts which are not allowed. Point (i) is obviously true for our first model (BRS). For FRS the only case which seems to cause trouble is depicted in fig. 3. There are three different ways of gluing the plaquettcs 1, 2, 3 and 4 together: (1,2),(3,4),(a);
(1,3),(2,4),(b);
(1,4),(2,3),(c).
(3.7)
Plaquettes (i, j ) are glued together. At the first look it is not obvious how to reach
672
B. Berg et al. / Random surfaces
i
Fig. 3. Three random surfaces. situation (c). In d >I 4 dimensions this is possible as follows: consider fig. 4, shif plaquette 2 of this figure in the direction (+)~4, then shift plaquette 1 two times ir direction + e 2 , finally shift plaquette 2 back in direction (T-)~ 4. In d--- 3 dimension: this does not work, and not all FRS can be reached with our present rules (Additional rules may, of course, be added.) A shift is rejected whenever it would modify the nature of the surface (e.g generates a spike on a fermionic surface) or has no inverse (according to our rule:
Fig. 4. A special random surface.
B. Berg et al. / Random surfaces
673
/",,,7
C Fig. 5. The simple way to generate a spike.
for surface deformation). Therefore our MCP2 has the character of a mixture between heat bath and Metropolis algorithm [14]. For FRS, one rejects shifts like moving in fig. 5 the plaquette 1 in direction 3. Tiffs would lead to a spike. One subtle point is that spikes may be generated in a way which has no analog in the case of a random walk. Consider again case (3.7) (c). W e have to reject a shift of plaquette 1 in +~2 direction, because a spike would be generated along the dotted link in fig. 3. The situation which would lead to problems with detailed balance is depicted in fig. 6. Plaquettes 2 and 3 are both connected to plaquette 1 but are not connected together whereas they are close by. A shift of plaquette 1 in direction 3 has no inverse and is thus rejected. This rule also applies to bosonic surfaces. For SRS, we have to introduce a "logical" lattice of link variables additionally to reject all shifts, which would touch links already occupied by the surface.
2
( Fig. 6. A problem with detailed balance.
674
B. Berg et aL / Random surfaces
BRS are slightly more subtle. Transitions which resolve a spike are only allowed if tile inverse transition would again create the spike. For instance shifting in fig. 1 one of the two plaquettes drawn in the (1,2) plane upward has to be rejected. Otherwise, as already noted in [2], detailed balance would be violated. In s u m m a r y : BRS and FRS can be simulated with a local upgrading procedure. O n l y i n f o r m a t i o n about nearest a n d next-nearest neighbours of a r a n d o m l y chosen plaquette is required. Introducing a logical lattice allows also to simulate SRS. Different models could be considered by defining new probabilities.
4. Conclusions W e explicitly worked out the ideas of ref. [2] for a non-trivial case. The M C P 2 allows M C simulations of very large r a n d o m surfaces. A major disadvantage is that p r o g r a m m i n g is very tedious. In ref. [11] the c o m p u t e r implementation has been carried out and the first numerical results obtained: (i). BRS d o not allow an efficient simulation because of a very low acceptance rate for p r o p o s e d shifts. (ii) F R S allow, however, precise numerical results and for SRS the efficiency is expected to increase even further. (iii) F u r t h e r applications of the method will be the subject of future work [15]. W e would like to thank E. Marinari, B. Pearson, J. B. Zuber and P. Windey for their stimulating interest.
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