Zeitschrift for P h y s i k B

Z. PhysikB25, 391-399 (1976)

© by Springer-Verlag 1976

Exact Distribution of Cluster Size and Perimeter for Two-Dimensional Percolation D. Stauffer Fachrichtung Theoretische Physik, Universit~it Saarbriicken, West Germany Received July 16, 1976 Exact series expansion data of Sykes et al. are used to calculate the average number c, and perimeter s, of clusters of size n__<20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation threshold Pc we find a sharply peaked distribution of perimeters s, with mean ( s , ) = ((1 -Pc)/Pc)n + O(n ~) and width (s 2 ) - ( s , ) 2 oc n 1"6 where a---1//~=0.39. This "perimeter" (s,) should not be interpreted as a cluster "surface" in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbers c, with n is consistent with the postulated asymmetry about Pc: log c, o c - n ; for n ~ o o with ~-~1 for PPc.

I. Introduction In the site percolation problem [1] one assumes that a random fraction p of lattice sites is occupied whereas the remaining fraction 1 - p is empty. Then "n-clusters" are defined as sets of nearest-neighbor-connected occupied sites. The perimeter s of a cluster is the number of empty sites which are nearest neighbors to occupied sites belonging to that cluster. We normalize the number c, of n-clusters as the number per occupied site; thus if no infinite cluster is present, i.e. for p below the percolation threshold Po each occupied site belongs to one cluster (if also single sites are regarded as clusters of size n = 1), and we have ~n-c.=l.

(1)

n=l

The number of clusters c. is given by c . = ~ c.~ where s

c.~ is the number (per occupied site) of clusters with size n and perimeter s. If g(n, s) counts the number of geometrically different configurations for such a cluster, then [-2]

c,,= g(n, s) p"- l ( 1 - p)L

(2)

The number of clusters c,, the perimeter s and other quantities measuring a cluster "surface" have received some recent attention [3-10]. The present work aims

to clarify some confusion, evident from these papers, about the behavior of the c,s and its physical interpretation ("surfaces"); in addition we will give numerical results for cluster numbers and perimeters, using the exact results ("series expansions") for the g(n, s), n < 20, given recently by Sykes et al. [11]. The next section discusses why the perimeter of a cluster should not be interpreted as a cluster surface in the usual meaning of the word, and why a finite ratio of perimeter s to size n for large clusters is nothing peculiar. Section III gives exact numerical results for the distribution of perimeters s around their mean value (s,) for intermediate n and tries to extrapolate the large-n behavior from these data. Similarly, Section IV tries to extrapolate from the exact c, the asymptotic cluster numbers for large n, e.g. in connection with the universality hypothesis [12] near the percolation threshold. Our results are summarized in Section V.

II..Surfaces, Perimeters, and Energies This section questions the identification [7, 10] of a cluster perimeter or cluster energy with a cluster surface.

392

D. Stauffer:Exact Distribution of Cluster Size

At zero temperature, in a three-dimensional ferromagnetic nearest-neighbor Ising spin ½ model, a roughly cylindrical " u p " domain consists of 100~o up-spins and no down spins, and the energy in the interior is the ground state energy which we normalize to zero. Thus broken bonds (antiparallel nearestneighbor spins) exist only at the boundary of the domain where it is surrounded by " d o w n " spins. The number of broken bonds determines thus the energy of the domain, and obviously this energy can be interpreted as a measure of the surface area of the domain. Alternatively, one can also define the perimeter of the domain as a measure of the domain surface area; qualitatively there should be little difference between energy and perimeter. In short, at T = 0 domains have surface areas proportional to their energy or perimeter; and therefore cluster energy or perimeter are then a good definition of the cluster "surface". (Such Ising clusters are defined as sets of up-spins connected by nearest-neighbor bonds.) But at all finite temperatures the situation is different since now internal broken bonds appear. A macroscopic up-domain in the Ising ferromagnet at temperatures between zero and the Curie temperature no longer has 100 ~o of its spins up but only, say, 99 ~o, whereas one spin in hundred points downward inside the domain. Each of these down spins, if isolated from other down spins, produces six broken bonds (i.e. 6 energy units) in a simple cubic lattice and adds one unit to the perimeter. (Here the perimeter is the number of down spins which are nearest neighbors to up-spins of the cluster or domain.) Thus one now has an internal energy and an internal part of the perimeter. In addition, at the outer boundary of the domain, nearly every bond is broken, thus contributing to the external part of the domain energy and perimeter. If the domain is sufficiently large, then the energy or perimeter in the interior are much larger than the additional energy or perimeter contributions of the boundary; and thus the total energy and perimeter will be, for large n, proportional to the size n of the domain: lim (s~>/n = a.

(3)

n~o

Thus large Ising clusters will have a perimeter or energy proportional to the number of spins. Similarly, large spherical raindrops of radius r have at room temperature a density in the interior which is lower than the maximum density of water at atmospheric pressure. If we approximate for clarity these raindrops by a lattice gas model, we have again at all finite temperatures below the critical temperature a certain fraction of interior bonds broken, giving thus a finite difference per molecule between the actual energy and the ground state energy. Again relation (3) is valid

(where the perimeter is now the number of empty lattice sites surrounding as nearest neighbors the water molecules of the raindrop). We see that the proportionality (3) between perimeter and size is not indicative of any peculiar property ("ramifiedness" [13]) of raindrops, domains, and clusters which would distinguish them from more spherical objects of nature. It merely states that for sufficiently large samples the interior properties are more important than the surface contributions. By now it should be obvious that the perimeter or the number of broken bonds is by itself not an appropriate measure of the surface area in the usual meaning of the word, since it includes all "interior surfaces". A spherical water droplet of radius r has, according to widespread convention, the surface area 4n r 2 and not a surface area varying as r 3. One does not usually count as surfaces of liquids the surfaces of the atomic nuclei and the "holes" the liquid may have in its interior due to thermal fluctuations. Of course one could define "surface" as the total perimeter s, or the total number of broken bonds; but with such a definition nearly every large object of nature would have a surface proportional to its volume. It seems more practical, instead, to define "surface" such that for usual objects (like raindrops or Ising domains below T~) it increases weaker than n with size n. For example, if (s,)=an+An~+

...,

a
(4)

one could identify A n* as a measure of the surface area. Or, more conveniently for some computer applications, the surface area could be measured by counting the number of broken bonds or missing neighbors in a connected layer surrounding the droplet (domain, cluster), i.e. the surface is identified with the e x t e r n a l energy or perimeter. Then for large Ising domain or raindrops one has surfaces proportional to the square of the radius, as it should be, for all temperatures below T~. The latter definition by external surfaces was used in [3] to evaluate the surface area of Monte-Carlo simulated Ising clusters below T~, which then was found to increase with n asymptotically as n ~d-~)/a in d dimensions. Domb et al. [7] on the other hand identified the total cluster energy as the cluster surface, found for large n this surface to vary (roughly) as n, and concluded that the clusters near T~are "ramified" As explained above, we regard this definition [7] as misleading; it is erroneous to compare, as done in [7], directly the two "surfaces" of [7] and [3]. In the percolation problem, the situation is quite analogous, only that the region of "finite temperatures below T~" now corresponds to concentrations p between the percolation threshold Pc and unity. There again, very large clusters can be expected to have a

D. Stauffer:Exact Distribution of Cluster Size

393

perimeter varying as the cluster size n, as in (3), even if these clusters are rather spherical (raindrop like) if viewed from the outside. The interior holes in the clusters give for very large n the dominating contribution to the perimeter. But such a perimeter proportional to the cluster size n does not indicate anything peculiar and may just correspond to the trivial "raindrop" example in the lattice gas. Domb [13] defined percolation clusters fulfilling relation (3) as "ramified"; and Leath [101 (who identifies the total perimeter with the cluster surface) uses the concept of" fractal dimensions" to describe clusters obeying (3). But as we saw above, with such a definition of" surface" nearly every large object in nature would be ramified with fractal dimensions. Therefore we avoid here such misleading definitions of "surface" and regard the ratio a in (3) simply as a measure of the internal disorder of large clusters, not unlike the energy in Ising models. (In fact, the analogy between (s,)/n in percolation theory at Pc and the energy of the Ising model at T~ seems to be quite good numerically [5].) In this sense, Duff and Canella [4] and Sur et al. [9] evaluated the number of broken bonds in percolation theory and avoided correctly its interpretation as a cluster surface. For small clusters, both in percolation theory and the Ising model, the situation is different since then internal broken bonds are less relevant or impossible. One cannot punch a hole into the interior of a cluster with only 5 sites on a simple cubic lattice. Thus small clusters are fully "compressed", i.e. have no holes in their interior. Their volume V, thus is given by

V,=n,

n~l,

(5a)

where the volume may be defined as the total number of sites within the smallest connected surface layer surrounding the whole cluster. For intermediate sizes, we expect near the percolation threshold the clusters to fill their interior with an increasing fraction of holes such that the cluster volume V, increases quicker then the number n of connected cluster sites. Thus V, counts not only the n cluster sites but also the holes inside the cluster, the small clusters within these holes, etc. Scaling suggests [3, 14, 15] for both percolation and Ising clusters:

V~ocn1+~/~,

l~n
(5b)

In this size range the shape of the cluster is getting more and more complicated the larger the cluster is. (In (5b), 6 is the "critical isotherm" exponent at p~, n~+~/~-n ~'2 in three dimensions and -~n~°5 in two dimensions; n~ is the "typical cluster size" oc ] p - p~[- ~~ and corresponds to clusters with a radius of the order of the correlation length ~;/~ is the exponent for the

percolation probability P o c ( p - p J , i.e. for the fraction of lattice sites belonging to the infinite percolating cluster; the socalled mean cluster size ~ n 2 c, diverges as [p-pc[ -~ with ~=fi(~-1).) Finally for large n, when the ratio n/V, of cluster sites to total sites within the cluster volume has decayed down to the percolation probability P, we expect a crossover [3, 15] to a different regime where

V, = n/P,

n >>n~

(5 c)

since very large clusters should be locally not much different from the infinite percolating network in their ratio of cluster sites to lattice sites per unit volume. If one likes one can call the regions of n corresponding to the predictions (5a, b, c) the compact regime, the hydra regime, and the Swiss cheese regime. At present no Monte Carlo confirmation of these predictions exists for V,; Leath [10] found in two dimensions for the mean square cluster radius: (RZ(n))ocn 1"1 and nL2 for p somewhat below Pc. (It is not clear at present what to expect below the percolation threshold for cluster, volumes and surfaces and whether crossover effects similar to (5a, b, c) exist also for the higher order terms in the perimeter versus n relation (4).) This section was an expanded version of arguments given shortly in [5] and in the references cited there (e.g. [3]). (A more precise discussion of Ising model clusters is given in [15]; it seems that some of the conceptual problems connected with Ising clusters discussed there disappear for percolation clusters.) In short, we concluded that perimeters and energies should not be confused with surface areas if a spherical raindrop at room temperature is supposed to have a surface area of 4 n (radius) 2.

IIl. Distribution of Cluster Perimeters

This section gives the shape of the probability distribution for the cluster perimeters s,, the resulting mean values (s,), and the width of the peak at this mean value. For the average perimeter (s,) the ansatz (4) was already proposed in [5] where a=l/fl6 (=0.39 in two dimensions) near the percolation threshold. Moreover, [51 suggested at the percolation threshold P=Pc: lim (s,)/n = a = (1 - Pc)/Pc.

(6)

Leath [101 later gave a more rigorous derivation of this relation. How can one test it ? Sykes et al. [11] gave exact expressions for the socalled perimeter polynomials [2] D, (q) in the triangular lattice (which we denote by TR; n=<14), the square

394

D. Stauffer: Exact Distribution of Cluster Size

lattice (SQ; n < 17), and the honeycomb lattice (HC; n<20):

6

c.=p'-~D,=p"

5

X~g(n,s)q ~,

q=l-p.

(7)

s

•

TR

•

As in (2) the g(n, s) are the number of different con4 figurations for a cluster with n sites and perimeter s. ~ 3 Thus from the expressions for D, = D. (q) one can read off directly the g(n, s) and calculate for any p the 2 probability distribution m, (s) for the perimeter:

m, (s) = c, jc, = g (n, s) p"- 1(1- p? / ~ g (n, s) p'- J(1 - p)* s

=g(n, s)(1 -p)~/~ g(n, s)(1 - p ? .

(8)

O(~

The first and second moment of this perimeter distribution function m. (s) give
(9a)

s


(9 b)

s

Since ~, m,(s)= 1 by definition, the width A, is deters mined by (A,) z =

-

(9 c)

Evaluation of finite polynomials thus gives the desired quantities for small and intermediate cluster sizes; by graphical extrapolation we try to find the asymptotic behavior for large clusters. (Total computer time on a TR 440 was 2 rain.) Figure la shows three examples for the perimeter distribution function m,(s), giving narrow peaks around which are roughly but not exactly independent of p. The width A, of these peaks at p =Pc is plotted in Figure l b, and we see that for large n:

A, ocn °,, 0.3 ¸

co_~0.8. An

e•

• •

0.2

•

m Q

0.1

•. 0 0.2

.

,

•I

•

15

, 20 O0

il

o i

~'5-

2o

~

25 li

30

35

1 SQ

•

•

0.5

2d---~o----~'5"

s

el

mo.1

15

20

2

s

•

o°~°~ •

0.2

010

~ 0.50

0.'75

' " 1.00

25

30

35

b

Fig. 2. Variation of C%)/n versus n " - t as a test of predictions (4, 6). The straight lines are fitted to obey (6) at the intercept

We are not aware of relations proposed between this new exponent co and the standard critical exponents fi, Y, and 8 of percolation theory. Figure 2 shows that the calculated mean values (s,) are compatible with our prediction (6). The percolation thresholds Pc are taken from Sykes et al. [ 11] : Pc(TR) = 1/2, Pc(SQ) = 0.593, Pc(HC) = 0.698, with the critical exponents taken as fi-0.138, y =2.43, cr= 1/fi 8 = 0.39, z ---2 + 1/8 = 2.054. The plots in Figure 2 show a systematic curvature; if one neglects that curvature and extrapolates to n = o• simply with a tangent to the last available data points, the resulting values for a in (3) are lower than the prediction (6) by a few percent. For the triangular lattice, where (6) predicts < s . ) / n ~ l exactly, we plotted ( s . ) / n - 1 double-logarithmically against n and found this difference to vary as n -k, 2_~0.7 for l_
5

HC,

Q

0.1 0

(10)

a

•

' 0.25

13o--1

8

s

/.
<7;"

Fig. 1. a Perimeter distribution function m, (s) for triangular (TR; n = 14), square (SQ; n = 17) and honeycomb (HC; n =20) lattices. The large dots correspond to P=Pc where the arrow indicates ( s , ) and the horizontal line gives 2A,; the small dots correspond to p=0.1 b Log-log plo{ of the width A2 ~ ( s ~ ) - ( s , ) 2 against n. The straight lines have slopes 1.58 (TR), 1.61 (SQ) and 1.67 (HC)

D. Stauffer: Exact Distribution of Cluster Size

395

Leath [ 10], who denotes the perimeter s as " b o u n d a r y " b, also mentiones perimeter distributions but does not give them explicitly. His function m is not proportional to a perimeter distribution function; therefore his Monte-Carlo determined mean values and width of his function m are not ( s , ) and An. The exponent co of the true width An does not give the exponents /~ and 7, contrary to the impression given in [10]. In conclusion, our prediction (6) was confirmed within a few percent; and the next-order term in the {s,) versus n relation has been estimated for the first time and found to be consistent with prediction (4) and o-= 1//~ 6.

7.0

6.5

HC

•

,o." • I / e 7

6.o ] d

r...'P

5.5

c2 d 5.oi 4.5

./

4.0

0.1

0.2

0.3

0.4

0.5

0.6

,

0.2

r

0.3

,

0.4-

,

0.5

,

0.6

Summation of the c,~ over all perimeters s gives the total number cn of n-clusters, which we evaluate here from the data of Sykes et al. [ i l l and which we then compare with earlier studies. For large clusters close to the percolation threshold Pc, one can apply scaling ideas [6, 13, 15-17] to the cluster size distribution c n = ~ cn~ and postulate [6] :

ct 0.03

v

~_ 0.02 0.01 a

O0

0.1

cn=q0 n-~f(e n~),

(11a)

8=ql(P~-p).

(lib)

Here qo and q~ are two positive constants and a = 1//~(5, T = 2 + 1/6 is the usual notation for critical exponents and cluster models [16]. At P=Pc, c,=qon-~f(O) decays as n -~ for not too small n [6, 8]; Figure 3a confirms again this simple power law decay. The function f = f ( x ) in (11 a) has been assumed [6] as analytic and universal, in analogy with two-scale-factor universality for usual critical phenomena [12]. ("Universal" quantities are dimensional invariants which are independent of the lattice type but depend on • its dimensionality. Also o- and r should be universal.) Below and at the percolation threshold Pc, the condition (1) must be fulfilled thus giving in the scaling region p ~ p / , n ~ oo (thus ~ > 0) for 0
0 : E n. c°m)- E n ncoc)= Z n{c.Co)-

Fig. 3 a and b. Variation of cluster numbers versus n - ~ = n-o.39. The small dots represent exact series results [11] with 4_
vanish [6]: oo

S x - l - ~ ( f ( x ) - f ( O ) ) dx=O, o

P
(12a)

a relation which goes back basically to Reatto [18] (in his model, the spontaneous magnetization must vanish for the paramagnetic side, leading to the equivalent of (12a)). Partial integration gives the alternative formulation

~ x - ~ f ' ( x ) dx=O,

P
(12b)

0

n oo

= qo ~ nl - ~(f(8 n~) - / ( 0 ) ) -~ qo S n l n

r,,

n-o-

s

n

n-ff

0.04

IV. Distribution of Cluster Sizes

n

l/

--

~(f(e n ~) - / ( 0 ) ) dn

0 oo

= (qo/a) 8(~- 2)/. £ x - 1+ (2-~)/. ( f ( x ) - f(O)) dx 0 oo

= (7 + fl) qo 8¢ ~ x - i - ~( f ( x ) - - f ( 0 ) ) dx. 0

Since for 0 < / ~ < 1 this e&term cannot be cancelled by any variation of q~ or qo analytic in p, it has to

It is evident from (12) that the function f ( x ) = f(e n ~) oc n ~c, cannot be a monotonic function with a derivative f'(x) never changing its sign. Instead, f(x) first increases from its value f(0) with increasing x, reaches a m a x i m u m at some Xma×, and then decreases to zero [6]. Thus the cluster numbers cn(p) as function of p at fixed n reach a m a x i m u m at some Pma~= Pc -- const, n- ~; and at fixed p
396

D. Stuaffer: Exact Distribution of Cluster Size

gated in [6] by analysis of Monte Carlo data and roughly agreed with these scaling predictions for p and p~. The universality assumption for the function f means that also the ratio f ( X = X m a x ) / f ( x = O ) is universal, i.e. lattice independent. This ratio equals, for large clusters, the ratio c.(p=pma~)/c.(p=pc ) which thus should approach for n ~oo a universal number independent of the type of lattice. Therefore we plotted c. as a function of p, found its maximum value c.(Pm~.), normalized it by the value of c. (p~) right at the percolation threshold, and plotted in Figure 3b the resulting ratios c.(Pm.x)/C.(p). The extrapolations are consistent with the universality idea and give for two dimensions: f(Xm,×)/f(O) = 4.5 _+0.2.

(13)

In fact, universality now can be used to determine quite accurately the percolation threshold Pc. For only p~(YR) is known exactly as 1/2 whereas p~(SQ)= 0.593+0.002 and pc(HC)=0.698±0.003 are merely numerical estimates from [11]. We may adjust p~(SQ) and pc(HC) such that the three curves TR, SQ, and HC in Figure3b overlap, and then find roughly Pc (SQ) = 0.591 +_0.001, Pc(HC) = 0.695 _+0.001, estimates which are compatible with but perhaps more accurate than those of Sykes et al. [11]. (The Monte Carlo analysis in [6] was not accurate enough for a better determination of the ratio (13).) Our results (12) and (13) are also a practical test for phenomenological assumptions about the cluster size distribution % For example, Leath [10] arrived in the scaling region n ~ o% p ~ Pc, at a result compatible with the scaling assumption (11) but with f ( x ) = exp(-x2), which violates requirement (12) and gives unity for the ratio in (13). (Actually his suggestion can be improved by equating his two exponents q5 and with our a; then f ( x ) = e x p ( - ( x - c o n s t ) 2 ) , and this assumption works better.) Another universal quantity R arises from a combination [12] of universal amplitudes B, C and D. Let C be defined by the "mean cluster size" slightly below Pc:

y~ ~2 e.= C(pc-p) -~.

P = 1 - ~ n. c , = B ( p - p y ; n

and D finally determines the "critical isotherm" p = Pc: at P=Pc.

n

From (5) and (11) we then find cao

B = (7 +fl) qo q~ ~ ( - x ) - X - ~ ( f ( O ) - f ( x ) ) 0

TR SQ HC

B

C

q 0 P~

P~

R

1.56 1.53 1.53

0.128 0.147 0.140

0.036 0.039 0.043

0.500 0.593 0.698

19.1 20.2 21.4

similarly to the derivation of (12); and analogously we get oo

C = (y + fi) qo q ; r S x -~ + r f ( x ) d x o

and D - 1/~= _ q o f (O) f ( -

dx

1/6)

with the Gamma function F. Since the function f and the exponents 13, c5 and 7 =13 ( 8 - 1 ) are assumed to be universal, the combination (14a)

R - - CDB '~-I

is universal since the two lattice-dependent parameters q0 and ql have cancelled out. Simpler for our purposes, the combination R' = C 1/a qo i B 1-1/a o~R1/a

(14b)

also should be universal. Numerical estimates are given in the table and are consistent with universality. The spread there of about 10 % can be attributed to the inaccuracy of the extrapolation in Figure 3a and for SQ and HC also to the inaccuracy in Pc. The significance of this universality test in the table is limited, however, since the variation of qo---0.07 and of the other factors entering (14b) is very weak. We gave this detailed presentation of more or less well known universality concepts [12] since no other amplitude-universality tests are given yet for percolation and since we want to encourage such tests. The cluster numbers c. have been assumed [5, 6, 10, 13, 17, 19, 20] to decay exponentially for large n if p ~=Pc: log(c,)< - n ~,

B is the amplitude of the percolation probability P:

1 - ~. n • c, e - " U = D - ~ / O H ~/a

Table 1. Test of universality. The quantity R' of (14b) should be the same for the three lattices TR, SQ, and HC. Values for p~, B, and C are taken from [11] whereas qo=n~c,, for large n is extrapolated from Figure 3a; 6 = 18.6

n~oo.

(15)

In the critical region p-~ Pc, this relation can be expected to be valid only for n>>lp-pcI -aa or lel n~> 1, and this size range is outside the presently investigated n < 20 region. Figure 4 gives an example of c, as a function of p where c, decays by many orders of magnitude; but that asymptotic-decay region is far away from Pc and thus not in the scaling regime. Thus an accurate determination of the scaling function f is still missing and cannot be given from the present analysis. Nevertheless, Monte Carlo results [6, 20] suggested for the asymptotic decay (15) that ~ ~ o--~0.4

D. Stauffer: Exact Distribution of Cluster Size

397

e•e

-1

10 •

5

•

-2

\

3

",,,,

\\

0.005 g

-6 -7

0.5

0.002

0.2

0.001

0.1

0.0005

1

0.0002

0.1 01.2 01,3 01.4 015 016 01.7 01.8 01.9 1LO

0.0001

P

Fig. 4. Variation of pn~c. (logarithmic scale) with p for n = 17 in the square lattice. The rise in the left half is somewhat steeper than the decay in the right half of the parabola-like curve. This figure gives a rough impression of the shape of the scaling function f(x) in ( l l a)

~(p
(16a)

in agreement with earlier Monte Carlo conclusions [20]. The same data for p = 0 . 7 5 are replotted in Figure 5 b in three different ways: As a function of n, as a function of l/n, and as a function of n a = n °'39. The middle choice, with 1/~ as variable, clearly is best and gives a straight line for all n: ~(p>p~)~½.

(16b)

This result, based on data far away from the percolation threshold (p-p~=0.25), is somewhat different from the Monte Carlo conclusion [6, 19] ~ (p > p~) -~ a-~ 0.4,

0.02

\~\o\

TR

0.01

0.00005

0.005

0.002

0 °00002

a

slightly above the percolation threshold, and ~-~i below p~. (Here the first result, c, ocn-~exp(-I~l n~), turned out [-6, 19] to fit the Monte Carlo data over the whole available range of - e n ~, not only for large - ~ n~; below pC, on the other hand, the maximum at ~n~=Xm,~ discussed above makes (15) invalid for small and intermediate e n °, and we have a simple asymptotic decay -.~exp( . . . . n) only for large en~.) By taking the extreme concentrations p=0.1 and p = 0.75 in the triangular lattice, we are clearly outside the scaling region near p~= 1/2 but now even for n < 20 we can study the asymptotic decay of c, with the present exact results. Figure 5a gives, roughly, log (n~c,) as a function of n, and we see a straight line decay for n > 5 at p=0.1 but a strong curvature at p=0.75; thus ~ ( p > p ~ ) < l but

0.05

o7 \

Pc

-8 -90

o.o

\ 0

I

I

5

10

]

t

0.005 0.002 -~

\

i

,,, I

\

~

"<

"-.,

•

2

15

n

,

n°.

"

\

0.001 0,0005 0.0002 0.0001

%. %

0.00005 0

5

10

,.

15 n

Fig. 5a and h. Logarithmic plot of the normalized cluster numbers

c,(p)/co(pc ) in the triangular lattice. The dots are the actual data, connected by lines for identification purpose only. a Variation with n for p=0.1, 0.25, 0.5 (--Pc), and 0.75, suggesting for large n a decay with ~(pp~)p~)-~ 1/2 even for small n

which was based on data near Pc: 0 < p - P c <0.05. Near Pc we cannot have the scaling function f = f(x) = f(e n") to vary as exp ( - Ix 1'/29 = exp ( - le [1/2,1/~) for small - x >0, for that assumption would violate the analyticity requirement for f(x) at x =0. But is

398

D. Stauffer: Exact Distribution of Cluster Size

is possible that at large x a transition occurs in the scaling function f from f = e x = e-Ixl for small and intermediate Ixl to f o c e x p ( - I x l 1/2~) for very large Ix], similar to the crossover between (5b) and (5c) for the cluster volumes above the percolation threshold. Such a crossover indeed was postulated by Binder [15] for clusters in the Ising model at zero magnetic field and finite temperatures; and Equation (16) of Kretschmer et al. [15] gives for these Ising clusters: ~=1 on the paramagnetic side and ~ = ( d - 1 ) / d on the ferromagnetic side in d dimensions. These Ising results are fully analogous to our result (16) here. From the approximation f(x)=eX---e -Ixl in the triangular lattice slightly above Pc, together with the q0 and C from our table, we find a "ferromagnetic susceptibility amplitude" C' =- ( p - pc)~ ~ n 2 c, of about n

0.0013, giving a universal ratio C/C',,~ 102, significantly larger than the ratio near 2 found by Sykes et al. [-The factor ql-~ 8.6 was determined from the Monte Carlo analysis [,19], and the sum ~ n 2 c, was evaluated as an integral (gamma function).] Apparently series expansion methods for the "susceptibility" in the "ferromagnetic region" p > Pc give less reliable extrapolations than Monte Carlo studies. Thus, in general, the exact expressions for the c, at small and intermediate n roughly confirm away from Pc the conclusions drawn earlier from Monte Carlo results closer to Pc [-6, 20].

width A , / ( s , ) thus vanishes as n-0.2. A physical interpretation of this behavior remains to be given; the width exponent is not directly related to the critical exponents p and ~, contrary to the impression from [,10]. 3) A universal (i.e. lattice-independent) ratio (13) for the cluster size distribution was found, which now allows quite accurate determinations for two-dimensional percolation thresholds. 4) Asymptotically the cluster numbers cn decay as exp( . . . . n) below and as exp( . . . . 1~) above the percolation threshold for n ~ oo. Although the main aim of these studies was a better understanding of cluster models for collective phenomena [,15, 16], technical applications might include the growth of metal vapor monolayers condensing on alkali halide surfaces, where the adsorption sites are the perimeter sites of clusters adsorbed already before [21]. Future Monte Carlo studies e.g. could study the form of the scaling function f ( x ) (roughly similar to Fig. 4); or with more accurate numbers c, at P=Pc for 102 < n < 103 one could determine for the first time a correction-to-scaling exponent in percolation theory. We thank Profs. K. Binder, C. Domb and P. Leath for discussion, critique, and information, and Prof. Binder also for a careful reading of the manuscript.

References V. Discussion

This paper first gave a qualitative discussion of how to define cluster surfaces, and it pointed out, in agreement with Duff and Canella [4] and Sur et al. [9] but in contrast to Domb et al. [7] and Leath [10] that perimeters and energies Should not be regarded as measures of a surface since they include internal surfaces. Very large clusters have quite necessarily a perimeter or energy proportional to the cluster volume but that property alone does not indicate that their overall shape is much different from a usual raindrop at room temperature. Only if also the true "external" surface varies as the cluster volume, then a peculiar cluster property would be found. Secondly, we analyzed exact two-dimensional "series expansion" results of Sykes et al. [-11] in order to study the distribution of sizes and perimeters; and we found the following main results: 1) The predictions (4, 6) of [-5] for the average cluster perimeter are confirmed. 2) The width of the peak in the perimeter distribution function increases as n °'s with cluster size; the relative

1. Kirkpatrick, S.: Rev. Mod. Phys. 45, 574 (1973) Essam, J.W., in: Phase Transitions and Critical Phenomena, edited by Domb, C., and Green, M.S., Vol. II, chapter 6. New York and London: Academic Press 1972 For a recent literature review on scaling theory of percolation see Stauffer, D., in: Second International Symposium on Amorphous Magnetism, to be edited by R.A. Levy, Rensselaer Polytechnic Institute, Troy, NY, USA, August 1976 2. de Gennes, P.G., Lafore, P., Millot, J. P.: J. Phys. Chem. Sol. 11, 105 (1959) 3. Binder, K., Stauffer, D.: J. Statist. Phys. 6, 49 (1972) 4. Duff, K.J., Canella, V.: AIP Conf. Proc. 10, 541 (1973) 5. Stauffer, D.: J. Phys. C8, L172 (1975) 6. Stauffer, D.: Phys. Rev. Lett. 35, 394 (1975) 7. Domb, C., Schneider, T., Stoll, E., J. Phys. A8, L90 (1975) Stoll, E.: Priv. Comm. 8. Quinn, G.D., Bishop, G.H., Harrison, R.J.: J. Phys. A8, L9 (1976) 9. Sur, A., Lebowitz, J.L., Marro, M., Kalos, M.H., Kirkpatrick, S.: J. Statist. Phys., to be published 10. Leath, P.L.: Bull. Am. Phys. Soc. 21, 386 (1976) and Phys. Rev. Lett. 36, 921 (1976) and Phys. Rev. B 14. to be published 11. Sykes, M.F., Gaunt, D.S., Glen, M.: J. Phys. A9, 87, 97, 715, 725 (1976) 12. Betts, D.D., Guttmann, A.J., Joyce, G.S.: J. Phys. C4, 1994 (1971) Aharony, A., Hohenberg, P.C.: Phys. Rev. B 13, 3081 (1976) See also [19]

D. Stauffer: Exact Distribution of Cluster Size 13. Domb, C.: J. Phys. C7, 2677 (1974) 14. Stauffer, D.: Z. Physik B22, 161 (1975) 15. Binder, K.: Ann. Phys. 98, 390 (1976) Kretschmer, R., Binder, K., Stauffer, D.: J. Statist. Phys. 15, 267 (1976) 16. Fisher, M.E.: Physics 3, 255 (1967) 17. Essam, J.W., Gwilym, K.M.: J. Phys. C4, L228 (1971) 18. Reatto, L.: Phys. Lett. 33A, 519 (1970) Reatto, L., Rastelli, E.: J. Phys. C5, 2785 (1972) 19. Stauffer, D.: J. Chem. Soc. Faraday II, 72, 1354 (1976) 20. Bakri, M.M., Stauffer, D.: Phys. Rev. B, to be published Mtiller-Krumbhaar, H., Stoll, E.P.: J. Chem. Phys., to be published

Note Added in Proof. A universality test more complete and accurate than Table 1 was made by J. Marro, preprint. A preprint ofC. Domb contains an analysis similar to our Figure 2. Our Figure 1 b gives A,2 and not A,.

399 21. Prutton: Surface Physics, Clarendon Press: Oxford 1975; p. 95 22. Dean, P, Bird, M.F.: National Physical Laboratory Report No. MA 61, Teddington, England, 1966 (unpublished)

D. Stauffer Fachrichtung 11.1 Theoretische Physik Universit~it des Saarlandes im Stadtwald D-6600 Saarbriicken 11 Federal Republic of Germany