Zeitschrift for P h y s i k B
Z. Physik B 28, 47-50 (1977)
© by Springer-Verlag 197:/
Percolation Cluster Sizes and Perimeters in Three Dimensions A. Flammang Fachrichtung 12.2 Elektrotechnik, Universit~t des Saarlandes, Saarbrticken, West Germany Received March 15, 1977 From exact perimeter polynomials of Sykes et al in d = 3 dimensions we determine the average perimeter ( s , ) of clusters, the width of the distribution about the average value, and the number c, of clusters containing n occupied sites each. The exponent ~, defined through l o g ( c , ) o c - ne for large n, is found to be consistent with the predictions ((p
P c ) = ( d - 1)/d. I. Introduction
II. Cluster Perimeters
"Clusters" in percolation theory are sets of occupied lattice sites connected directly or indirectly via nearest-neighbor bonds. In the site percolation problem each lattice site is randomly occupied or empty with probabilities p and 1 - p respectively. An ncluster is a cluster to which n occupied sites belong. For concentrations p above some percolation threshold Pc besides many finite clusters also an infinite cluster appears in an infinite lattice. Reviews of percolation theory and its applications were given e.g. in
The perimeter s, of an n-cluster is the number of empty sites which are nearest neighbors to at least one of the n cluster sites. Clusters of the same size n can have different perimeters, and thus we calculate the average perimeter
[i].
[6]
The phase transition behavior near Pc is governed by very large clusters, and the average number c, of nclusters is not yet known exactly for n above 20. But for smaller sizes n, exact "series expansion" expressions ("perimeter polynomials") were given for two [2], three [3] and more [4] dimensions. For dimensionality d = 2, these expressions were analyzed before [5] in terms of cluster properties; but of course the main experimental interest is in threedimensional percolation. Thus we analyze here the perimeter polynomials for d = 3 from [3]. Our methods are the same as in [5], and thus we present here only the results. Section II discusses cluster "perimeters", Section III gives cluster numbers, and the concluding Section IV compares the two- and threedimensional results. An appendix lists cluster numbers c, for the square and simple cubic lattices.
( s , ) =n-(1 -pc)/pc+O(n a)
( n ~ oo);
(1)
the intercept in this figure was taken as (1-P~)/Pc, and then a straight line is fitted on the data points. The data seem to be consistent with this fit to (1) for large n. Figure 2a gives an example for the probability distribution of perimeters about their average value ( s , ) ; and Figure 2b shows the squared width A2 of this distribution. The data can be fitted for large n in the diamond lattice to a power law:
A, ocn °~,
co-~ 0.83.
(2)
The data from the other three lattices are less good and give somewhat higher apparent exponents o~, as Figure2b shows. (We used, from [3], for the percolation thresholds: pc(D)= 0.428, pc(SC)= 0.310, pc(BCC)
A. Flammang: Percolation Cluster Sizes and Perimetersin Three Dimensions
48 n 12-
FCC
o .
I
I
0,25
0,5
Fig. 1. Plot of the ratio of average perimeter and cluster size against n~-l~-n -w2. The solid lines show a fit on prediction (1)
I
n~-1
0,75
-1000
m
-100
,.25
,4 /~/i
...... --
1.20
p=0.1 p = 0./,28
~,15fi 1i i.I0 1/
I~
,o /j Ii - -
'25
~30
J35
I.s
2
4
6
8
10 12 1/.
n
Fig. 2. a Example for the distribution of perimeters at n = 14 in the diamond lattice. The solid lines connects points for p = 0.428, the dashed line those for p=0.1, b Log-log plot of the m e a n square width of the perimeter distribution versus cluster size
=0.245, pc(FCC)=0.198; and for the critical exponents: /3=0.4, 7=1.66, 6=1+7//3=5.15, ~=1//~6 =0.485, T = 2 + 1 / 6 = 2 . 1 9 4 . The cluster size n varies from Unity up to 14(D), ll(SC), 10(BCC) and 9(FCC).)
III. Cluster Numbers
The number c, of n-clusters is shown in Figure 3 for the diamond lattice only which gave the best results since there the maximum value of n is largest, Pc is near 1/2, and the series are smooth.
It has been speculated [7] that for large n near Pc one has c,(p)ocn-~f(en ~) with eocp-pc , where the scaling function f remains finite at zero argument. Then the ratio c,(p)/c,(pc ) simply gives the function f Moreover, by looking at these ratios, some of the deviations from asymptotic behavior cancel out for small n; c, varies as n -~ only for large n. Thus the ratios c,(p)/c,(pc ) may be appropriate for an analysis of c, even rather far away from the percolation threshold. Figure3a gives the ratios c,(p)/c,(pc ) for various concentrations p, showing clearly a different behavior above and below pc=0.43. Below Pc, for p=0.1, a
A. Flammang: Percolation Cluster Sizes and Perimeters in Three Dimensions D
49
1086
10-1 iO ', \
0,428
~
1
b
'1
'5
0,8 "
'1
0,6"
'5
'10
~10
n"
~. n2/3 D
0,4-
0,2-
ho4 0,10,0 8-
10-4
0,0 6 0,04 -
10-5
0,75 0,02
a/
11
15
110
10-6 ~5
115 ---~n
t10
115
120 n
Fig. 3. a Logarithmic plot of c.(p)/c.(pc) versus n in the diamond lattice at variconcentrations. The straight line through the data for p = 0.1 indicates the validity of (3). (On the left axis, the larger numbers correspond to p=O.1 and the smaller numbers to p =0.75. For the other three concentrations the right axis is valid.) b Logarithmic plot of c,(p)/c,(pO for p=0.75 in the diamond lattice, versus n, versus n 2/3, and versus n~-l/n. The straight line in the plot versus n 2/3 indicates the validity of (4)
straight line seems to fit the data for large n in this logarithmic plot, m e a n i n g log(c,)oc - n
(p
(3)
in agreement with some M o n t e Carlo simulations [8]. But above Pc. log(c,) varies with some power on f smaller than unity. This effect is analyzed m o r e quantitatively in F i g u r e 3 b where the same data (p=0.75) again are plotted logarithmically three times: versus n, v e r s u s t22/3 and versus n¢--t/n. The plot v e r s u s y~2/3 gives the best fit to a straight line, suggesting log(c,)oc - n 2/3
(P>Pc, n ~ oo).
(4)
(We tried without m u c h success to find for d = 3 the analog of Equation(13) in [5] for d = 2 . )
lations and perimeter polynomials [5]. Just as for d = 2 we found in three dimensions the average perimeter to be proportional to n, with ( s , ) / n ~ (1 -Pc)/Pc at Pc for large n, A n d the cluster n u m b e r s c, seem to have an asymptotic behavior log(c,)oc - n ~
(n~ ~)
with (-~1
for P
The results of this note largely confirm results found earlier for two dimensions from M o n t e Carlo simu-
(Sb)
and g~-x2
for P>Pc.
In two dimensions [5] data were consistent with ( ( p < p c ) - I and ( ( p > p c ) = l / 2 . Thus we can reasonably suggest below Pc: (p > pc) = (d - 1)/d.
IV. Conclusion
(5a)
(5 c)
These results (5b, 5c) are in full agreement with the theoretical prediction (16) of [9] for clusters in the Ising model.
A. F l a mma ng: Percolation Cluster Sizes and Perimeters in Three Dimensions
50 Table l. Cluster Nu mbers in 2 and 3 Dimensions p (%)
n (square lattice) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
15 20 24 26 30 34 38 42 46 50 55 59 60 65 70 75
783E-4 819E-4 801E-4 780E-4 720E-4 645E-4 562E-4 475E-4 391E-4 313E-4 226E-4 167E-4 154E-4 975E-5 567E-5 293E-5
170E-4 209E-4 222E-4 222E-4 212E-4 191E-4 164E-4 134E-4 105E-4 781E-5 502E-5 331E-5 295E-5 155E-5 714E-6 275E-6
617E-5 940E-5 112E-4 117E-4 120E-4 114E-4 101E-4 844E-5 662E-5 488E-5 305E-5 193E-5 170E-5 830E-6 345E-6 116E-6
238E-5 448E-5 600/?7-5 658E-5 728E-5 729E-5 671E-5 570E-5 450E-5 330E-5 200E-5 122E-5 106E-5 484E-6 182E-6 537E-7
965E-6 225E-5 340E-5 391E-5 467E-5 494E-5 473E-5 411E-5 328E-5 240E-5 143E-5 844E-6 729E-6 313E-6 109E-6 288E-7
406E-6 117E-5 200E-5 241E-5 311E-5 349E-5 347E-5 309E-5 249E-5 182E-5 106E-5 610E-6 522E-6 211E-6 675E-7 160E-7
175E-6 628E-6 121E-5 153E-5 214E-5 254E-5 263E-5 241E-5 197E-5 144E-5 827E-6 464E-6 394E-6 151E-6 450E-7 964E-8
774E-7 344E-6 748E-6 998E-6 151E-5 190E-5 206E-5 193E-5 160E-5 117E-5 665E-6 365E-6 307E-6 112E-6 313E-7 612E-8
347E-7 191E-6 471E-6 660E-6 108E-5 145E-5 163E-5 158E-5 133E-5 975E-6 548E-6 294E-6 246E-6 861E-7 224E-7 401E-8
158E-7 108E-6 301E-6 443E-6 787E-6 112E-5 131E-5 131E-5 112E-5 828E-6 461E-6 243E-6 201E-6 676E-7 165E-7 173E-8
723E-8 616E-7 194E-6 300E-6 579E-6 873E-6 107E-5 ll0E-5 959E-6 712E-6 393E-6 204E-6 168E-6 541E-7 125£'-7 190E-8
334E-8 354E-7 126E-6 206E-6 430E-6 689E-6 886E-6 938E-6 831E-6 621E-6 341E-6 173E-6 142E-6 440E-7 959E-8 135E-8
156E-8 205E-7 828E-7 142E-6 321E-6 548E-6 738E-6 805E-6 726E-6 547E-6 298E-6 149E-6 122E-6 363E-7 750E-8 98E-8
73E-8 I20E-7 547E-7 984E-7 242E-6 439E-6 618E-6 696E-6 640E-6 485E-6 263E-6 130E-6 106E-6 303E-7 594E-8 72E-8
34E-8 700E-8 363E-7 687E-7 183E-6 354E-6 522E-6 606E-6 568E-6 434E-6 235E-6 114E-6 922E-7 256E-7 476E-8 54E-8
16E-8 412E-8 242E-7 481E-7 140E-6 287E-6 443E-6 530E-6 507E-6 391E-6 211E-6 101E-6 812E-7 218E-7 386E-8 41E-8
8E-8 243E-8 162E-7 339E-7 107E-6 233E-6 377E-6 467E-6 456E-6 354E-6 190E-6 901E-7 721E-7 187E-7 316E-8 31E-8
p(%)
n (simple cubic lattice) 3
4
5
6
7
8
9
10
11
373E-4 396E-4 317E-4 212E-4 123E-4 109E-4 632E-5 288E-5 116E-5 412E-6
156E-4 208E-4 185E-4 127E-4 724E-5 634E-5 349E-5 144E-5 509E-6 152E-6
709E-5 120E-4 120E-4 859E-5 482E-5 4i8E-5 220E-5 832E-6 261E-6 669E-7
341E-5 738E-5 826E-5 617E-5 344E-5 297E-5 150E-5 522E-6 146E-6 325E-7
170E-5 471E-5 593E-5 464E-5 257E-5 221E-5 107E-5 347E-6 873E-7 170E-7
874E-6 459E-6 309E-5 208E-5 440E-5 334E-5 361E-5 288E-5 200E-5 159E-5 170E-5 135E-5 800E-6 615E-6 240E-6 172E-6 546E-7 354E-7 93E-7 53E-7
246E-6 142E-5 259E-5 235E-5 130E-5 110E-5 483E-6 126E-6 236E-7 31E-7
133E-6 991E-6 204E-5 194E-5 108E-5 909E-6 388E-6 949E-7 162E-7 19E-7
1
10 15 20 25 30 31 35 40 45 50
2
5 3 1 E - 3 105E-3 3 7 7 E - 3 886E-3 2 6 2 E - 3 644E-4 178E-3 422E-4 118E-3 254E-4 108E-3 227E-4 7 5 4 E - 4 141E-4 4 6 7 E - 4 726E-5 2 7 7 E - 4 342E-5 156E-4 146E-5
No simple interpretation was given for the width exponent co in (2). It should be mentioned that Leath [10] has cautioned against the use [5] of these perimeter polynomials for extrapolation to large cluster sizes n since these series are rather short for d = 2 (n~20) and even shorter for d = 3 (n<14). Thus we made no attempt here to analyze the even shorter series (n__<7)in more than three dimensions [4]. We thank D. Stauffer for suggesting this work and help with the manuscript.
Appendix For the convenience of readers who want to make their own analysis of clustering or want to compare their Monte Carlo results with exact expressions Table 1 gives for selected concentrations p the cluster number c, in the square lattice (from Ref. 5; c, = n u m b e r per site) and the simple cubic lattice (this work; cn=number per occupied site), as calculated from the perimeter polynomials of Refs. 2 and 3. Here i E - k is an abbreviation for i x 10 -k. Note Added in Proof. A preprint of H. Kunz and B. Souillard found, on the basis of exact inequalities, results which are fully consistent with our predictions (5 b) and (5 c).
References 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 De Gennes, P.G., La Recherche 7, 919 (1976) 2. Sykes, M.F., M. Glen: J. Phys. A 9, 87 (1976) 3. Sykes, M.F., Gaunt, D.S., Glen, M.: J. Phys. A 9, 1705 (1976) 4. Gaunt, D.S., Sykes, M.F., Ruskin, H.: J. Phys. A 9, 1899 (1976) 5. Stauffer, D.: Z. Physik B 25, 391 (1976) 6. Stauffer, D.: J. Phys. C 8, L 172 (1976) Leath, P.L.: Phys. Rev. I~etters 36, 921 (1976) and Phyg Rev. B 14, 5046 Domb, C.: J. Phys. A 9, L 141 (1976) 7. Stauffer, D.: Phys. Rev. Letters 35, 394 (1976) 8. Bakri, M.M., Stauffer, D.: Phys. Rev. B 14, 4215 (1976) Miiller-Krumbhaar, H., Stoll, E.P.: J. Chem. Phys. 65, 4294 (1976) 9. Kretschmer, R., Binder, K., Stauffer, D.: J. Statist. Phys. 15, 267 (1976) 10. Leath, P.L.: talk at Mid-Winter Solid State Research Conference, January 1977, Laguna Beach, Cal., USA (unpublished) A. F l a m m a n g Fachrichtung 12.2 Elektrotechnik Universitiit des Saarlandes Im Stadtwald D-6600 Saarbrticken 11 Federal Republic of Germany
