Zeitschrift fQr P h y s i k B

Z. Physik B 30, 173 - 176 (1978)

© by Springer-Verlag 1978

Speculations on the Cluster Radius Below the Percolation Threshold D. Stauffer Institut ffir Theoretische Physik der Universit~it zu K61n, West Germany Received February 13, 1978 We suggest the average radius of percolation clusters with s sites to vary below Pc as s ~°, where vo is the exponent for the mean radius of self-avoiding walks. This result gives the desired asymptotic behavior of the correlation function for percolation (" connectivity") and is consistent with Leath's Monte Carlo data.

1. Introduction

Recently several papers [1-73 have appeared which seem to give compatible results on the number and structure of percolation clusters: i) The average numbers n~ of percolation clusters with s sites each decay for s ~ at fixed concentration p (+Pc) as: logn, o c - s ~, with ~ = l - 1 / d above the percolation threshold Pc, and with ~ = 1 below Pc, in d dimensions [1, 2, 4-63. ii) The perimeter-to-size ratio approaches (1-p)/p for large clusters above Pc but not below Pc [2, 3, 5] and is the same for the infinite cluster [4]. iii) The "density" within a large cluster, i.e. the probability of a site within the outer boundary to be connected to that cluster, approaches the percolation probability Po~, i.e. the density of the infinite network, for all p above p~ [2, 8]; thus for s ~ m the cluster volume (in suitable units) is s/Po~ and the cluster radius R s varies as s TM above Pc I-2, 8]. iv) At Pc the cluster volume varies as s ~+~/~ and thus the cluster radius R~ as s ~+~/~/~ [7, 8]. The present paper tries to fill a remaining gap: What is the cluster radius R s below the percolation threshold, i.e. with what exponent p of s does R s increase for s--+oo at fixed p < Pc. In Section 2 we suggest that very large clusters behave like self-avoiding walks (SAW) on a scale larger than the correlation length ~, and that R~ therefore is proportional to the mean square radius of an SAW [9]. In Section 3 we use this assumption to calculate the correlation function for

the percolation problem and we show that the analogy with SAW's leads to reasonable results. Further tests of our speculations are suggested in Section 4.

2. Cluster Radius

We assume [10, 1 !] that very large clusters below Pc consist of many rather compact regions of linear dimension ¢; each of these regions contains s~ ~ (Pc-P)-a ~ sites, where ~ o c ( p c - p ) -~ is the coherence length. (We use the standard notation for the critical exponents /~, 7, 6, ~/, v; e.g. P o o ~ ( p - p y for the probability P~ of a site to belong to the infinite network. Also we assume dv =/~(6 + 1) = ~ + 2/~ = (2 - r/) v + 2/L) These rather compact regions are then connected by narrow channels, as in our figure. Of course, the reality (e.g. Fig. 3 in [l 1] or Fig. 2 in [5]) looks less simple, but we hope that our approximation is good enough to determine the qualitative behavior, i.e. the asymptotic exponents. If in this picture we omit those channels which make the whole cluster multiply connected (dashed lines in our figure) then the cluster becomes a chain of narrow channels connecting the more compact st regions. Such a chain behaves like a SAW on a continuum having l=s/s~ links. We know [9] that the mean square radius (R~z) of a SAW with l links varies as (/vo)2 with V o - 3 / ( 2 + d ) being a good approximation for dimensionality d<4, at least for SAW's on a lattice. Thus our cluster radius R~ is

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174

D. Stauffer: Speculations on the Cluster Radius Table 1. Predicted variation with dimensionality d of the exponent p for the cluster radius R~ocs°, Equations (1), (3), (4). For percolation exponents see Stanley [13]

d

p
p=p~

p>p~

1 2 3 4 5 6

1 0.75 0.59 0.50 0.50 0.50

1 0.53 0.40 0.31 0.27 0.25

1 0.50 0.33 0.25 0.20 0.17

Fig. 1. Schematic structure of a cluster much larger than the coherence length ~, for p
proportional to (s/s~) ~°, or

R~ocsP,

p(p
(1)

for s ~ c e at fixed p, i.e. for s/s~>> 1. This equation is our main result. We hope it to be valid for all p below Pc and not only near Pc, just as properties (i) to (iii) of the introduction are valid for all p below Pc and above Pc, respectively. Only predictions for powers of p - P c are restricted to p near Pc. To find the powers of p - P c entering the cluster radius for p--, Pc, we make a scaling assumption:

Rs(p)= ~(p). e(s/s¢)= IPc-Pl -~" f~ [(Pc-P)

sl/flfi]

(2a)

in analogy to the scaling assumption [12, 5]

G (P) = s- (2 +1/~)f [(G - P) sl/~~]

(2 b)

for the average cluster numbers, where f, R and /~ are suitable scaling functions of a single variable. Right at p =Pc all powers of P-Pc have to cancel out, giving Rs(Pc)OCs~/P~=s (l+l/~)/d, which is property (iv) of the introduction [8]; or

p(p=G)=(l+~)/d,

(3)

if R~ocs p, as in Equation (1). Above Pc, property (iii) of the introduction gives R~ = (s/Po~)l/daCSi/d(p -= G)- ~/d apart from prefactors of order unity; thus P (P > Pc) = 1/d.

(4)

These very large clusters above Pc seem to behave like a spherical Swiss cheese [8] :Inspite of its internal holes the total volume of the cheese is proportional to the mass s of cheese and to the d-th power of the cheese radius R~. This Swiss cheese or raindrop model [8] breaks down for P
at least at Pc the external perimeter is not proportional to the excess perimeter as we suggested in [3].) Instead the SAW picture of (1) is supposed to replace the Swiss cheese picture: For s ~ o e at fixed p slightly b el ow Pc we have R s oc (Pc - P)- ~[(Pc - P) sl/~~]~~~o from (2a), i.e.

Rs ocs ~° • (pc-p) ~ ° - ~

(5a)

and

P(P
(5 b)

In other words, by assuming the asymptotic s-dependence of R s to be given by (1), we found the asymptotic behavior of the scaling function in (2a), and this result in turn gave the dependence of R~ on p - Pc in (5 a). Our table collects the resulting estimates for the radius exponent p for dimensionalities between 1 and 6, using critical exponents as tabulated e.g. in [9] and [13]. Leath [14] found by Monte Carlo simulation of clusters in the square lattice (pc= 0.59): p(p=0.55)=0.57;

p(p=0.50)=0.61.

(6)

A typical cluster size in these computer simulations is s=300; then the scaling variable z = ( p c - p ) s 1/~ is z(p = 0.55) = 0.4 and z(p = 0.50) = 0.9. We may roughly identify with s~ that cluster size s where the ratio G(P)/G(Pc) reaches a maximum at fixed p below Pc; then z~ = (Pc- P) s~/~ is [15] about 0.5. Thus the Monte Carlo data are just in the transition region from z ~ z¢ (cluster radius ~ , Equation (3); p = 1 + 1/6 =0.53) to z>>z¢ (cluster radius >>~, (5); p=v0=0.75). Thus Leath's exponents, which are just in between, are consistent with our speculation but do not prove it. (Ref. 11 also made a SAW approximation to percolation; but that paper assumed correlations to spread within a cluster of size ¢ like a SAW, i.e. it dealt with distances < 4 ; the present note looks at correlations between different s¢-regions, i.e. it deals with distances ~. Thus we are dealing here with a problem different from the one discussed in [11].)

D. Stauffer: Speculations on the Cluster Radius

175

3. Correlation Function As a further test of our results we now calculate the correlation function near the percolation threshold, i.e. the probability of two distant sites to belong to the same cluster [-16, 171. This discussion is a generalization and improvement of an earlier attempt (Section 3 of [17]). We neglect here the effects due to the infinite network above G. Furthermore we take the SAW exponent vo to be smaller than unity and thus exclude one-dimensional percolation 1-13] from our considerations. Let Ds=D,(r ) be the average density of a cluster, i.e. the probability of site r to belong to a cluster of size s if the origin belongs to that cluster. The cluster radius may then be defined e.g. by

R 2 = ~ r 2 Ds(r) ddr/SD,(r) dar.

for P+Pc at fixed r.) The nontrivial question is: Are our assumptions consistent with the expected asymptotic decay, log G(r)oc - r/l,

r > 4,

(9c)

both above and below Pc, inspite of the asymmetric behavior of G and R s about Pc Above Pc we use the droplet picture [3] for large clusters (R~> ~), that the density profile within the cluster (r> 4, just as near a liquid-gas surface. (In this sense the correlation length is the thickness of the interface region.) This exponential decay is now assumed to be valid also for clusters of size near se. Then D,(r),,~ Pooexp(- ( r - R,)/{), and (9 a) simply gives

(7)

G(r)oc ~, Poo e-(r-R*)/~ n, soce -~/~ Since above G for very large dusters D~=P~ in the cluster interior (property (iii) in the introduction), we make the scaling assumption

D s (r) = Poo" D (r/R,, s/s~) = [pc-pl p. [)[r/Rs, (Pc-P) sl/~a] = s -1/a. D[r/R~, ( G - P ) sl/P~]

(8a)

with the normalization

(8b)

D(0, oo)= 1.

(In Ref. 17 we neglected erroneously the dependence of D~ on s/s~.) The origin belongs to a s-cluster with a probability G s; thus a site at distance r from the origin belongs with probability D~Gs to the same s-cluster. Summation over all cluster sizes s then gives the correlation function G(r), i.e. the probability of two arbitrary sites at distance n to belong to the same cluster [171 :

G(r) = ~ Ds(r) G s.

(9a)

8

with the main contribution coming from s near s~. Equation (10) already is the desired result above Pc, but it is not a test of the SAW picture below Pc. Below Pc our SAW picture requires the cluster density profile Ds(r ) to be proportional to the density profile of a SAW with l~s/s¢ links; the latter density is known [-9] to decay asymptotically with r as exp[--(r/Rl)+*/(1-~°)]. Thus in our cluster picture the very large clusters below Pc have a density profile D~(r) oc exp [ - (r/Rs) 1/(1- vo)],

~+oo

G(r)oc(pc-p) 2p ~ z -1-2fl . f ( z ) . D[r/(~ R(z)), z] dz 0

=(Pc __p)2fl. g(r/~.)oc{-(d-2 +,)g(r/O.

(11)

r >>R~

apart from preexponential factors. If we take into account in (9a) only the exponentially varying terms, use R s = B ( p ) s ~° from (1), and l n G = - A ( p ) s from property (i) of the introduction (s~oo), then we get

G(r)oc S ds e x p [ - A s - r 1/(1- vo)B- 1/(1-~o)s- vo/(1-vo)]. (12a) The argument of the exponential in (12a) has for v0 < 1 a maximum at

A = r 1/(1 - vo) Vo(1 - Vo)- 1B - 1/(1-

With the scaling assumptions (2.8) and the abbreviation z = ( p c - p ) s 1/~ we get from Equation (9a) the correlation function apart from numerical factors:

(10)

(r>> ~)

s

vo) S -

1/(1 -

Vo),

i.e. at s = C(p) r, with C(p) = const. A ~°- 1B- 1. Thus the main contribution to the integral in (12a) comes from very large clusters if r--,oo (more precisely, from s/s~ ~ r/i). And apart from preexponential factors this integral is

(9b) G(r)ocexp( - const • r),

Thus the general scaling assumptions made before lead to a general scaling result for the correlation function, as expected. (As in Ref. 17 we recover G(r)oc r 2-~-d at P= Pc, and again G(r)occonst + [p-pc[ 2~

(12b)

with constocA(p) C(p)oc A ~°/B oc(pc - p)V oc 1/4

(p --+pC) (12c)

176

according to (2b), (5a). Thus again (9c) is recovered, as desired. The approximations used here are too inaccurate to allow a determination of preexponential factors. But we have seen that the SAW approximation for large clusters, together with the profile of SAW's, (11), and the simple exponential decay of the numbers ns of very large clusters, gives the desired exponential decay of the correlation function, (9 c). {Our approach neglected the difference in the average density profile obtained at the site r if the origin belongs to any part of the cluster, and the profile obtained if the origin is the cluster center. For the present calculations we have checked that this difference does not matter apart from numerical factors. But this difference is important in future Monte Carlo simulations of cluster density profiles where it is more interesting to keep the cluster center fixed at the origin.}

4. Conclusions

The main result of this paper is Equation (1), that 1/trge clusters below Pc (but not above Pc) behave like self-avoiding walks and thus have an average radius varying with the number s of sites in the cluster as s v°, thus connecting percolation theory with the "critical" exponents of SAW's. The remaining equations in Section 2 are evaluations of the powers of P - P o whereas Section 3 showed this SAW picture to be consistent with the expected behavior of the correlation function. Monte Carlo tests similar to References 14 and 7 could check the validity of our speculation, (1). In order to be safely in the region of very large clusters (s >>s~oc(pc - p)-P~), where the average cluster numbers ns are extremely low, it may be useful to work with a fixed cluster size s and to allow only the shape of the cluster to fluctuate randomly [-18] with probability oc(1 - p)t, where t is the perimeter of the cluster. If this factor ( l - p ) ~ is neglected, i.e. for p=0, one then is simulating "animals" [3] on the lattice and thus could check whether the present SAW picture also applies for the average animals.

Note Added in Proof. Preliminary Monte Carlo simulations on the square lattice suggest that the present SAW picture, Equation (1), does not apply to the average "animals" where we found p~½ instead.

D. Stauffer: Speculations on the Cluster Radius

Such Monte Carlo calculations then could tell us whether the present conclusions are correct: that percolation clusters much larger than the coherence length behave like raindrops above Pc ("Swiss cheese regime") and like "ramified" self-avoiding walks below Pc (" hydra regime"). We thank the authors of References 1, 2, 4, 5 for their preprints, K. Binder for important and helpful suggestions, and A. Weinkauf for a critical reading of the manuscript.

References 1. Schwartz, M.: Phys. Rev. B17 (1978) to be published 2. Kunz, H., Souillard, B.: Phys. Rev. Letters 40, 133 (1978) and J. Statist. Phys., to be published 3. Stauffer, D.: J. Statist. Phys. 18, 125 (1978) 4. Stoll, E., Domb, C.: J. Phys. A 11, L57 (1978) 6. Hoshen, J., Stauffer, D.: in preparation for Phys. Rev. B 5. Leath, P.L., Reich, G. R.: preprint 7. Harrison, R.J., Bishop, G.J., Quinn, G.P.: as cited by Stanley [13] 8. Stauffer, D.: Z. Physik B 25, 391 (1976) 9. McKenzie, D.S.: Phys. Reports 27, 35 (1976) 10. Binder, K.: Ann. Phys. 98, 390 (1976) and Sol. State Comm. 24, 401 (1977), footnote 3 11. Stanley, H.E., Birgeneau, R.J., Reynolds, P.J., Nicoll, J.F.: J. Phys. C9, L553 (1976), Fig. 3; also de Gennes, P.G.: J. Physique 37, L 1 (1976) 12. Stauffer, D.: Phys. Rev. Letters 35, 394 (1975) 13. Stanley, H.E.: J. Phys. A10, L211 (1977) Reynolds, P.J., Stanley, H.E., Klein, W.: J. Phys. A 10, L 203 (1977) Klein, W., Stanley, H.E., Redner, S., Reynolds, P.J.: J. Phys. A 11, L 17 (1978) Stauffer, D., Jayaprakash, C.: Phys. Letters 64A, 433 (1978) 14. Leath, P.L.: Phys. Rev. B 14, 5046 (1976) 15. Wolff, W., Stauffer, D.: Z. Physik B 29, 67 (1978) 16. Dunn, A.G., Essam, J.W., Loveluck, J.M.: J. Phys. C8, 743 (1975) 17. Stauffer, D.: Z. Physik B 22, 161 (1975) 18. Binder, K., Stauffer, D.: J. Statist. Phys. 6, 49 (1972) D. Stauffer Institut fiir Theoretische Physik der Universit~it zu KiSln Ziilpicher Strage 77 D-5000 K~iln Federal Republic of Germany