P. R. A. Campos, L. F. C. Pessoa, and F. G. Brady Moreira ... B 40, 10 986 1989 , the activation of a bond between two first-neighbor occupied sites is also.
PHYSICAL REVIEW B
VOLUME 56, NUMBER 1
1 JULY 1997-I
Cluster-size statistics of site-bond-correlated percolation models P. R. A. Campos, L. F. C. Pessoa, and F. G. Brady Moreira Departamento de Fı´sica, Universidade Federal de Pernambuco, 50670-901 Recife-Pernambuco, Brazil ~Received 31 October 1996! We consider the cluster-size statistics of two types of site-bond-correlated percolation. In model A ~B! @see Phys. Rev. B 40, 10 986 ~1989!#, the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of both ~either! of their own nearest neighbors, along the line joining them. The critical concentration for models A and B, obtained from simulations on square lattices of sizes up to 200032000, is found to be 0.740560.0005 and 0.614560.0005, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent ~site or bond! percolation. @S0163-1829~97!06325-X#
In the ordinary site percolation problem, sites on a lattice are occupied at random with probability p and two firstneighbor occupied sites are connected by an open bond independently of the occupancy of other nearest-neighboring sites. In the site-bond-correlated ~SBC! percolation model we consider here, sites are occupied in the same way as in the ordinary site percolation model but the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of their own nearest neighbors, along the line joining the two sites. The SBC model was originally introduced as a model for the randomly diluted magnetic systems KNip Mg12p F3 ,1 whose properties are not well described by the usual model. We may distinguish two sorts of site-bond correlations according to the way in which the active sites participate in the correlation process @see Eqs. ~2! and ~3! of Ref. 2#. In model A ~B! the activation of a bond between two neighboring occupied sites, say, i and i1 d , is also dependent on the occupancy of both ~either! of their own nearest-neighbor sites, located at i2 d and i12 d , along the line joining them but in opposite directions. Note that both models restrict the site-bond correlation to be active only along the direction of the bond being considered. Such a directionality effect constitutes a marked difference between the SBC model and other correlated dilution models. For instance, in bootstrap and high-density models3–5 all neighboring sites of a given site take part in the correlation. In this work we present results of Monte Carlo calculations on the cluster-size distribution for both versions of the SBC percolation model. For the evaluation of cluster numbers we use the cluster multiple labeling technique first introduced by Hoshen and Kopelman,6 where slight modifications are needed to handle the site-bond correlation. The simulations are performed in two-dimensional square lattices of sizes up to 200032000 and periodic boundary conditions. Through the calculation of the reduced mean cluster size, we estimate the percolation thresholds for both models. Then we apply a finite-size-scaling analysis in a three-parameter space to obtain the critical exponents b , g , and n . Our results are consistent with both models lying in the universality class of usual percolation. In what follows, we introduce the computational procedure used within the context of cluster-size statistics and present and discuss our results. Then we conclude with a short summary. 0163-1829/97/56~1!/40~3!/$10.00
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As for ordinary independent site percolation, in the SBC percolation each site is occupied randomly with probability p, and we look for clusters of occupied sites connected by open bonds. In the presently studied model B ~A!, however, for a bond between two neighboring occupied sites being open it is required that at least one ~both! of their own nearest neighbors along the line joining the two occupied sites are also occupied. For model A, the mean number of one cluster per lattice site is given by n 1 ~ p ! 5 p ~ q 2 12 pq1p 2 q 2 ! 2 ,
~1!
while for model B we have n 1 ~ p ! 5 p ~ q 2 12 pq 2 ! 2 ,
~2!
where q512 p is the probability of unoccupied sites, and an s cluster means a cluster of s connected sites. The above exact relations are to be compared with n 1 (p)5pq 4 for ordinary site percolation on a square lattice,7 corresponding to a configuration where all nearest neighbors of an occupied site are empty. Equations ~1! and ~2! were found to be consistent with numerical simulation results, providing a check on the latter. Figure 1 shows the results of simulations for the reduced mean cluster size S(L,p), defined by S ~ L,p ! 5
(s s 2 n s~ p ! ,
~3!
where n s (p) is the mean number of clusters of size s, per lattice site, and the summation is over all cluster sizes except the largest cluster. This function increases rapidly in the region of transition and diverges at the percolation threshold p c . From the position of the maximum of S(L,p) we obtain the estimates p c ~ A ! 50.740560.0005,
~4!
p c ~ B ! 50.614560.0005,
~5!
for models A and B, respectively. The above value for p c (A) compares well with previous estimates,8–10 whereas this is the first estimate for the threshold p c (B) of the model 40
© 1997 The American Physical Society
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FIG. 1. Reduced mean cluster size for ~a! model A and ~b! model B of SBC percolation, for concentrations near the percolation thresholds. Data from simulations on square lattices of size L52000 with periodic boundary conditions.
B of SBC percolation. These results are to be compared with p c 50.592 746 ~Ref. 11! for ordinary site percolation on the square lattice. In order to obtain the critical exponents and other estimations of the percolation thresholds, we explore the following finite-size-scaling functions: f ~ z ! 5L 2 g / n S ~ L, p ! ,
~6!
g ~ z ! 5L b / n P ~ L, p ! ,
~7!
h ~ z ! 5R ~ L, p ! ,
~8!
z5 ~ p2p c ! L 1/n .
~9!
where
Here, the percolation probability P(L, p) is the fraction of sites belonging to the largest cluster, and the spanning probability R(L,p) is the probability that there exists a spanning cluster on a lattice of linear size L. To determine the ratio 1/n we calculated numerically the derivative of the function h(z), defined in Eq. ~8!. The derivative dh/dp is plotted in Fig. 2~a! as a function of the concentration p, for model A, and for several values of the system size L. It presents a sharp peak at a given value of the concentration p * , which is in good agreement with the value of p c obtained from the position of the maximum of the reduced mean cluster size. Moreover, let
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FIG. 2. ~a! The derivative dh/dp as a function of p, for three values of L. ~b! Plot of log(dh/dp) at p c vs logL; the lattice sizes used are L5500, 800, 1000, 1200, 1500, and 2000. The slope is an estimate of 1/n.
dh dp
U
5 p5p c
dh dz dz d p
U
5 p5p c
dh dz
U
L 1/n .
~10!
p5p c
From the value of dh/d p, at p5p c , we can measure the exponent 1/n as the slope of the straight line fitted to the data points in a log-log plot as indicated in Fig. 2~b!, since, at the percolation threshold, dh/dz is a constant independent of L. For this analysis, we performed simulations on square lattices of linear sizes L5500, 800, 1000, 1200, 1500, and 2000, and with periodic boundary conditions. For each value of the concentration p, the number of runs ranged from 500 for L5500 to 150 for L52000. For both models, we get for the exponent y p 51/n the value y p 50.7960.05, which should be compared with y p 5 43 , the exact result7 for uncorrelated percolation. The values of the ratios g / n and b / n come out when the above scaling laws are satisfied, that is, when one obtains the best fits for the curves described by the scaling functions for different sizes of lattices. We succeeded for both models, after various attempts: It was assumed the central value of our estimation ~see above! for the exponent y p 51/n , and the other two exponents and the percolation thresholds were varied until the data for lattice sizes L51000, 1500, and 2000 fall on a single smooth curve. Indeed, the variation of p c was restricted to those values inside the limits indicated by the error bars in Eqs. ~4! and ~5!. The results obtained and presented here are consistent with both SBC percolation models being in the same universality class of ordinary percolation.
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TABLE I. Critical parameters calculated for model A and model B of site-bond-correlated percolation on the square lattice. For estimates of the critical parameters for a selection of bootstrap percolation models, see Ref. @5#.
Model A Model B
FIG. 3. The scaling functions ~a! f (z) and ~b! g(z) as a function of z ~see text!, for model A. The data are for three different lattice sizes L.
In Fig. 3~a! we plot the simulational data for model A, showing the z dependence of the scaling function f (z) @see Eq. ~6!# on square lattices with L51000, 1500, and 2000. We see a quite good data collapse for all three lattices simulated, allowing us to determine an accurate value for the ratio g / n 51.83560.045. In Fig. 3~b! we plot the corresponding results for the scaling function g(z) @see Eq. ~7!#. Again, we obtained a satisfactory data collapse for all three lattices simulated, yielding b / n 50.10260.002. The same procedure used for model A was applied for model B, and our final estimates for the exponents and for
1
J. A. O. de Aguiar, F. G. Brady Moreira, and M. Engelsberg, Phys. Rev. B 33, 652 ~1986!. 2 A. A. P. da Silva and F. G. Brady Moreira, Phys. Rev. B 40, 10 986 ~1989!. 3 G. R. Reich and P. L. Leath, J. Stat. Phys. 19, 611 ~1978!. 4 J. Chalupa, P. L. Leath, and G. R. Reich, J. Phys. C 12, L31 ~1979!. 5 J. Adler, Physica A 171, 453 ~1991!. 6 J. Hoshen and R. Kopelman, Phys. Rev. B 14, 3438 ~1976!.
b/n
g/n
pc
0.10260.002 0.10560.001
1.83560.045 1.7760.02
0.740560.0005 0.614560.0005
the percolation thresholds are given in Table I. The calculated values for b / n and g / n agree, within the error bars, 43 with the exact results b / n 5 485 50.104 and g / n 5 24 51.792 ~Ref. 7! for uncorrelated percolation. In summary, Monte Carlo experiments on the cluster-size distribution for two versions of the site-bond-correlated percolation model, on the square lattice, were used to calculate the critical percolation concentrations, the scaling power exponent, y p 51/n , and the critical exponents g / n and b / n for the mean cluster size and for the strength of the infinite cluster, respectively. It should be noted that all the calculations were performed without assuming any previous knowledge of the critical parameters. For model B the threshold was found to be only about 4% higher than the percolation threshold for ordinary independent site percolation on a square lattice, whereas, for model A, in which all previous MC simulations on SBC percolation have concentrated, the percolation threshold is 25% higher than that for the usual percolation. These results show that the influence of site-bond correlation on the percolation threshold is much less pronounced in model B than that observed in model A. The marked behavior of the scaling functions in the transition region permitted an accurate estimate of the critical exponents. The present estimated values for the critical exponents, compared with the exact results for ordinary independent ~site or bond! percolation, support the conclusion that all these percolation models are in the same universality class. We thank Adauto de Souza for discussions. This work was supported in part by CNPq and FINEP.
D. Stauffer, Introduction to Percolation Theory ~Taylor & Francis, London, 1985!; D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. ~Taylor & Francis, London, 1992!. 8 N. S. Branco, S. L. A. de Queiroz, and R. R. dos Santos, Phys. Rev. B 42, 458 ~1990!. 9 L. M. de Moura and R. R. dos Santos, Phys. Rev. B 45, 1023 ~1992!. 10 N. S. Branco and K. D. Machado, Phys. Rev. B 47, 493 ~1993!. 11 R. M. Ziff, Phys. Rev. Lett. 69, 2670 ~1992!. 7
