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The transport exponent in percolation models with additional loops
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PHnlOA
Physica A 211 (1994) 1-12
The transport exponent in percolation models with additional loops F. Babalievski HLRZ at the KFA Jiilich, D-52425 Jiilich, Germany Inst. General and Inorganic Chem., 1113 Sofia, Bulgaria
Received 21 June 1994; revised I July 1994
Abstract Several percolation models with additional loops were studied. The transport exponents for these models were estimated numerically by means of a transfer-matrix approach. It was found that the transport exponent has a drastically changed value for some of the models. This result supports some previous numerical studies on the vibrational properties of similar models (with additional loops).
1. Introduction There are many systems where the ordinary percolation model [ 1 ] could be applied as a zero (or, say, first) approximation. Indeed there are quite a few systems where it might be a final approximation. That is one of the reasons, for the proliferation of the percolation archetype during the last two decades into a family o f percolation models. Some of them differ on the connectivity rule imposed (directed percolation, polychromatic percolation, AB percolation...); alterations were made of the underlaying lattice, continuum (off-lattice) models were introduced as well. The continuum models can be mapped [2] onto a random lattice with a broad distribution of bond lengths. This led to several studies of bond percolation models on square (cubic) lattice where the distribution of bond "strengths" was taken to have the same form as the distribution of bond lengths in the respective random lattice. The 'thermal-like' (the static) critical properties of these models were studied to a large extent, and it was shown, for the most o f the cases that they belong to the ordinary percolation universality class. Some other classes were distinguished as well - the directed percolation is an example. So the accumulated numerical and analytical results 0378-4371/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved S S D 1 0 3 7 8 - 4 3 7 1 (94)00173-1
F. Babalievski / Physica A 211 (1994) 1-12 offer the possibility to predict (to some extent) the static exponent values for a newly "generated" percolation model. The case with the s.c. dynamical exponents (see below for definitions) appears to be quite different. A lot of attempts were made [ 1 ] to find a connection between the static and the dynamic exponents but a "convincing" result is still missing. So, it appears that the accumulation of results about the value of the transport exponent for diverse percolation models, could be of use for clarifying the connection between static and dynamic exponents, as well as for justifying the connections between the dynamic exponents. Some recent studies [3,4] on vibrations on fractal networks gave the search direction for this study.
2. Background As is known [5] the density of vibrations on a fractal structure could be described (qualitatively) by localized excitation called fractons. The frequency dependence follows a power law N ( w ) cx w ds-1 resembling the phonon spectra expression, but with a new exponent (ds) called fracton (spectral) dimension "sitting in for" the Euclidian space dimension in the power term. It could be shown [ 6,7 ] that the spectral dimension could be expressed in terms of the mass fractal dimension and one of two other dynamic exponents: the transport exponent and the walk exponent, ds=2df/dw;
dw=t/~,+df-(d-2),
(1)
d and df being the euclidian and fractal dimension, and the walk exponents (tw) is defined as the scaling exponent for the mean end-to-end distance ( R ( t ) ) travelled of a random walker on a fractal structure - R ( t ) cx t 1/~w. The experimental techniques developed for measuring phonon spectra could be applied in many cases to fractons - situations contrasting with the need for developing conceptually new theoretical and numerical methods for studying the properties of fractal structures. Experimental systems whose geometry resembles the incipient percolation cluster are some silica aerogels (a first approximation of course...). The experimental work done so far showed that these systems have a value of the spectral dimension which is much higher than the value for vector elasticity on 3d percolation cluster. It was seen [8] under the microscope that the floppy ends of such aerogels tend to connect each other in this way enhancing with loops their percolating-cluster-like geometry. This inspired some numerical works to obtain the spectral and the walk exponent for percolation models with additional loops [ 3,4]. The results showed a drastic change of the respective values comparing with the ordinary percolation. Stoll and Courtens [3] calculated numerically the spectral dimension (ds) for a set of "extracted" spanning percolation clusters in 2d for three ranges of connectivity: (I)
F. Babalievski / Physica A 211 (1994) 1-12
3
first neighbors only, (II) first and second neighbors, and (HI) first and third neighbors. For the ordinary percolation (model ( I ) ) , they obtained a value near to 4/3 (namely 1.29), but for the models (II) and (III) the values were respectively 1.53 and 1.79. The numerical method consisted in obtaining each individual eigenvector and eigenfrequency of a dynamical matrix for 45 realizations of 68 x 68 clusters. In a recent work [4] Nakanishi studied another percolation model enhanced with loops. After numerical calculations of the eigenspectrum of the transition probability matrix, he found that the spectral dimension ds and the walk dimension dw change suddenly - as soon as the floppy ends of an incipient percolation cluster are connected together by adding new particles to the cluster in a way to increase the number of loops in the cluster. The systems' sizes were of the order of 100 for the 2d case, and 30 for the 3d case. The aim of the present work is twofold: first, to compare the percolation conductivity for two models of percolation with additional loops, and then to check the connection between the transport and spectral exponent for the models studied (by Stoll & Courtens and by Nakanishi).
3.
Models
Several site percolation models with additional loops were studied in order to reveal these changes of percolation structure which lead to a change of the transport exponent.
3.1. One step bootstrap percolation (BP1) The model was obtained by adding to the initial percolation structure new sites by the rule that such site should have more than one initially occupied neighbor. The building of such model needs two steps: (i) create an ordinary site percolation structure (ii) check for every unoccupied site if it has at least two occupied (on the first step) neighbors, if it is true for a certain site then turn this site into occupied
A completed bootstrap percolation model [ 10] could be obtained by repeating step two (using the "previous step" configuration for neighbors counting) until there is no more unoccupied site with more than one occupied neighbor. (Some authors [ 11 ] prefer to call this model diffusion percolation keeping the term bootstrap for the opposite procedure - discarding sites with "too many neighbors".) The second step (even done only once) leads to a drastic change of the density of occupied sites: Ap = (1 - p ) [ 1
- (1
_p)4 --4p(1 _p)3]
F. Babalievski
1.00
O O
I
I
I
Physica A 211 (1994) 1-12
I
0.75
z
zx
~'~ 0.50 O .~,-t
L=100 ........ L=200
L-ltV
I ~ ,I/ ~ I..¢~
× . . . . L=400
/
O 0.25
o - - - L=800 it~---
L=1200
o - - - L=2000 0 0
z
0.00 25.50
i
26.00
26.50
27.00
27.50
28.00
28.50
i
29.00
i
29.50
30.00x10 -2
p (site occupation probability) Fig. 1. Spanning probability curves for percolation threshold determination for the BP1 model in two dimensions. The insert presents the extrapolation of the finite size percolation thresholds to infinity.
in two dimensions, and in 3d it is:
A p = (1 - - p ) [ 1
-- (1 _ p ) 6
--6p(1
_p)5]
(It is worth to note that there are no free and linear terms in the polynomial presentations of these expressions.) It is clear that the structures obtained after one bootstrap iteration with initial concentration equal to the ordinary percolation threshold are not fractals, so one has to look for a threshold initial concentration. The threshold concentration in 2 and 3d were estimated by the standard procedure for obtaining the spanning probability curves for different system sizes and then extrapolating to infinity the "mean points" of these curves [fig 1,2]. What "mean point" should mean is clear for 2d ordinary percolation, it should be the 50% point of this experimental curves [9]. For 3d this quantity was estimated recently [ 12] to 0.42 .... I tried different techniques [9], but mainly kept within the Gaussian assumption. I accepted that the spanning curves follow the cumulative Gaussian distribution, and the mean of this distribution is "the mean point" to be extrapolated to infinity. (Or better to say, extrapolating to zero of L -1/~ - see the inset of Figs. 1 and 2). Contrasting with ordinary percolation models~ the "smooth curve" passing through the experimental points was not linear but quadratic polynomial in L -1/~. The same plots were made using as argument the concentrations after the bootstarp iteration (final concentrations), but results did not differ much. The percolation thresholds for two and three dimensions are p~d = 0.283 4- 0.002 and pc3d = 0.0942 ± 0.003. Expressed in final
F. Babalievski / Physica A 211 (1994) 1-12
" • 1.00
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.
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L = 2 0 0 - 4 3 0 : 1 0 0 s.p.p. 0.00 8.00
is: e r f ( ( p a 0 ) / a l ) / 2 + 1/2
L = 7 0 , 1 0 ) : 2 0 0 s.p p.
8.50
9.00
9.50
10.00
10.50
(a " g a u s dan" a s s m p t n . )
11.00
11.50x 10 2
p (site occupation probability) Fig. 2. Spanning probability curves for percolation threshold determination for the BP1 model in 3d. The insert presents the extrapolation of the finite size percolation thresholds to infinity.
concentrations these values are 0.51.. and 0.28.. - appreciably smaller than the values for square and cubic lattice (0.5927460.. and 0.3116.. respectively [9] ) in the ordinary percolation.
3.2. Spanning clusters with increased connectivity This model coincides with the Stoll & Courtens model (II) - see Section 2.2 and Ref. [ 3 ]. The starting configuration was a strip-like piece of a square lattice which sites were randomly "occupied". First, the spanning clusters (the clusters which connect the long edges of the strip) for the ordinary (site) percolation model (OPM) at threshold were extracted. Unlike the Stoll&Courtens work the "samples" are strip-like here, so there are a lot of different spanning clusters which connect the long edges of the strip [ 13]. The resistor networks were created connecting by unit resistors the nearest and the next-nearest neighbors (the second neighbors) among the occupied sites which belong to the spanning clusters. The difference with a site percolation model with first and second neighbor connections is that the spanning clusters are determined on the base of first neighbor connections only and the interaction spreads to the second neighbors after removing the nonspanning clusters. This model is intended to resemble the experimental situation where after the preparing of the infinite percolation cluster (e.g. a gel formation or a catalyst producing) its dangling ends may connect each other increasing in this way the connectivity and adding new loops to the percolation structure.
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F. Babalievski / Physica A 211 (1994) 1-12
3.3. Spanning clusters with additional sites
This model could be described as a combination of the previous two. As in the previous section first the spanning clusters have to be extracted. Then instead of increasing the connectivity range for the perimeter sites new sites are occupied in a way to connect the sites from different dangling ends of the spanning clusters. This is a two step procedure as for the previous models: (i) extract the spanning clusters for the OPM at percolation threshold (i.e. turn the nonspanning clusters' sites into unoccupied) (ii) check for every unoccupied site if it has at least two occupied (on the first step) neighbors, if it is true for a certain site then turn this site into occupied
The first step is the same as the first step for the Stoll&Courtens model and the second step coincide with the second step of the BP1 procedure and also is made only once. The obtained structures are similar to the models studied by Nakanishi [4] - see Section 2. He imposed additional restrictions for adding a new site in step 2 of the above procedure. It is checked how large would be the closed loops after occupying the questioned site. Depending on the result the proposal for occupying a site in step 2 may be rejected. The results obatined in his paper for the spectral and walk dimension do not change much (see Section 5.2) when changing the loops size control. Therefore, the size of loops added was not restricted for the prevailing part of the structures simulated here. Some samples were prepared with an additional restriction for adding a new occupied site to the spanning cluster - the new site should not close the smallest (4 site) loop. This restriction differs slightly from the Po = 6 rule in Ref. [4] where the addition is rejected when the minimal closed loop for every pair of initially occupied neighbors equals to 4. This difference was left for simplifying the loops identifying function. These two models will be referred further as the basic and the restricted model respectively.
4. Numerical method The percolation conductivity of the structures examined here was studied numerically by means of the transfer-matrix (TM) approach to the Random Resistor Network (RRN) problem. The resistor networks consisted in unit resistors which bridged the nearest neighbors (and the next-nearest neighbors for the model) in the final structures described above. As known, the TM approach to the RRN problem was originally proposed by Derrida and Vannimenus [14] and then refined by several groups [ 15-18]. The modification proposed in [ 18 ] was adapted for the purposes of the present study. Further modifications were made in order to meet the requirements of the spanning cluster models - the
F. Babalievski / Physica A 211 (1994) 1-12
7
clusters have to be extracted before the TM calculations. The details of the algorithm are presented elsewhere [ 19] - I will recall here only the main features of the TM approach. It makes use [ 14,15] of the general idea of phenomenological renormalization introduced by Nightingale [20] for calculation of critical exponents near second order phase transition. The quantities of interest are calculated for very long and relatively narrow strips. Then by means of scaling of the width of these strips one could extract the critical exponents. In the case of the 2d percolation transport exponent the TM method could be described briefly as follows: First, the conductivities of strips from the percolation structure (usually at p = Pc) are calculated (exactly). The width (L) of these strip is chosen having in mind that their transverse conductivity should scale as L - t / v where t and v are the transport and the correlation length exponents respectively. The conductivity of a certain strip is obtained by means of a "virtual building" of the resistor network inside the frames of two electrodes attached to the long edges of the strip. Initially the space between the electrodes is empty. Then the resistors "take their places" (one after one) on the underlaying lattice filling it column by column starting (say) from the left end of the strip. The resistors are added in a way to ensure required properties of the final structure [ 19]. After adding every resistor the conductivity between the electrodes is recalculated, so, after adding all of the resistors in the strip the conductivity between electrodes is the transverse conductivity of the strip. (Neglecting the "edge" effect is reasonable, when the strip length is much greater than its width.) The second step consists in performing finite-size scaling analysis to extract the value of t / v . Three difficulties in obtaining this value will be mentioned here : the correctionto-scaling terms (important for small widths), the precision of the percolation threshold determination, and third, the increasing of relative error for the strip conductivity when the strip width is increased (the computational effort increases too, with a power of L significantly larger than one). It is accepted that correction to scaling could be expressed as O'L = L - t / v ( 1 + a L - ° ' . . . ) ,
(2)
with a ~ - 1 and to ~ 1 for the ordinary percolation. An additional "electrode" effect appeared for the models studied here. The upper and lower lattice layers of the strip are filled with probability one, to represent the electrodes. Then the chance for adding new sites in the nearest layers inside the strip is higher than in the other bulk layers, because the mean number of occupied neighbors is higher due to the electrodes. This effect alone would lead to a positive value of a, and (generally said) different value for to. The competition between these effects made the deviations in the region of small sizes smaller and more difficult to describe. The practical solution was simply to discard the small widths in determining the transport exponent. The other special points of the data analysis will be discussed in the next section. The pseudo random number generator(PRNG) used in this study was drand48. It was tested together with the programs by obtaining the known values for percolation
8
F. Babalievski / Physica A 211 (1994) 1-12
threshold for square and cubic lattice and the values for the transport exponent for ordinary percolation. The ordinary percolation simulations were made by changing one number in one instruction in the programs for simulating the modified models: i f (NoNeib > Number) { t h e n add a now s i t e . . .
}
For the models with additional loops the Number was equal to 1 and when comparing with ordinary percolation it was set larger than the lattice coordination number. (Thus making impossible the addition of a new site.) "NoNeib" contained the number of neighbors in the initial percolation structure. This number was obtained by means of a "moving sandwich" construction by the following procedure: Three vertical lattice layers (L1,L2,L3) of the strip are created according to the OPM. The variable "NoNeib" is calculated for every empty site in the middle layer. If the above condition is fulfilled the site changes into occupied. Then the final structure obtained for the middle layer is passed to the conductivity measuring part of the program as input data. The moving of the "sandwich" is ensured by the following instructions: LI = L2; L 2 = L3; L 3 = a n e w
OPM
configuration
The calculations were made on the SUN SPARC cluster at HLRZ. The computational efforts were approximately equivalent to 800-1000 hours of SPARC10 CPU time (including the percolation threshold estimations). The preliminary attempts to run the programs on a CM5 machine remained unfinished.
5. Results
5.1. The BP1 model This is the model where the most computational efforts were invested. It was the only model studied in 3d. The value of t / v was found (both in 2 and 3d) to be very close to the OPM value but the difference seemed statistically significant. Despite the efforts, the accuracy of the percolation threshold determination remained insufficient to allow the neglect of the threshold uncertainty contribution to the above difference. The description of the results for the percolation thresholds was given in the previous section, indeed these values were made more exact in parallel with conductivity calculations. The prevailing part of the calculations were made slightly above and below the suspected value for the percolation threshold. In two dimensions the conductivity calculations were done at p = 0.2833, 0.2830 and 0.2820. Despite of estimating the percolation threshold to be around 0.283, the scaling plots (Fig. 3) for p = 0.2833 and 0.2830 showed deviations from the straight line for larger widths which means (most probably) that 0.283 is above Pc. It seems that Pc is closer to 0.282, because the conductivity scaling plot at this value showed smaller deviations from straight line (on the log-log scale, of course). The estimation for the t / v value was obtained by averaging the exponents extracted from every of these three lines.
F. Babalievski / Physica A 211 (1994) 1-12
9
0.1
I
p=0.2833(2d) o 1.81/x^2.046 ..... p=0.2820(2d) +
>.. ~ .....
p=0.944 n
~7"'%
p=O.O940(3d) x 3.0/x^2.402
-~.. ~ "'~..
0.01
• El
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'~-
"'*-... X
o.ool
""~"-.
~
--,..
0.0001
I e-O05
,
10
,
,
I
100 Strip Width
Fig. 3. Finite-size scaling plots for conductivity at the percolation threshold for the BP1 model in 2d and 3d.
Direct fitting with the correction to scaling form (Eq. ( 2 ) ) was not used because of the electrode effect mentioned above. The deviations from pure power law was evaluated by doing the fit first with all points and then discarding the smaller width and repeating the fit, then discarding the next smaller width, and so on until two points are left. The same procedure was repeated but starting from the larger width. Some fits were made with the middle widths only. The strip widths ranged up to 100 and the strip lengths were of the order of several millions. After discarding the widths below 10, the remaining points were fitted with a simple power law XL = aL -1-t/v where XL is the conductance per unit strip length. (To obtain the conductivity one has just to multiply by L.) The fits gave t/v = 1.021, 1.017, 1.046 for p = 0.2833, 0.2830 and 0.2820 respectively. The least-squares fitting procedure determined error bars less than 0.01, so the main source of uncertainty was the percolation threshold estimation. The method was tested (see below) by determining the OPM t/v. The resulting exponent 0.99 -t- 0.01 showed that the error bars are underestimated (the accepted estimate is 0.9745 4-0.0015 [ 17] ). Hence as a final result for the BPlmodel's t/v in 2d should be given 1.024-0.03 without excluding the chance of coinciding with the OPM value. In three dimensions the same procedure was followed. Two values were taken for p: one above (p = 0.0944) and one below (p = 0.0940) the percolation threshold estimate. The bar width was varied between 6 and 30 for both concentrations. The bar length was 106 for L = 6 to 11 and 4 × 104 for widths larger than 11. The runtime scaled approximately as L zS. For L = 30 it was 30 hours SPARC10 CPU time. A run with L = 40 done for p = 0.0940 took 80 hours CPU time. The value 0.0944 seem
F. Babalievski / Physica A 211 (1994) 1-12
10
m
~'~. ....
~
. . . . . . . .
~ 1+ - - S A S mod:0.446
lO't
~ ........ S A S - I o o p s 4 : 0 . 5 6 4
c.:O.983(estd.) I 10 "2
.~.,q
I0 )
0
10 ~
10 z
Strip Width Fig. 4. Finite-size scaling plots for conductivity at the percolation threshold for three spanning cluster models: spanning clusters with connectivity increased to the next-nearest neighbor model (SNNN); spanning clusters with additional sites (SAS) and a restricted variant of this model (SAS-loops4). The ordinary percolation model (OPM) set of data is included for comparison.
to be significantly above the threshold because the slope for larger strip widths start to decrease. For the largest two widths this slope is smaller than the OPM value and for averaging over all points (bar widths) the value for t/v was 2.41 .... The scaling plot (Fig. 3) at p = 0.0940 gave straight line for widths above 6. Despite estimation that the percolation threshold seems to be closer to 0.0944 than to 0.0940 the conductivity scaling at 0.0940 gave a straight line without deviations for larger widths. The slope of this line gave t/v = 2.402 ± 0.003 with error bars given by the least-squares fit procedure. The best known to me estimate for 3d OPM t/v is 2.276 i 0.012 [21]. The correlation length exponent (u) was estimated both in 2 and 3d using the scaling behavior of the width of the transient region of the spanning probability curves [ 1 ]. Within the error bars of several percents it coincided with the OPM value.
5.2. The spanning cluster models The simulations were made in two dimensions only. The strips were shorter here (around 2 x 105) as a consequence of the need the spanning clusters to be extracted beforehand (see [ 19] for details). The results given on Fig. 4 were obtained after averaging over several runs (typically 5 or 10) with the exception of the two largest widths L = 200 and L = 300. The strips for these widths were 105 long. One run was done for L = 300 and the two runs for L = 200 were given as different points. The error bars for these largest widths were estimated using the records for the conductivity of every 1/5 of the strip length. (As a rule the conductivity of the first 1/5 had always been discarded throughout this study in order to reduce the "finite-length" effect on the results.) The errors for smaller widths were set equal to the standard deviation of the averaging over different runs.
F. Babalievski / Physica A 211 (1994) 1-12
ll
As seen on the figure the extracted values of t/p for these models are significantly different from the OPM ( ( t / p ) o e ~ t = 0.9745.. + 0.0015; Ref. [ 17] ). (The simulations of the OPM were done here in order to check three possible sources of systematic errors. In addition to checking the PRNG and the algorithm the influence of shorter strip lengths on the results was examined. The value obtained here: t/p = 0.983... 4- 0.015 is very close to the already known.) The Stoll&Courtens model t/~, was estimated to 0 . .5~ "~ +0.02 The error bars were .... -0.05" set much larger than given by the least-squares fit procedure because of the deviations from a straight line for the largest width. The same deviation was found for the other two models. This led me to additional computations for L = 200 and L = 300. In spite of the the poor statistics it seems that the change of the slope around L = 100 is relatively small. So the final estimates for the model with additional sites (Section 2.3) is 0 .446 ... +0.01_0.05for the basic model and 0. . .~ta . . . . . . +0.02 0.09. for the restricted model. A comparison with the results in Refs. [3,4] can be made via Eq. (1). The coincidence in the results is really surprising for the Stoll & Courtens model. The fracton dimension (ds) for this model was found [3] close to 1.53, and putting the obtained here (t/v) value into Eq. (1) give ds = 1.543 .... The comparison with the results in Ref. [4] with the two models of Section 3.3 (spanning clusters with additional sites) is more difficult because the models do not coincide exactly with any of the three variants studied in [4]. The value for ds obtained from the first (the basic) model t/l, estimation is ds = 1.619... which is close to the result in [4]for the loop addition parameter Po = 6: 1.63. Following the definitions in [4] the restricted model corresponds to a mixture of different loop addition parameters, so the value obtained by means of Eq. (1), ds = 1.54... seems to conflict with the expectation to be found in the range 1.63-1.69 (Po = 6 to 12 - see Ref. [4] ). This is due probably to the larger error bars: only four widths (9,13, 20 and 40) were used for obtaining the t/~, value for the restricted model. I can not exclude the possibility that this excellent coincidence for the other two models may be a numerical artefact and that using larger samples may give smaller (t/u) 's and hence bigger values for ds.
6. Conclusion
The obtained results for the transport exponent of several percolation models with loops divide these models in two classes regarding the change of this value. The transport exponent for one-step-bootstrap percolation (BP1) models is very close or coincide with the ordinary percolation value. The main result for the spanning cluster models consists in significantly changed transport exponent in comparison with ordinary percolation. The connections between the dynamical exponents (Eq. (1)) seems to hold for this class of models. The obtaining of results for the spectral and walk dimension for the vectorial elasticity models with loops would assist further work on the connections between the percolation critical exponents.
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F. Babalievski / Physica A 211 (1994) 1-12
The local connectivity modifications for percolation structures may lead to a significant change of the transport exponent even in the case of uniform distribution of the resistors. It is important that such modifications should take place on the incipient percolation cluster in this way preserving the fractal dimension of the conducting configuration.
Acknowledgement I would like to thank to H.J. Herrmann for suggesting this problem and for helpful discussions and comments. Special thanks to R. Ziff for the e-mail discussion on the spanning probability curves. S. Melin and H. Puhl helped me to use computational facilities at HLRZ. I am grateful to W. Vermoehlen for his help during my stay at HLRZ and for the final E~TEX-ing of the manuscript after my departure. This work was supported by the European Communities commission on science Grant No. CIPA3511PL920176.
References [ 1] D. Stauffer and A. Aharony, Introduction to percolation theory (second edition) (Taylor & Francis, London, 1992, second printing 1994); M. Sahimi, Applications of percolation theory (Taylor & Francis, London, 1994). [21 B. Halperin, S. Feng and P. Sen, Phys. Rev. Lett. 54 (1985) 2391. [3] E. Stoll and E. Courtens, Z. Phys. B 81 (1990) 1. [4] H. Nakanishi, Physica A 196 (1993) 33. [5] S. Alexander, E. Courtens and R. Vacher, Physica A 195 (1993) 286. [6] H.J. Herrmann, Phys. Rep. 136 (1986) 153. [7] S. Havlin, in: Random fluctuations and pattern growth, H.-E, Stanley and N.Ostrovski, eds. (Kluwer, Dordrecht, 1988). [8] R. Maynard, private communication. [9] R. Ziff, Phys. Rev. Lett. 69 (1992) 2670. [10] J. Chalupa, P. Leath and G. Reich, J. Phys. C 12 (1979) L31; M. Aizenman and J. Lebowitz, J. Phys. A 21 (1988) 3081; J. Adler, Physica A 171 (1991) 453. [11] M. Sahimi and T. Ray, J. Phys. 1 1 (1991) 685. [12] D. Stauffer, J. Adler and A. Aharony, J. Phys. A Lett. (in press). [13] R. Monetti, E. Albano, Z. Phys. B 90 (1993) 351. [14] B. Derrida and J. Vannimenus, J. Phys. A 15 L557 (1982). [15] B. Derida, D. Stauffer, H.J. Herrmann and J. Vannimenus, J. Phys. 44 (1983) L701. [16] B. Derrida, J. Zabolitzky, J. Vannimenus and D. Stauffer, J. Stat. Phys. 36 (1984) 31. [17] J. Normand, H.J. Herrmann and M. Hajjar, J. Stat. Phys. 52 (1988) 441. [18] E Babalievski, Z. Phys. B 84 (1991) 429. [ 19] E Babalievski, HLRZ Preprint 97/1993. [20] M.P. Nightingale, Physica A 83 (1976) 561. [21] D. Gingold and C.J. Lobb, Phys. Rev. B 42 (1990) 8220.
PHnlOA
Physica A 211 (1994) 1-12
The transport exponent in percolation models with additional loops F. Babalievski HLRZ at the KFA Jiilich, D-52425 Jiilich, Germany Inst. General and Inorganic Chem., 1113 Sofia, Bulgaria
Received 21 June 1994; revised I July 1994
Abstract Several percolation models with additional loops were studied. The transport exponents for these models were estimated numerically by means of a transfer-matrix approach. It was found that the transport exponent has a drastically changed value for some of the models. This result supports some previous numerical studies on the vibrational properties of similar models (with additional loops).
1. Introduction There are many systems where the ordinary percolation model [ 1 ] could be applied as a zero (or, say, first) approximation. Indeed there are quite a few systems where it might be a final approximation. That is one of the reasons, for the proliferation of the percolation archetype during the last two decades into a family o f percolation models. Some of them differ on the connectivity rule imposed (directed percolation, polychromatic percolation, AB percolation...); alterations were made of the underlaying lattice, continuum (off-lattice) models were introduced as well. The continuum models can be mapped [2] onto a random lattice with a broad distribution of bond lengths. This led to several studies of bond percolation models on square (cubic) lattice where the distribution of bond "strengths" was taken to have the same form as the distribution of bond lengths in the respective random lattice. The 'thermal-like' (the static) critical properties of these models were studied to a large extent, and it was shown, for the most o f the cases that they belong to the ordinary percolation universality class. Some other classes were distinguished as well - the directed percolation is an example. So the accumulated numerical and analytical results 0378-4371/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved S S D 1 0 3 7 8 - 4 3 7 1 (94)00173-1
F. Babalievski / Physica A 211 (1994) 1-12 offer the possibility to predict (to some extent) the static exponent values for a newly "generated" percolation model. The case with the s.c. dynamical exponents (see below for definitions) appears to be quite different. A lot of attempts were made [ 1 ] to find a connection between the static and the dynamic exponents but a "convincing" result is still missing. So, it appears that the accumulation of results about the value of the transport exponent for diverse percolation models, could be of use for clarifying the connection between static and dynamic exponents, as well as for justifying the connections between the dynamic exponents. Some recent studies [3,4] on vibrations on fractal networks gave the search direction for this study.
2. Background As is known [5] the density of vibrations on a fractal structure could be described (qualitatively) by localized excitation called fractons. The frequency dependence follows a power law N ( w ) cx w ds-1 resembling the phonon spectra expression, but with a new exponent (ds) called fracton (spectral) dimension "sitting in for" the Euclidian space dimension in the power term. It could be shown [ 6,7 ] that the spectral dimension could be expressed in terms of the mass fractal dimension and one of two other dynamic exponents: the transport exponent and the walk exponent, ds=2df/dw;
dw=t/~,+df-(d-2),
(1)
d and df being the euclidian and fractal dimension, and the walk exponents (tw) is defined as the scaling exponent for the mean end-to-end distance ( R ( t ) ) travelled of a random walker on a fractal structure - R ( t ) cx t 1/~w. The experimental techniques developed for measuring phonon spectra could be applied in many cases to fractons - situations contrasting with the need for developing conceptually new theoretical and numerical methods for studying the properties of fractal structures. Experimental systems whose geometry resembles the incipient percolation cluster are some silica aerogels (a first approximation of course...). The experimental work done so far showed that these systems have a value of the spectral dimension which is much higher than the value for vector elasticity on 3d percolation cluster. It was seen [8] under the microscope that the floppy ends of such aerogels tend to connect each other in this way enhancing with loops their percolating-cluster-like geometry. This inspired some numerical works to obtain the spectral and the walk exponent for percolation models with additional loops [ 3,4]. The results showed a drastic change of the respective values comparing with the ordinary percolation. Stoll and Courtens [3] calculated numerically the spectral dimension (ds) for a set of "extracted" spanning percolation clusters in 2d for three ranges of connectivity: (I)
F. Babalievski / Physica A 211 (1994) 1-12
3
first neighbors only, (II) first and second neighbors, and (HI) first and third neighbors. For the ordinary percolation (model ( I ) ) , they obtained a value near to 4/3 (namely 1.29), but for the models (II) and (III) the values were respectively 1.53 and 1.79. The numerical method consisted in obtaining each individual eigenvector and eigenfrequency of a dynamical matrix for 45 realizations of 68 x 68 clusters. In a recent work [4] Nakanishi studied another percolation model enhanced with loops. After numerical calculations of the eigenspectrum of the transition probability matrix, he found that the spectral dimension ds and the walk dimension dw change suddenly - as soon as the floppy ends of an incipient percolation cluster are connected together by adding new particles to the cluster in a way to increase the number of loops in the cluster. The systems' sizes were of the order of 100 for the 2d case, and 30 for the 3d case. The aim of the present work is twofold: first, to compare the percolation conductivity for two models of percolation with additional loops, and then to check the connection between the transport and spectral exponent for the models studied (by Stoll & Courtens and by Nakanishi).
3.
Models
Several site percolation models with additional loops were studied in order to reveal these changes of percolation structure which lead to a change of the transport exponent.
3.1. One step bootstrap percolation (BP1) The model was obtained by adding to the initial percolation structure new sites by the rule that such site should have more than one initially occupied neighbor. The building of such model needs two steps: (i) create an ordinary site percolation structure (ii) check for every unoccupied site if it has at least two occupied (on the first step) neighbors, if it is true for a certain site then turn this site into occupied
A completed bootstrap percolation model [ 10] could be obtained by repeating step two (using the "previous step" configuration for neighbors counting) until there is no more unoccupied site with more than one occupied neighbor. (Some authors [ 11 ] prefer to call this model diffusion percolation keeping the term bootstrap for the opposite procedure - discarding sites with "too many neighbors".) The second step (even done only once) leads to a drastic change of the density of occupied sites: Ap = (1 - p ) [ 1
- (1
_p)4 --4p(1 _p)3]
F. Babalievski
1.00
O O
I
I
I
Physica A 211 (1994) 1-12
I
0.75
z
zx
~'~ 0.50 O .~,-t
L=100 ........ L=200
L-ltV
I ~ ,I/ ~ I..¢~
× . . . . L=400
/
O 0.25
o - - - L=800 it~---
L=1200
o - - - L=2000 0 0
z
0.00 25.50
i
26.00
26.50
27.00
27.50
28.00
28.50
i
29.00
i
29.50
30.00x10 -2
p (site occupation probability) Fig. 1. Spanning probability curves for percolation threshold determination for the BP1 model in two dimensions. The insert presents the extrapolation of the finite size percolation thresholds to infinity.
in two dimensions, and in 3d it is:
A p = (1 - - p ) [ 1
-- (1 _ p ) 6
--6p(1
_p)5]
(It is worth to note that there are no free and linear terms in the polynomial presentations of these expressions.) It is clear that the structures obtained after one bootstrap iteration with initial concentration equal to the ordinary percolation threshold are not fractals, so one has to look for a threshold initial concentration. The threshold concentration in 2 and 3d were estimated by the standard procedure for obtaining the spanning probability curves for different system sizes and then extrapolating to infinity the "mean points" of these curves [fig 1,2]. What "mean point" should mean is clear for 2d ordinary percolation, it should be the 50% point of this experimental curves [9]. For 3d this quantity was estimated recently [ 12] to 0.42 .... I tried different techniques [9], but mainly kept within the Gaussian assumption. I accepted that the spanning curves follow the cumulative Gaussian distribution, and the mean of this distribution is "the mean point" to be extrapolated to infinity. (Or better to say, extrapolating to zero of L -1/~ - see the inset of Figs. 1 and 2). Contrasting with ordinary percolation models~ the "smooth curve" passing through the experimental points was not linear but quadratic polynomial in L -1/~. The same plots were made using as argument the concentrations after the bootstarp iteration (final concentrations), but results did not differ much. The percolation thresholds for two and three dimensions are p~d = 0.283 4- 0.002 and pc3d = 0.0942 ± 0.003. Expressed in final
F. Babalievski / Physica A 211 (1994) 1-12
" • 1.00
[
I
5
I
zx - -
L=20
• ........ 1.,=30 0.80 O O
o
9,~
0.60
o •~
x
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L'Uv
V
0.40
0 O Z
-
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.
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L=100
o
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•
L=400
L=20-5(3 1000 sa~aRl_esp_er_1,9_i_n_t...... The fit f.
O ~
I_,=50
9.S5
0.29
L = 2 0 0 - 4 3 0 : 1 0 0 s.p.p. 0.00 8.00
is: e r f ( ( p a 0 ) / a l ) / 2 + 1/2
L = 7 0 , 1 0 ) : 2 0 0 s.p p.
8.50
9.00
9.50
10.00
10.50
(a " g a u s dan" a s s m p t n . )
11.00
11.50x 10 2
p (site occupation probability) Fig. 2. Spanning probability curves for percolation threshold determination for the BP1 model in 3d. The insert presents the extrapolation of the finite size percolation thresholds to infinity.
concentrations these values are 0.51.. and 0.28.. - appreciably smaller than the values for square and cubic lattice (0.5927460.. and 0.3116.. respectively [9] ) in the ordinary percolation.
3.2. Spanning clusters with increased connectivity This model coincides with the Stoll & Courtens model (II) - see Section 2.2 and Ref. [ 3 ]. The starting configuration was a strip-like piece of a square lattice which sites were randomly "occupied". First, the spanning clusters (the clusters which connect the long edges of the strip) for the ordinary (site) percolation model (OPM) at threshold were extracted. Unlike the Stoll&Courtens work the "samples" are strip-like here, so there are a lot of different spanning clusters which connect the long edges of the strip [ 13]. The resistor networks were created connecting by unit resistors the nearest and the next-nearest neighbors (the second neighbors) among the occupied sites which belong to the spanning clusters. The difference with a site percolation model with first and second neighbor connections is that the spanning clusters are determined on the base of first neighbor connections only and the interaction spreads to the second neighbors after removing the nonspanning clusters. This model is intended to resemble the experimental situation where after the preparing of the infinite percolation cluster (e.g. a gel formation or a catalyst producing) its dangling ends may connect each other increasing in this way the connectivity and adding new loops to the percolation structure.
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F. Babalievski / Physica A 211 (1994) 1-12
3.3. Spanning clusters with additional sites
This model could be described as a combination of the previous two. As in the previous section first the spanning clusters have to be extracted. Then instead of increasing the connectivity range for the perimeter sites new sites are occupied in a way to connect the sites from different dangling ends of the spanning clusters. This is a two step procedure as for the previous models: (i) extract the spanning clusters for the OPM at percolation threshold (i.e. turn the nonspanning clusters' sites into unoccupied) (ii) check for every unoccupied site if it has at least two occupied (on the first step) neighbors, if it is true for a certain site then turn this site into occupied
The first step is the same as the first step for the Stoll&Courtens model and the second step coincide with the second step of the BP1 procedure and also is made only once. The obtained structures are similar to the models studied by Nakanishi [4] - see Section 2. He imposed additional restrictions for adding a new site in step 2 of the above procedure. It is checked how large would be the closed loops after occupying the questioned site. Depending on the result the proposal for occupying a site in step 2 may be rejected. The results obatined in his paper for the spectral and walk dimension do not change much (see Section 5.2) when changing the loops size control. Therefore, the size of loops added was not restricted for the prevailing part of the structures simulated here. Some samples were prepared with an additional restriction for adding a new occupied site to the spanning cluster - the new site should not close the smallest (4 site) loop. This restriction differs slightly from the Po = 6 rule in Ref. [4] where the addition is rejected when the minimal closed loop for every pair of initially occupied neighbors equals to 4. This difference was left for simplifying the loops identifying function. These two models will be referred further as the basic and the restricted model respectively.
4. Numerical method The percolation conductivity of the structures examined here was studied numerically by means of the transfer-matrix (TM) approach to the Random Resistor Network (RRN) problem. The resistor networks consisted in unit resistors which bridged the nearest neighbors (and the next-nearest neighbors for the model) in the final structures described above. As known, the TM approach to the RRN problem was originally proposed by Derrida and Vannimenus [14] and then refined by several groups [ 15-18]. The modification proposed in [ 18 ] was adapted for the purposes of the present study. Further modifications were made in order to meet the requirements of the spanning cluster models - the
F. Babalievski / Physica A 211 (1994) 1-12
7
clusters have to be extracted before the TM calculations. The details of the algorithm are presented elsewhere [ 19] - I will recall here only the main features of the TM approach. It makes use [ 14,15] of the general idea of phenomenological renormalization introduced by Nightingale [20] for calculation of critical exponents near second order phase transition. The quantities of interest are calculated for very long and relatively narrow strips. Then by means of scaling of the width of these strips one could extract the critical exponents. In the case of the 2d percolation transport exponent the TM method could be described briefly as follows: First, the conductivities of strips from the percolation structure (usually at p = Pc) are calculated (exactly). The width (L) of these strip is chosen having in mind that their transverse conductivity should scale as L - t / v where t and v are the transport and the correlation length exponents respectively. The conductivity of a certain strip is obtained by means of a "virtual building" of the resistor network inside the frames of two electrodes attached to the long edges of the strip. Initially the space between the electrodes is empty. Then the resistors "take their places" (one after one) on the underlaying lattice filling it column by column starting (say) from the left end of the strip. The resistors are added in a way to ensure required properties of the final structure [ 19]. After adding every resistor the conductivity between the electrodes is recalculated, so, after adding all of the resistors in the strip the conductivity between electrodes is the transverse conductivity of the strip. (Neglecting the "edge" effect is reasonable, when the strip length is much greater than its width.) The second step consists in performing finite-size scaling analysis to extract the value of t / v . Three difficulties in obtaining this value will be mentioned here : the correctionto-scaling terms (important for small widths), the precision of the percolation threshold determination, and third, the increasing of relative error for the strip conductivity when the strip width is increased (the computational effort increases too, with a power of L significantly larger than one). It is accepted that correction to scaling could be expressed as O'L = L - t / v ( 1 + a L - ° ' . . . ) ,
(2)
with a ~ - 1 and to ~ 1 for the ordinary percolation. An additional "electrode" effect appeared for the models studied here. The upper and lower lattice layers of the strip are filled with probability one, to represent the electrodes. Then the chance for adding new sites in the nearest layers inside the strip is higher than in the other bulk layers, because the mean number of occupied neighbors is higher due to the electrodes. This effect alone would lead to a positive value of a, and (generally said) different value for to. The competition between these effects made the deviations in the region of small sizes smaller and more difficult to describe. The practical solution was simply to discard the small widths in determining the transport exponent. The other special points of the data analysis will be discussed in the next section. The pseudo random number generator(PRNG) used in this study was drand48. It was tested together with the programs by obtaining the known values for percolation
8
F. Babalievski / Physica A 211 (1994) 1-12
threshold for square and cubic lattice and the values for the transport exponent for ordinary percolation. The ordinary percolation simulations were made by changing one number in one instruction in the programs for simulating the modified models: i f (NoNeib > Number) { t h e n add a now s i t e . . .
}
For the models with additional loops the Number was equal to 1 and when comparing with ordinary percolation it was set larger than the lattice coordination number. (Thus making impossible the addition of a new site.) "NoNeib" contained the number of neighbors in the initial percolation structure. This number was obtained by means of a "moving sandwich" construction by the following procedure: Three vertical lattice layers (L1,L2,L3) of the strip are created according to the OPM. The variable "NoNeib" is calculated for every empty site in the middle layer. If the above condition is fulfilled the site changes into occupied. Then the final structure obtained for the middle layer is passed to the conductivity measuring part of the program as input data. The moving of the "sandwich" is ensured by the following instructions: LI = L2; L 2 = L3; L 3 = a n e w
OPM
configuration
The calculations were made on the SUN SPARC cluster at HLRZ. The computational efforts were approximately equivalent to 800-1000 hours of SPARC10 CPU time (including the percolation threshold estimations). The preliminary attempts to run the programs on a CM5 machine remained unfinished.
5. Results
5.1. The BP1 model This is the model where the most computational efforts were invested. It was the only model studied in 3d. The value of t / v was found (both in 2 and 3d) to be very close to the OPM value but the difference seemed statistically significant. Despite the efforts, the accuracy of the percolation threshold determination remained insufficient to allow the neglect of the threshold uncertainty contribution to the above difference. The description of the results for the percolation thresholds was given in the previous section, indeed these values were made more exact in parallel with conductivity calculations. The prevailing part of the calculations were made slightly above and below the suspected value for the percolation threshold. In two dimensions the conductivity calculations were done at p = 0.2833, 0.2830 and 0.2820. Despite of estimating the percolation threshold to be around 0.283, the scaling plots (Fig. 3) for p = 0.2833 and 0.2830 showed deviations from the straight line for larger widths which means (most probably) that 0.283 is above Pc. It seems that Pc is closer to 0.282, because the conductivity scaling plot at this value showed smaller deviations from straight line (on the log-log scale, of course). The estimation for the t / v value was obtained by averaging the exponents extracted from every of these three lines.
F. Babalievski / Physica A 211 (1994) 1-12
9
0.1
I
p=0.2833(2d) o 1.81/x^2.046 ..... p=0.2820(2d) +
>.. ~ .....
p=0.944 n
~7"'%
p=O.O940(3d) x 3.0/x^2.402
-~.. ~ "'~..
0.01
• El
"-,i~.
× "'0.
n × "",¢,.... ~
'~-
"'*-... X
o.ool
""~"-.
~
--,..
0.0001
I e-O05
,
10
,
,
I
100 Strip Width
Fig. 3. Finite-size scaling plots for conductivity at the percolation threshold for the BP1 model in 2d and 3d.
Direct fitting with the correction to scaling form (Eq. ( 2 ) ) was not used because of the electrode effect mentioned above. The deviations from pure power law was evaluated by doing the fit first with all points and then discarding the smaller width and repeating the fit, then discarding the next smaller width, and so on until two points are left. The same procedure was repeated but starting from the larger width. Some fits were made with the middle widths only. The strip widths ranged up to 100 and the strip lengths were of the order of several millions. After discarding the widths below 10, the remaining points were fitted with a simple power law XL = aL -1-t/v where XL is the conductance per unit strip length. (To obtain the conductivity one has just to multiply by L.) The fits gave t/v = 1.021, 1.017, 1.046 for p = 0.2833, 0.2830 and 0.2820 respectively. The least-squares fitting procedure determined error bars less than 0.01, so the main source of uncertainty was the percolation threshold estimation. The method was tested (see below) by determining the OPM t/v. The resulting exponent 0.99 -t- 0.01 showed that the error bars are underestimated (the accepted estimate is 0.9745 4-0.0015 [ 17] ). Hence as a final result for the BPlmodel's t/v in 2d should be given 1.024-0.03 without excluding the chance of coinciding with the OPM value. In three dimensions the same procedure was followed. Two values were taken for p: one above (p = 0.0944) and one below (p = 0.0940) the percolation threshold estimate. The bar width was varied between 6 and 30 for both concentrations. The bar length was 106 for L = 6 to 11 and 4 × 104 for widths larger than 11. The runtime scaled approximately as L zS. For L = 30 it was 30 hours SPARC10 CPU time. A run with L = 40 done for p = 0.0940 took 80 hours CPU time. The value 0.0944 seem
F. Babalievski / Physica A 211 (1994) 1-12
10
m
~'~. ....
~
. . . . . . . .
~ 1+ - - S A S mod:0.446
lO't
~ ........ S A S - I o o p s 4 : 0 . 5 6 4
c.:O.983(estd.) I 10 "2
.~.,q
I0 )
0
10 ~
10 z
Strip Width Fig. 4. Finite-size scaling plots for conductivity at the percolation threshold for three spanning cluster models: spanning clusters with connectivity increased to the next-nearest neighbor model (SNNN); spanning clusters with additional sites (SAS) and a restricted variant of this model (SAS-loops4). The ordinary percolation model (OPM) set of data is included for comparison.
to be significantly above the threshold because the slope for larger strip widths start to decrease. For the largest two widths this slope is smaller than the OPM value and for averaging over all points (bar widths) the value for t/v was 2.41 .... The scaling plot (Fig. 3) at p = 0.0940 gave straight line for widths above 6. Despite estimation that the percolation threshold seems to be closer to 0.0944 than to 0.0940 the conductivity scaling at 0.0940 gave a straight line without deviations for larger widths. The slope of this line gave t/v = 2.402 ± 0.003 with error bars given by the least-squares fit procedure. The best known to me estimate for 3d OPM t/v is 2.276 i 0.012 [21]. The correlation length exponent (u) was estimated both in 2 and 3d using the scaling behavior of the width of the transient region of the spanning probability curves [ 1 ]. Within the error bars of several percents it coincided with the OPM value.
5.2. The spanning cluster models The simulations were made in two dimensions only. The strips were shorter here (around 2 x 105) as a consequence of the need the spanning clusters to be extracted beforehand (see [ 19] for details). The results given on Fig. 4 were obtained after averaging over several runs (typically 5 or 10) with the exception of the two largest widths L = 200 and L = 300. The strips for these widths were 105 long. One run was done for L = 300 and the two runs for L = 200 were given as different points. The error bars for these largest widths were estimated using the records for the conductivity of every 1/5 of the strip length. (As a rule the conductivity of the first 1/5 had always been discarded throughout this study in order to reduce the "finite-length" effect on the results.) The errors for smaller widths were set equal to the standard deviation of the averaging over different runs.
F. Babalievski / Physica A 211 (1994) 1-12
ll
As seen on the figure the extracted values of t/p for these models are significantly different from the OPM ( ( t / p ) o e ~ t = 0.9745.. + 0.0015; Ref. [ 17] ). (The simulations of the OPM were done here in order to check three possible sources of systematic errors. In addition to checking the PRNG and the algorithm the influence of shorter strip lengths on the results was examined. The value obtained here: t/p = 0.983... 4- 0.015 is very close to the already known.) The Stoll&Courtens model t/~, was estimated to 0 . .5~ "~ +0.02 The error bars were .... -0.05" set much larger than given by the least-squares fit procedure because of the deviations from a straight line for the largest width. The same deviation was found for the other two models. This led me to additional computations for L = 200 and L = 300. In spite of the the poor statistics it seems that the change of the slope around L = 100 is relatively small. So the final estimates for the model with additional sites (Section 2.3) is 0 .446 ... +0.01_0.05for the basic model and 0. . .~ta . . . . . . +0.02 0.09. for the restricted model. A comparison with the results in Refs. [3,4] can be made via Eq. (1). The coincidence in the results is really surprising for the Stoll & Courtens model. The fracton dimension (ds) for this model was found [3] close to 1.53, and putting the obtained here (t/v) value into Eq. (1) give ds = 1.543 .... The comparison with the results in Ref. [4] with the two models of Section 3.3 (spanning clusters with additional sites) is more difficult because the models do not coincide exactly with any of the three variants studied in [4]. The value for ds obtained from the first (the basic) model t/l, estimation is ds = 1.619... which is close to the result in [4]for the loop addition parameter Po = 6: 1.63. Following the definitions in [4] the restricted model corresponds to a mixture of different loop addition parameters, so the value obtained by means of Eq. (1), ds = 1.54... seems to conflict with the expectation to be found in the range 1.63-1.69 (Po = 6 to 12 - see Ref. [4] ). This is due probably to the larger error bars: only four widths (9,13, 20 and 40) were used for obtaining the t/~, value for the restricted model. I can not exclude the possibility that this excellent coincidence for the other two models may be a numerical artefact and that using larger samples may give smaller (t/u) 's and hence bigger values for ds.
6. Conclusion
The obtained results for the transport exponent of several percolation models with loops divide these models in two classes regarding the change of this value. The transport exponent for one-step-bootstrap percolation (BP1) models is very close or coincide with the ordinary percolation value. The main result for the spanning cluster models consists in significantly changed transport exponent in comparison with ordinary percolation. The connections between the dynamical exponents (Eq. (1)) seems to hold for this class of models. The obtaining of results for the spectral and walk dimension for the vectorial elasticity models with loops would assist further work on the connections between the percolation critical exponents.
12
F. Babalievski / Physica A 211 (1994) 1-12
The local connectivity modifications for percolation structures may lead to a significant change of the transport exponent even in the case of uniform distribution of the resistors. It is important that such modifications should take place on the incipient percolation cluster in this way preserving the fractal dimension of the conducting configuration.
Acknowledgement I would like to thank to H.J. Herrmann for suggesting this problem and for helpful discussions and comments. Special thanks to R. Ziff for the e-mail discussion on the spanning probability curves. S. Melin and H. Puhl helped me to use computational facilities at HLRZ. I am grateful to W. Vermoehlen for his help during my stay at HLRZ and for the final E~TEX-ing of the manuscript after my departure. This work was supported by the European Communities commission on science Grant No. CIPA3511PL920176.
References [ 1] D. Stauffer and A. Aharony, Introduction to percolation theory (second edition) (Taylor & Francis, London, 1992, second printing 1994); M. Sahimi, Applications of percolation theory (Taylor & Francis, London, 1994). [21 B. Halperin, S. Feng and P. Sen, Phys. Rev. Lett. 54 (1985) 2391. [3] E. Stoll and E. Courtens, Z. Phys. B 81 (1990) 1. [4] H. Nakanishi, Physica A 196 (1993) 33. [5] S. Alexander, E. Courtens and R. Vacher, Physica A 195 (1993) 286. [6] H.J. Herrmann, Phys. Rep. 136 (1986) 153. [7] S. Havlin, in: Random fluctuations and pattern growth, H.-E, Stanley and N.Ostrovski, eds. (Kluwer, Dordrecht, 1988). [8] R. Maynard, private communication. [9] R. Ziff, Phys. Rev. Lett. 69 (1992) 2670. [10] J. Chalupa, P. Leath and G. Reich, J. Phys. C 12 (1979) L31; M. Aizenman and J. Lebowitz, J. Phys. A 21 (1988) 3081; J. Adler, Physica A 171 (1991) 453. [11] M. Sahimi and T. Ray, J. Phys. 1 1 (1991) 685. [12] D. Stauffer, J. Adler and A. Aharony, J. Phys. A Lett. (in press). [13] R. Monetti, E. Albano, Z. Phys. B 90 (1993) 351. [14] B. Derrida and J. Vannimenus, J. Phys. A 15 L557 (1982). [15] B. Derida, D. Stauffer, H.J. Herrmann and J. Vannimenus, J. Phys. 44 (1983) L701. [16] B. Derrida, J. Zabolitzky, J. Vannimenus and D. Stauffer, J. Stat. Phys. 36 (1984) 31. [17] J. Normand, H.J. Herrmann and M. Hajjar, J. Stat. Phys. 52 (1988) 441. [18] E Babalievski, Z. Phys. B 84 (1991) 429. [ 19] E Babalievski, HLRZ Preprint 97/1993. [20] M.P. Nightingale, Physica A 83 (1976) 561. [21] D. Gingold and C.J. Lobb, Phys. Rev. B 42 (1990) 8220.