Percolation as a fractal growth problem

Physica A 170 (1990) 64-8(1 North-Holland P E R C O L A T I O N AS A F R A C T A L G R O W T H P R O B L E M L. P I E T R O N E R O Dipartimento di Fisica, Universita di Rorna "'La Sapienza", P.le A. Moro 2, 00185 Roma, Italy A. S T E L L A Dipartimento di Fisica " G . Galilei", Universita di Padova, Via Marzolo 8, Padova, lmly Received 22 June 1990 We consider percolation in two dimensions as a fractal growth problem, and apply to it the theory of fractal growth based on the fixed-scale transformation approach developed for diffusion-limited aggregation and the dielectric breakdown model. This represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group. We compute the fractal dimension of the percolating cluster including terms up to third order. The result is D = 1.8830 for the square lattice and D = 1.8650 for the triangular lattice. These values are in excellent agreement with the universal exact result D = 91/48 = 1.8958 and also show the potential of this new method for standard problems. 1. Introduction The theory of fractal growth based on the fixed-scale transformation (FST) approach [1] exploits a new invariance property in addition to scale invariance on which the renormalization group ( R G ) is based. The FST method was developed having in mind irreversible growth problems that are intrinsically critical (for recent conference proceedings see, e.g., ref. [2]) like diffusionlimited aggregation ( D L A ) [3] and the dielectric breakdown model ( D B M ) [4]. This new approach clarifies the origin of fractal structures in these models and it allows the computation of the fractal dimension in a rather systematic way as is also confirmed by the recent application to the D B M in three dimension [5]. The FST is defined with respect to the dynamical growth process at the same scale and reflects the expected invariance of the fractal properties tested in different regions of the space where growth occurs. The corresponding fixed point is then used at all scales in order to compute the fractal dimension. This implies that the growth process to be used in this transformation should be the 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland) L. Pietronero and A. Stella / Percolation as a fractal growth problem 65 one corresponding to a generic scale of coarse graining in the self-similar regime. This does not necessarily correspond to the growth process used at the minimal scale. This problem was not considered in ref. [1] but recently we have introduced a method that allows the inclusion of this point into the theory [6]. In order to test these concepts, which are not involved in the usual renormalization group, we have decided to apply them to the well known problem of percolation [7] for which much more information is available with respect to D L A and the DBM. In order to do this it is convenient to interpret percolation as a growth process [8] and to focus on the fractal dimension of the percolating cluster. A basic difference between percolation and D L A or D B M is that these last two are intrinsically critical, while percolation requires the fine tuning of being exactly at Pc in order to give rise to a fractal structure on all length scales. The FST theory is based on intrinsically critical growth processes and this condition is satisfied only at the percolation threshold. Using as input the value of Pc for the given lattice topology it is then possible to compute the fractal dimension of the percolating cluster in a straightforward way. Including terms up to third order in the calculations we obtain D = 1.8830 for the square lattice and D = 1.8650 for the triangular lattice. These values are in very good agreement with the universal exact result D = 91/48 = 1.8958 [9] and represent an important test for the validity of these concepts. In the following paper we are also going to apply the FST method to the problem of invasion percolation [10]. 2. The fixed-scale transformation In this section we present a brief summary of the elements of the FST theory [1] that are necessary for the percolation problem. Suppose that a fractal structure of dimension D is generated by an irreversible growth process, e.g., in d = 2. It is useful to consider the intersection of this structure with a line perpendicular to the growth direction. This intersection leads to a set of points with fractal dimensions D'= D- 1. (2.1) By analyzing this set of points with a box-covering procedure we assign a black dot to a box if this contains some point of the set and a white dot otherwise. By decreasing the scale of the box by a factor of two each black box is divided into two sub-boxes. The possible configurations of these pairs are denoted as type 1 (one black sub-box and one white sub-box) and type 2 (both sub-boxes black). The corresponding probabilities of occurrence in the process of fine graining 66 L. Pietronero and A . Stella / Percolation as a fractal growth problem are indicated by C~ and C 2, respectively. The average number of black sub-boxes that appear at the next level of fine graining from one black box is (n) = ~ n;C; = C I + 2C 2 , (2.2) where n; indicates the number of black sub-boxes in the configuration of type i. It is easy to show that the fractal dimension of the whole structure is related to the values of C 1 and C 2 by [1] In ( n ) D = 1+ - In 2 (2.3) The problem is then to define the asymptotic distribution of elementary configurations {(7;}. To this purpose we have to define an appropriate selfconsistent iterative scheme and study the fixed point for this distribution. For a self-similar structure this asymptotic distribution should not only be invariant with respect to a change of scale but also with respect to the dynamical evolution at the same scale. The renormalization group would be based primarily on the first invariance property while the FST is based on the second one. This new type of approach seems to be more natural for problems of fractal growth and it does not have the conceptual difficulties that applications of R G ideas to these problems seem to have encountered up to now [11]. It is important to notice, at this stage, that the C; defined above clearly have the meaning of average probabilities for all cells on the line. Thus, one can expect that a self-consistent determination of them, as realized through the fixed-point condition of the FST, should be rather successful. The fixed-scale transformation is defined with respect to the dynamical evolution at the same scale. This gives rise to an iterative equation of type, see ref. [1], ~C~C]k+"] 2k+I)]{ = (M:I; M~I;)( C~k)] , (2.4) where k is the index of the iteration and M;,j defines the conditional probability that a configuration of type i is followed by one of type j, in the growth direction. Note that this probability refers to a frozen structure that has already grown to its asymptotic state and in which no further growth will occur in the future. The practical calculation of M;.j can be done by starting with a configuration of type i and considering the growth processes that can lead to a subsequent configuration j. For example, M~, 2 = (growth processes leading from 1 to 2 ) = p ~ +(1-p,~)pt3 + ' " , (2.5) L. Pietronero and A. Stella / Percolation as a fractal growth problem 67 where p~ is the probability that the first non-trivial growth process leads to a configuration of type 2, while Pt3 corresponds to the second-order term and so on. In order to define the growth process completely it is necessary to specify the boundary conditions whose fluctuations are often quite relevant. Each type of boundary condition gives rise to a different set of values for the matrix elements M~,j and this complicates the structure of the iteration given by eq. (2.4). If only one type of boundary condition is used the fixed point of this iterative equation is simply [1] C1= M1,2, ~ -1 l+~z,l / (2.6) , and from this, via eqs. (2.2) and (2.3), one obtains the fractal dimension D. In order to include the fluctuations of the boundary conditions it is necessary to know the probability that the pair of sites corresponding to the initial configuration i has a void of size h as a neighbour. A simple approximation for this effect is to assume that, if h > 0 , then one replaces it by h---~oo. This corresponds to the open-closed approximation (1), which is especially suitable for percolation because, contrary to the cases of DLA or BDM, in this case the growth probability is local and does not depend on the distance of the next branch. So, in practice, we have only two types of boundary conditions: closed if the starting pair is immediately followed by an occupied site (h = 0) or open in all other cases. The probability to have a closed configuration is [1, 12] c2 Po = P ' ( A = O) = 1 + ' (2.7) while the probability for the open configuration is P= = ~ P'(A,,) = 1 - Po. (2.8) n=l Within this scheme one has to consider two types of matrix elements, Mi,op~ and M i,j, d corresponding, respectively, to open and closed boundary conditions. The iterative process then becomes [1] (["~(k+l))={[PomCll,"[-Pooml,pl] ~2('(k+1) \ [eoM~12, + P~Mi,~ l [Pom21,1 "[- Po~M2,Pl])(C~k)) [PoMz,,z + p~M~,~ l \ C~k) and the explicit solution for the fixed-point distribution is [1] (2.9) 68 L. Pietronero and A . Stella / Percolation as a fractal growth problem { M c' 1,2 + 2 M 2 ' 1 ° P 2,1 -M~'2-2M~ -- 23 M 2 P I - I c 3[\1 2Az'Jt C1 = ~ ) 2 - 4 M ~ L 2, 1A]'/2 ~,1 } 2A (2.1o) A = M d1,2 + MCl2.1 -- ~-'"" 3 g A//°p 1.: + M~.P,), from which the fractal dimension D can be obtained. 3. Bond percolation on the square lattice In order to apply the method of the fixed-scale transformation it is most convenient to define a dynamic growth process that generates the percolating cluster. This can be achieved by the following process [8]. Given a starting configuration of sites (even a single one) connected by occupied bonds one considers all the bonds of the p e r i m e t e r of this structure and for each bond a stochastic process takes place (only once) and with probability p the bond becomes part of the structure while with probability (1 - p) it will be e m p t y for ever. It is easy to see that a cluster generated by this dynamical process has the same statistical weight as in the percolation process [7, 8]. One should note, however, that even for p = Pc it is not obvious that this process will generate the infinite percolating cluster. There is in fact a finite probability (strictly equal to one, for p ~< Pc) that this growth process will generate a finite cluster that is in general present at the percolation threshold in the standard percolation problem. Since the fractal dimension is only well defined for the infinite cluster we have to c o m p l e m e n t this growth process with a condition of connectivity in order to m a k e sure that we are actually dealing with the infinite cluster. We are going to see in the following two different ways of implementing this connectivity condition. A n o t h e r problem is the meaning of the growth process for p #- Pc and the difference between site and bond percolation. The FST method as discussed in ref. [l] was originally formulated assuming that the growth rules considered are a priori scale invariant. Recently, we have c o m p l e m e n t e d this work with a scheme that allows us to decide whether a growth process is scale invariant and, if not, one can derive which will be the asymptotic (scale invariant) growth process that corresponds to a particular microscopic process [6]. This analysis also suggests that b o n d growth processes rather than site ones, are good candidates to display scale invariance in the sense required by the FST transformation. For percolation one naturally expects that the scale-invariant growth process is the only one corresponding to the percolation threshold: P = Pc = 0.5 for bond percolation on the square lattice [7]. We can now define the FST transformation via the matrix element Mi. p L. Pietronero and A. Stella / Percolation as a fractal growth problem 69 corresponding to the probability that a configuration of type i is followed, in the growth direction, by a configuration of type j. As shown in fig. 1 the scheme to compute these matrix elements is similar to the one used for D L A and DBM in ref. [1]. In fig. 1 one starts with a frozen configuration of type 1 (encircled) in which no growth can occur. The case illustrated refers to the open configuration in the sense that there is no nearby side branch to the structure that is considered. Later we are going to consider also the case of closed configurations in which the structure considered is immediately followed by another branch. We intend to describe the growth in relation to the infinite (connected) percolating cluster and the growth probability is only considered in the column above the starting configuration [1]. One way to implement the condition of connectivity is to assume a priori the existence of an occupied line of points above the occupied site of the I ~ I (A) I At o r d e r I JH~O IW vI I , (B) H nd order I i (C) "~ IIIrdorder Fig. 1. Probability tree for the calculation of the FST matrix elements. In the case shown one starts from a frozen configuration of type 1 (encircled). Connectivity is guaranteed by a pre-existing vertical line. W h e n a bond is being considered in the growth process, this is characterized by an arrow. With probability p this process will lead to the occupation of the bond and the corresponding site, while with probability 1 - p the bond is eliminated forever. In such a case it will be denoted by a cross. L. Pietronero and A. Stella / Percolation as a fractal growth problem 7{} starting configuration. This assumption is sometimes also used in numerical simulations of dynamic or epidemic percolation [8]. Later on we are going to implement the connectivity condition with a different m e t h o d and the results will be extremely close for the two methods. The first non-trivial growth process considered is shown by the arrow in the top configuration of fig. 1. This bond will be occupied with probability p (we omit the index c for simplicity). This will lead to a configuration of type 2 following the starting one of type 1. In the opposite case, with a probability 1 - p the bond does not b e c o m e occupied and it will remain e m p t y for ever. In this case the bond will be denoted by a cross. The order of the calculation is defined by the n u m b e r of different arrows that are necessary to reach the initially e m p t y site above the starting configuration. The events that lead to the occupation of this site up to third order are shown in fig. 1. The corresponding matrix elements are M~P2(I) = p = 0 . 5 , (3.1) op p)p2 = M1,2(II ) = p + (1 M~P2(III) = p + (1 - 0.625, p)p2 + (1 (3.2) - p)Zp3 = 0.65625, (3.3) where the index " o p " refers to the open configuration (or boundary conditions) and the R o m a n numeral in parentheses refers to the order of the calculation. In fig. 2 we show the scheme of calculation for the dynamical evolution conditional to a starting configuration of type 2. For simplicity we omit in this figure the starting connected line and when a configuration of type 2 is reached this is indicated by an encircled 2. The situation is slightly more complex than in fig. 1 because the third-order processes also include a three-step loop that goes out from the column considered but finally leads to the occupation of the relevant site. The matrix elements are in this case M2.P2(I) = p + (1 - p)p (3.4) = 0.75, (3.5) M2P2(II) = M2P2(I) + (1 - p)Zp2 = 0.8125, M Y2(III) = M P2(II) + (1 - p)3p3 + (1 - p) + (1 - p y p 5 + (1 - p)4p3 = 4 + (1 - p f p 4 0.849605. (3.6) We can turn now to the analysis of the closed-boundary conditions. This means that the starting configuration is immediately followed on the right by an L. Pietronero and A. Stella / Percolation as a fractal growth problem ~ /,-., 71 /(l.p) ---Z~ Oq • T ¢-'\. / • [] oq oq [~ oq Fig. 2. Similar to fig. 1 but starting from a configuration of type 2. occupied point and t h e r e f o r e one has to consider also the growth processes that originate from this point. For the rest the situation is analogous to the previous (open) case. F r o m an analysis of fig. 3 we obtain cl M1,2(I ) = p = 0 . 5 , (3.7) M~,2(II) = p + (1 - p)pZ + (1 - p)Zp3 + (1 - p)Zp2 = 0.71875, (3.8) M~l,2(III) = p + (1 - p)p2 + (1 - p)Zp3 + (1 - p)Zp3 + (1 - p)3p4 + (1 - p ) 3 p 3 + (1 - p ) 3 p Z = 0.7422, where the index "cl" refers now to closed b o u n d a r y conditions. (3.9) 72 L. Pietronero and A. Stella / Percolation as a fractal growth problem ~-N" [] [i--6] o [-~-6] o "-% [] [] ~--N• / "" 7 " ".3 / ,/((l-p) :ai \ • • [] Fig. 3. If the growing structure is immediately followed on the right side by an occupied site we say that this is a situation with "closed" boundary conditions. In this case more processes can occur with respect to the "open" boundary condition case shown in fig. 1. I n fig. 4 w e s h o w t h e c a s e o f t h e c l o s e d b o u n d a r y c o n d i t i o n w i t h a s t a r t i n g c o n f i g u r a t i o n o f t y p e 2. T h i s l e a d s to M~12(I) = p + (1 - p ) p = 0 . 7 5 , (3.1o) M~I2(II ) = p + (1 - p)p + (1 _p)ZpZ + (1 _p)3p3 + (1 _p)3pZ = 0 . 8 5 9 3 7 , (3.11) M~I,2(III) = p + (1 - p ) p + (1 - p)2p2 + (1 - p)3p3 + (1 - p)~p2 + (1 _ p)4p4 + (1 _ p)4p3 + ( l _ p)4p2 = 0 . 8 7 1 0 9 . (3.12) B y i n s e r t i n g t h e f o u r m a t r i x e l e m e n t s ( c o r r e s p o n d i n g to a g i v e n o r d e r ) i n t o e q . L. Pietronero and A. Stella / Percolation as a fractal growth problem ~-~ 73 O [] ~~o N o~ N to-61 ~ ~o~ li-61 • ~ N • • ~t¢~-p) ~-1 [!1 Fig. 4. Analogous to fig. 3, in the sense that we have "closed" boundary conditions. The starting configuration is of type 2 here. (2.10) and using eqs. (2.2) and (2.3) one can now compute the fractal dimension for the various orders considered. The results are D ( I ) = 1.7370, (3.13) D ( I I ) = 1.8643, (3.14) D ( I I I ) = 1.8830. (3.15) Comparing these results with the exact value D = 91/48 = 1.8958 [9], one can see that they are very accurate and are rapidly converging to it. Already the first-order value, whose calculation is extremely simple, gives a value that is 74 L. Pietronero and A . Stella / Percolation as a fractal growth problem within ten per cent of the exact result. The third-order result is instead within one per cent. These results support, therefore, the validity of the new approach based on the fixed-scale transformation tested on a problem for which the exact result is known. The degree of accuracy of the above results is usually not met by analytical renormalization group calculation of the critical exponents. We come back now to the p r o b l e m of the connectivity condition, which in the previous calculation was ensured by the assumption of a starting line of occupied points. It is actually possible to implement this condition by a different method and we are now going to see that this leads to results that are extremely close to the previous ones. This consists in considering that the starting configuration should be connected to a line at a certain distance, this distance corresponds now to the order of the calculation. For example, fig. 5 shows the case of open boundary conditions with a starting configuration of type 1. We then consider the possible configurations that connect the starting structure to a dashed line at the distance of two bonds. In this sense the case shown corresponds to a second-order calculation. As one can see in the left-hand structure of fig. 5 the bond above the starting black site must be occupied otherwise the structure cannot be connected. Note that the white site in the starting configuration can never b e c o m e black because our calculation refers to the probability of evolution conditional to a given starting configuration. Therefore, we have to consider only the three bonds indicated by the thin lines in the box on the left-hand side of fig. 5. This gives a total of eight configurations but out of these we consider only those that connect the starting structure with the final line. Each of these configurations should then be weighted by its probability of occurrence. For a configuration with n occupied bonds and n' e m p t y ones this probability is pn(1 _p)n,. Since in our case P = Pc = 0.5 the statistical weight is the same for all the configurations. Looking now at the five possible configurations shown in fig. 5 we can see that the configurations that contribute to the matrix elements M 1,2 are only (c), (d) and (e). In fact (a) leaves an e m p t y site above the starting configuration while (b) leads to two separate clusters so that the two occupied sites on the (a) (b) (c) (d) (e) Fig. 5. The figure on the left-hand side (encircled) shows the initial configuration (the first growth process is trivial) for the calculation of second-order matrix elements with a different connectivity condition. In this case connectivity is guaranteed by the fact that we only consider the connected configurations with their statistical weight. Only the cases (c), (d) and (e) contribute to M~,~2 as discussed in the text. L. Pietronero and A. Stella / Percolation as a fractal growth problem 75 right-hand side would not be part of the percolating cluster. In summary, there are five connected configurations, three of which lead to a configuration of type 2 above the starting configuration of type 1. Therefore, we have M~P2(II) -- -35= 0.6. (3.16) By repeating this analysis starting with a configuration of type 2 we obtain M2,P2(II) = ~ = 0.75. (3.17) If we shift the final line to a distance of three bonds from the starting configuration we can compute the matrix elements up to third order in a similar way. The calculation is slightly more elaborate and gives MI,P2(III) = ~ = 0.73077, (3.18) ME,P2(III) = ~ : 0.83333. (3.19) The case of closed configurations corresponds to having an extra occupied site on the right of the starting configuration. The enumeration of all connected configurations is, in this case, substantially more elaborate. In order to understand the main effect of this extra occupied site we can consider, e.g., the matrix element M~,P2 for the case of open boundary conditions. This can be written as s2 M l P 2 - N1 q- N 2 ' (3.20) where N 2 is the number of configurations that lead to the occupation of both sites above the starting configuration while N 1 is the number of configurations that leave one of these sites empty. Now in the case of closed boundary conditions this empty site may be occupied by a branch that originates from the new black site. The leading term for the probability of occurrence of this event is P2 (see, for an analogy, figs. 3 and 4). Therefore for closed boundary conditions we have to replace N 1 and N 2 by N,1 = NI(1 _ p 2 ) , (3.21) N; = N 2 + N,p 2 , (3.22) and the matrix element then becomes simply 76 L. Pietronero and A. Stella / Percolation as a fractal growth problem Md N2 ~,2 N I + N~ " (3.23) For closed boundary conditions one obtains in this way M]~2(II) = 0 . 7 , (3.24) M ~ 2 ( I I I ) = 0.79808, (3.25) M ~ 2 ( I I ) = 0.8125, (3.26) M ~ I 2 ( I I I ) = 0.875. (3.27) The values of the fractal dimensions corresponding to this different type of calculation are D ( I I ) : 1.8182, (3.28) D ( I I I ) = 1.8914. (3.29) C o m p a r i n g these values with those obtained with the previous method one should keep in mind that the order has a different meaning in the two cases. The best value obtained for the two cases (third order) are extremely similar and also in very good a g r e e m e n t with the exact result. This implies that implementing the connectivity condition by a line is actually a very good method and allows rather simple calculations. 4. Percolation on the triangular lattice In the previous section we have c o m p u t e d the fractal dimension of the percolating cluster in two dimensions using as input the topology of the lattice and the value of the bond percolation threshold for the square lattice Pc = 0.5. Since the fractal dimension is expected to be a universal quantity, the application to bond percolation problems with different lattice topologies, and thus different Pc, is an important test of the potential of the method. This test also involves the expected asymptotic character of bond percolation in the FST context. We therefore repeat the calculation for the triangular lattice for which the bond percolation threshold has a very different value, p~. = 0.34729, than the value for the square lattice. Universality implies that by changing the lattice L. Pietronero and A. Stella / Percolation as a fractal growth problem 77 topology and the associated value of Pc one should recover the same result for the ffactal dimension D. In order to apply the FST theory to the triangular lattice (fig. 6a) it is convenient to make a distortion of the original lattice into the one shown in fig. 6b #1. The topology is preserved by this distortion and the FST method is only dependent on the connectivity properties of the bonds. One recovers, therefore, a square lattice with the addition of diagonal bonds with alternating directions. The calculation can be done in a similar way to that of section 3 and it is only slightly more complicated because of the presence of these diagonal bonds. Connectivity is implemented by the assumption of a starting line of occupied sites. One may notice that with respect to the square lattice the value of Pc is smaller but the bond connectivity is larger. In view of the similarity with the case of the square lattice we only report here the final results for the triangular lattice. These are M~P2(I) = 0.57397, (4.1) M~,v2(II) = 0.65889, (4.2) M~Pz(III) = 0.67145, (4.3) M2P2(I) = 0.72193, (4.4) M2P2(II) = 0.77736, (4.5) M2,%(III ) = 0.78556, (4.6) /'7"-, "-.dJ /'T'--,. "-M.,/ "--.L,/ /.--.... (a) (b) Fig. 6. In practice it is convenient to look at the triangular lattice (a) as slightly distorted (b). The lattice topology of (a) and (b) is the same and (b) allows a treatment along the lines used for the square lattice with modification of the extra diagonal bonds. *~ We thank A. Erzan for a discussion about this point. 78 L. Pietronero and A. Stella / Percolation as a fractal growth problem M~12(I) = 0.64795, (4.7) M~12(11) = 0.75212, (4.8) M~2(111) = 0.77573, (4.9) M~2(I) = 0.77022, (4.10) M2~2(II) = 0.83821, (4.11) M2~2(II1) = 0.85362. (4.12) The corresponding values for the fractal dimension D at various orders are D(I) = 1.7777, (4.13) D(II) = 1.8506, (4.14) D(III) = 1.8650. (4.15) By comparing these results to those of the square lattice (eqs. (3.13)-(3.15)) one can see that, despite the non-universal values of Pc being different by more than 30%, the values of D at third order differ only by about 1%. This provides a strong support to the FST theory in identifying universal exponents using as input the value of Pc and the topology of the lattice. 5. Conclusions The percolation problem is one for which an extensive literature exists concerning applications of the renormalization group, in particular within the real-space context [7-9]. If we compare our results (as shown in fig. 7 together with the results for invasion percolation that will be derived in the following paper [10]) with analogous ones of the analytical renormalization group approaches of similar complexity, we conclude that this FST method has in general higher accuracy, and is more systematic. As mentioned above what characterizes FST with respect to renormalization group transformations, is that the first is based on an invariance property other than scale invariance. This invariance is that of the fractal properties as tested at different sections of the growth pattern. The FST fixed point is such that this invariance is satisfied and is obtained with a self-consistent determination of the C i. L. Pietronero and A . Stella / Percolation as a fractal growth problem 1.90/ 79 EXACT: D = 9 1 / 4 8 = 1 . 8 9 5 8 .... ! ID 1.80 ~ 1.70 I C E TRIANGULARLATTICE , INVASIONPERCOLATION ( SQUARELATTICE) / I i i I H HI i IV ORDEROFTHECALCULATION Fig. 7. Values obtained with the FST method for square, triangular and invasion percolation (ref. [10], following paper) as a function of the order of the calculation. This picture gives an idea about the accuracy and the convergence properties of the method. Of course, dealing with scaling, the FST approach is also based on scale invariance. The method indeed should work optimally when the coarse graining level, and the corresponding growth rules [6] are asymptotic enough to be a good representation of the fractal properties of the structure. To this purpose it is not a priori sufficient that we choose to deal with intrinsically critical clusters (as in D L A or D B M ) , or put ourselves at Pc, as in percolation. The asymptoticity of the growth rules could in principle require a m o r e or less drastic redefinition of the rules themselves, with respect to those of the model at hand. We have an example of this in percolation, where a straightforward application of FST to site percolation would be much less successful than in the bond case [13]. This is clear, however, because site growth evolves into bond growth at large scale [6]. The problem of correctly identifying an asymptotic growth rule in a given problem is not a trivial one, in general, and is analogous to that of determining the fixed point in a renormalization group transformation. Indeed, one should require invariance of the growth rules when passing from a certain level of coarse-grained description to a new one, rescaled with respect to the original one [6]. 80 L. Pietronero and A. Stella / Percolation as a ]ractal growth problem References [1] L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861; Physica A 151 (1988) 207. [2] A. Aharony and J. Feder, eds., Fractals in Physics, Physica D 38 (1989) (special volume). L. Pietronero, ed., Fractals' Physical Origin and Properties (Plenum, New York, London, 1989). [3] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [4] L. Niemeyer, L. Pietronero and H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1033. [5] A. Vespignani and L. Pietronero, Fixed scale transformation approach applied to DLA and DBM in three dimensions, preprint. [6] R. De Angelis, M. Marsili, L. Pietronero and A. Vespignani, Scale invariance of growth rules in fractal growth models, to be published. [7] D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, London, 1985). [8] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989). [9] Y. Gefen, A. Aharony, B. Mandelbrot and S. Kirkpatrick, in: Disordered Systems and Localization, C. Castellani, C. Di Castro and L. Peliti, eds. (Springer, Berlin, 1981). M. den Nijs, J. Phys. A: Math. Gen. 12 (1979) 1857; Phys. Rev. B 27 (1983) 1674. [10] L. Pietronero and W.R. Schneider, Physica A 170 (1990) 81, following paper, this volume. [11] L.P. Kadanoff, Physics Today (February 1986) 6; Physica A 163 (1990) 1. [12] L. Pietronero, Physica A 163 (1990) 316. [13] L. Pietronero and A. Stella, unpublished.