Percolation in random-Sierpin ´ ski carpets: A real space renormalization group approach Michel Perreau, 1 Joaquina Peiro, 2 and Serge Berthier 2 1 Laboratoire de Magne ´tisme des Surfaces, Universite ´ Paris 7, 2 place Jussieu, F-75251 Paris Cedex 05, France 2 Laboratoire d’Optique des Solides, Universite ´ Paris 6, 4 place Jussieu, F-75252 Paris Cedex 05, France Received 3 June 1996 The site percolation transition in random Sierpin ´ski carpets is investigated by real space renormalization. The fixed point is not unique like in regular translationally invariant lattices, but depends on the number k of segmentation steps of the generation process of the fractal. It is shown that, for each scale invariance ratio n , the sequence of fixed points p n, k is increasing with k , and converges when k → toward a limit p n strictly less than 1. Moreover, in such scale invariant structures, the percolation threshold does not depend only on the scale invariance ratio n , but also on the scale. The sequence p n, k and p n are calculated for n =4, 8, 16, 32, and 64, and for k =1 to k =11, and k = . The corresponding thermal exponent sequence  n, k is calculated for n =8 and 16, and for k =1 to k =5, and k = . Suggestions are made for an experimental test in physical self-similar structures. S1063-651X9604510-2 PACS numbers: 64.60.Ak, 61.43.Hv I. INTRODUCTION Phase transitions in fractals are of special interest since they are structures of noninteger dimension in which the critical behavior can be compared with the analytical con- tinuations obtained from the renormalization group methods. Transitions in finite ramification order fractals i.e., in which any bounded part of the structure can be isolated by cutting a finite number of bonds have been extensively stud- ied. Gefen, Aharony, and Mandelbrot 1 proved in this case that transitions occur only at zero temperature. Then the Si- erpin ´ ski gasket model triangular self-invariant fractal has been solved exactly for the Potts model, the Ising model, percolation, and electric conductance Gefen et al. 2. More recently, Yang 4 found an exact expression for the partition function of the Ising model on some Sierpin ´ ski car- pets with finite ramification order. However, few results are known on infinite ramification order fractal lattices, those in which transitions can occur at a nonzero temperature. Gefen, Aharony, and Mandelbrot 3 found some approximations for the critical exponents in Sierpin ´ ski carpets of various fractal dimension, for the Ising model, and the conductivity transition, using real space renormalization methods. It appears that the critical expo- nents depend not only on the fractal dimension, but also on other geometrical parameters like lacunarity and connectiv- ity. Thus the self-invariant lattices do not follow the univer- sal behavior observed in translationally invariant lattices. Interesting questions arise from these works: i what is the role of the fractal dimension in the phase transition? ii What are the relevant parameters and what are their physical meaning? iii How does the renormalization group work when translation invariance is replaced by scale invariance? This paper is a contribution to these questions. We inves- tigate one of the most simple second order phase transition: the percolation transition, in a general class of fractals: the random Sierpin ´ ski carpets RSCdefined later using a real space renormalization group method. Percolation is a geometrical second order phase transition occurring in random lacunary media. Considering a set of elements randomly distributed on the sites of a lattice, the percolation transition is the property of these elements to become connected in a cluster of infinite size when their concentration is large enough. The concentration at which such connections occur is called the percolation threshold. The real space renormalization group has been extensively used to study the critical properties of the percolation transi- tion in regular i.e., translationally invariant lattices, leading to the successful calculation of several physical characteris- tics of lacunary materials, such as their dielectric functions 5. Few works, however, have been devoted to the percola- tion transition in random fractals which are a special class of lacunary lattices frequently used as models for disordered materials. Random Sierpin ´ ski carpets RSC are a general class of regular-random fractals 7 generated by a segmentation pro- cess like the well known Sierpin ´ ski carpet, but generalized to any scale invariance ratio n and to a random choice of q conserved subsquares among the n 2 generated at each seg- mentation step. They are diluted fractal lattices quite differ- ent from the diluted regular square lattice. Indeed many con- figurations which occur into the diluted square lattice do not exist in a scale invariant fractal structure with the same con- centration. We thus expect a quite different behavior of the percolation parameters, as already suggested by previous works 6. Here, the critical properties of the percolation transition into RSC are investigated with a real space renormalization group method. More generally, this paper brings some new insights about the relation between renormalization group and fractality, both involving scale invariance, but with a different point of view. From the combination of these two related but somewhat different aspects of the scale invariance rise some interesting percolation properties of scale invariant fractal structures which should find experimental applica- tions as suggested in the conclusion. II. THE FRACTAL STRUCTURES RSC are built as follows. An initial square is divided into n 2 subsquares, only q of them are conserved at random. This PHYSICAL REVIEW E NOVEMBER 1996 VOLUME 54, NUMBER 5 54 1063-651X/96/545/45906/$10.00 4590 © 1996 The American Physical Society