Anisotropy and percolation threshold in a multifractal support

Abstract

Recently a multifractal object, Q mf , was proposed to allow the study of percolation properties in a multifractal support. The area and the number of neighbors of the blocks of Q mf show a non-trivial behavior. The value of the probability of occupation at the percolation threshold, pc, is a function of ρ, a parameter of Q mf which is related to its anisotropy. We investigate the relation between pc and the average number of neighbors of the blocks as well as the anisotropy of Q mf . I IntroductionDue to the work of many physicists, and particularly to the contributions of Dietrich Stauffer, percolation theory has became a powerful tool in Science describing phenomena in many areas as geology, biology, magnetism, or social phenomena [1,2]. Despite the enormous success of percolation it has been a theory studied in a support (lattice) that has a single dimension. The only references relating percolation and multifractality concern to the multifractal properties of some quantities of the spanning cluster at the percolation threshold [3,4].Recently, a model to study percolation in a multifractal was proposed [5] in the literature. In fact, the authors have created an original multifractal object, Q mf , and an efficient way to estimate its percolation properties. In this work we study in detail the method to estimate p c for this multifractal and discuss the relation between p c , some topologic characteristics, and the anisotropy of Q mf .The multifractal object we develop, Q mf , is an intuitive generalization of the square lattice [5]. Suppose that in the construction of the square lattice we use the following algorithm: take a square of size 1 and cut it symmetrically with vertical and horizontal lines. Repeat this process n-times; at the n th step we have a regular square lattice with 2 n × 2 n cells. The setup algorithm of Q mf is quite similar, the main difference is that we do not cut the square in a symmetric way. In section 2 we explain in detail this algorithm.The development of Q mf has a twofold motivation. Firstly, there are systems like oil reservoirs that show multifractal properties [6] and are good candidates to be modeled by such object. Secondly, there is indeed a much more general scope: we want to study percolation phenomena in lattices that are not regular, but that are multifractal in the geometrical sense. It is important to know how site percolation transition happens in lattices in which the cells vary in size and also in the number of neighbors.In this work we analyse some geometric and topologic properties of the percolation cluster generated on Q mf at the percolation threshold. The paper is organized as follows: in Section II we present the process of construction of Q mf and the algorithm for the estimation of p c , in Section III we show the numerical simulations concerning the percolation threshold p c and the topologic properties of Q mf ; and finally in Section IV we present our final remarks and comments. II The modelIn this section we show the process of building the multifrac...

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