Percolation on a multifractal

Abstract

We investigate percolation phenomena in multifractal objects that are built in a simple way. In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent beta. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation.

Anisotropy and percolation threshold in a multifractal support

“…At the n th -step of the algorithm the partition of the square in blocks follows the binomial rule: The number of elements of a k-set is C n k . In reference [5] we see that as n → ∞ all the k-sets determine a monofractal whose dimension is…”

“…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.…”

“…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.…”

“…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 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.…”