Percolation transition in fractal dimensions

Abstract

We provide a Monte Carlo analysis of the moments of the cluster size distributions built up from random occupation of deterministic Sierpinski fractals. Features of the site percolation transition in non-integer dimensions are investigated for two fractal dimensions lying between 1 and 2. Data collapses of the moments when going from an iteration step of the fractal to the next are associated with a real space renormalization procedure and show that a constant gap scaling hypothesis is satisfied. Nevertheless, scaling corrections occuring in the behavior of the thresholds with the size of the lattices are stronger that in the standard percolation case; we point out that, in the case of fractals, a contribution to these corrections can be interpreted as a topological effect of the convergence towards the thermodynamical limit and described by a size dependent shift of the percolation threshold. The bounds of the infinite limit thresholds that we are able to provide are in disagreement with the predictions of Galam and Mauger in the case of translationally invariant lattices, and suggest a new behavior in the case of percolation in non-integer dimension.