Percolation on infinitely ramified fractal networks

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff (DH), topological Hausdorff (DtH), and spectral (ds) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers (DH, DtH, ds), rather than solely by the spatial dimension (d). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension DtH, whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension DH. The effect of the network connectivity on the site percolation threshold is revealed.