Percolation dimension of fractals

Percolation Dimension of Fractals

“Percolation dimension” is introduced in this note. It characterizes certain fractals and its definition is based on the Hausdorff dimension. It is shown that percolation dimension and “boundary dimension” are in a sense independent from the Hausdorff dimension and, therefore, provide an additional tool for classification of fractals. A unifying concept of fractal was introduced by Mandelbrot t...

Fractals Meet Fractals: Self-Avoiding Random Walks on Percolation Clusters

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results yield estimates of critical exponents, governing the scaling laws of disorder averages of the co...

Fractals, Dimension, and Formal Languages

We consider classes of sets of r-adic expansions of reals speciied by means of the theory of formal languages or automata theory. It is shown how these speciications are used to calculate the Hausdorr dimension and Hausdorr measure of such sets. Since the appearence of Mandelbrot's 11 book \Fractals, Form, Chance and Dimen-sion" Fractal Geometry as a means providing a theory describing many of ...

Random fractals and scaling limits in percolation

Magnetization Distribution on Fractals and Percolation Lattices

We study the magnetization distribution of the Ising model on two regular fractals (a hierarchical lattice, the regular simplex) and percolation clusters at the percolation threshold in a two dimensional imbedding space. In all these cases, the only fixed point is T = 0. In the case of the two regular fractals, we show that the magnetization distribution is non trivial below T ∗ ≃ A/n, with n t...