Percolation in random-Sierpin-acuteski carpets: A real space renormalization group approach.

Abstract

The site percolation transition in random Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpets is investigated by real space renormalization. The fixed point is not unique like in regular translationally invariant lattices, but depends on the number k of segmentation steps of the generation process of the fractal. It is shown that, for each scale invariance ratio n, the sequence of fixed points ${\mathit{p}}_{\mathit{n},\mathit{k}}$ is increasing with k, and converges when k\ensuremath{\rightarrow}\ensuremath{\infty} toward a limit ${\mathit{p}}_{\mathit{n}}$ strictly less than 1. Moreover, in such scale invariant structures, the percolation threshold does not depend only on the scale invariance ratio n, but also on the scale. The sequence ${\mathit{p}}_{\mathit{n},\mathit{k}}$ and ${\mathit{p}}_{\mathit{n}}$ are calculated for n=4, 8, 16, 32, and 64, and for k=1 to k=11, and k=\ensuremath{\infty}. The corresponding thermal exponent sequence ${\ensuremath{\nu}}_{\mathit{n},\mathit{k}}$ is calculated for n=8 and 16, and for k=1 to k=5, and k=\ensuremath{\infty}. Suggestions are made for an experimental test in physical self-similar structures. \textcopyright{} 1996 The American Physical Society.