PERCOLATION IN RANDOM-SIERPINSKI CARPETS - A REAL-SPACE RENORMALIZATION-GROUP APPROACH

The site percolation transition in random Sierpinski carpets is invest igated by real space renormalization. The fixed point is not unique li ke in regular translationally invariant lattices, but depends on the n umber It of segmentation steps of the generation process of the fracta l. It is shown that, for each scale invariance ratio n, the sequence o f fixed points p(n,k) is increasing with k, and converges when k-->inf inity toward a limit p(n) strictly less than 1. Moreover, in such scal e invariant structures, the percolation threshold does not depend only on the scale invariance ration, but also on the scale. The sequence p (n,k) and p(n) are calculated for n = 4, 8, 16, 32, and 64, and for k = 1 to k = 11, and k = infinity. The corresponding thermal exponent se quence nu(n,k) is calculated for n = 8 and 16, and for k = 1 to k = 5, and k = infinity. Suggestions are made for an experimental test in ph ysical self-similar structures.