On the spreading dimension of percolation and directed percolation clusters

Abstract

The authors study the number AN of sites that are accessible after N steps at most on clusters at the percolation threshold. On a Cayley tree AN is of order N2 if the origin belongs to a larger cluster, whereas its average over all clusters is of order N. This suggests that the intrinsic spreading dimension d , defined by AN approximately Nd, is equal to two for fractal percolation clusters in space dimensions d>or=6 and depends on d for d<6. For directed percolation clusters they argue that d is related to usual critical exponents by d=( beta + gamma )/ nu /sub ///. Monte Carlo data that support this relation are presented in two dimensions. Analogous results are derived for lattice animals d=2 on the Cayley tree and d=1/ nu /sub /// for directed animals in any dimensions.