On the spreading dimension of percolation and directed percolation clusters

This is the authors' abstract. We don't add key points for this paper.

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.