Directed compact percolation near a wall. II. Cluster length and size

Abstract

The mean cluster size and length for the unbiassed growth of compact clusters near a dry wall are considered. In the case of the cluster size below pc an exact expression is obtained for a seed of arbitrary width and distance from the surface. It is found that the critical exponent gamma =1 for any finite distance from the surface. Crossover to the bulk value gamma =2 as the distance from the surface tends to infinity is observed. This extends an existing result for the exponent beta of the percolation probability which changes from a value of 2 in the presence of a surface to 1 in the bulk limit. The value Delta =3 of the scaling size exponent is unchanged by the introduction of the surface..