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

Abstract

For paper I, see ibid., vol.27, p.3743-50 (1994). 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. The cluster size above pc and the mean cluster length are investigated using differential approximants from which we conjecture that these functions satisfy second-order differential equations. Accepting this conjecture gives a mean size exponent the same as below pc and a logarithmic divergence of the mean length from both sides of the critical point. The latter result together with scaling theory predicts that the exponent v/sub /// has the value 2, the same as for the bulk problem.