DIRECTED COMPACT PERCOLATION NEAR A WALL .1. BIASED GROWTH

The directed compact percolation cluster model of Domany and Kinzel is considered in the presence of a wall which is parallel to the growth direction and hence restricts the lateral growth of the cluster in one direction. The critical exponents are found to depend on whether the wall is wet or dry. In the former case the model is solved exactly for all the standard percolation functions and the critical behaviour is found to be the same as that for cluster growth with no wall present. With this boundary condition the cluster is completely attached to the wall and the model may also be viewed as one of symmetric compact cluster growth. In the case of a dry wall the cluster may repeatedly leave and return to the wall as it grows and in this case the percolation probability has been derived exactly by Lin and found to have a critical exponent different from that of the bulk. Lin's result is rederived and an exact formula for the percolation probability is found for a more general model in which the cluster growth is biased either towards or away froin the wall. It is found that the unbiased case is special in that any bias away from the wall recovers the bulk critical exponent and a bias towards the wall produces a problem in the same class as the wet-wall model.