Jw. Essam et D. Tanlakishani, DIRECTED COMPACT PERCOLATION NEAR A WALL .1. BIASED GROWTH, Journal of physics. A, mathematical and general, 27(11), 1994, pp. 3743-3750
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 clu
ster growth. In the case of a dry wall the cluster may repeatedly leav
e 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 critic
al exponent different from that of the bulk. Lin's result is rederived
and an exact formula for the percolation probability is found for a m
ore 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 expone
nt and a bias towards the wall produces a problem in the same class as
the wet-wall model.