DIRECTED PERCOLATION NEAR A WALL

Series expansion methods are used to study directed bond percolation c lusters on the square lattice whose lateral growth is restricted by a wail parallel to the growth direction. The percolation threshold p(c) is found to be the same as that for the bulk. However, the values of t he critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by beta(1) = 0.7338 +/- 0.0001 and gamma(1) = 1.8207 +/- 0.0004 respecti vely. On the other hand the exponent Delta(1) = beta(1) + gamma(1) cha racterizing the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness leng th, which is the scale for the cluster length distribution, has an exp onent which we estimate to be nu(1 parallel to) = 1.7337+/-0.0004 and is also unchanged. The exponent tau(1) of the mean cluster length is r elated to beta(1) and nu(1 parallel to) by the scaling relation nu(1 p arallel to) = beta(1) + tau(1) and using the above estimates yields ta u(1) = 1 to within the accuracy of our results. We conjecture that thi s value of tau(1) is exact and further support for the conjecture is p rovided by the direct series expansion estimate tau(1) = 1.0002 +/- 0. 0003.