Directed percolation near a wall

Series expansion methods are used to study directed bond percolation clusters 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 the 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 respectively. On the other hand the exponent Delta(1) = beta(1) + gamma(1) characterizing the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent 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 related to beta(1) and nu(1 parallel to) by the scaling relation nu(1 parallel to) = beta(1) + tau(1) and using the above estimates yields tau(1) = 1 to within the accuracy of our results. We conjecture that this value of tau(1) is exact and further support for the conjecture is provided by the direct series expansion estimate tau(1) = 1.0002 +/- 0.0003.