Analysis of mean cluster size in directed compact percolation near a damp wall

We investigate the behaviour of the mean size of directed compact, percolation clusters near a damp wall in the low-density region. Where sites in the bulk are wet (occupied) With probability p while sites on the wall are We with probability p(w). Methods used to find the exact solution for the dry case (p(w) = 0) and the wet case (p(w) = 1) turn out to be inadequate for the damp case. Instead we use a series expansion for the p(w) = 2p case to obtain a second-order inhomogeneous differential equation satisfied by the mean site which exhibits a critical exponent gamma-2, in common with the wet all result. For the more general case of p(w) = rp. with r rational. we use a modular arithmetic method for finding ordinary differential equations (ODEs) and obtain a fourth-order homogeneous ODE satisfied by the series. The ODE is expressed exactly in terms of r. We find that in the damp region 0 < r < 2 the critical exponent gamma(damp) = 1, which is the same as the dry wall result.