Directed compact percolation near a wall: III. Exact results for the mean length and number of contacts

Existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall. The results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below p(c). The results for the cluster length satisfy previously conjectured differential equations. A closed expression is conjectured for the mean size above p(c) in terms of a hypergeometric function.