Surface shape and local critical behaviour in two - dimensional directed percolation

Two–dimensional directed site percolation is studied in systems directed along the x–axis and limited by a free surface at y=±Cx k. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z where z=ν /ν is the dynamical exponent. The tip–to–bulk order parameter correlation function is calculated in the mean–field approximation. The tip percolation probability and the fractal dimensions of critical clusters are obtained through Monte–Carlo simulations. The tip order parameter has a nonuniversal, C– dependent, scaling dimension in the marginal case, k=1/z, and displays a stretched exponential behaviour when the perturbation is relevant. The k–dependence of the fractal dimensions in the relevant case is in agreement with the results of a blob picture approach.