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=+or-Cxk. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z, where z=v/sub ////v 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 non-universal, 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.

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