Crossover and critical exponents for percolation in the semi-infinite plane

The percolation problem in the semi-infinite plane is discussed in terms of the two competing length scales xi (p), the correlation length, and d, the distance from the surface. A crossover hypothesis for the percolation probability P(d, xi (p)) is proposed and two limiting regimes are identified. For xi (p)< >d a new critical exponent, beta s , is required. Using simple RSRG procedures, a sequence of approximations to beta s / beta are obtained and these are seen to show good agreement with a recent Monte Carlo simulation of the system. Finally the authors indicate how the techniques are applied to the semiinfinite three-dimensional case.