Asymmetric directed percolation on the square lattice

Abstract

The author considers directed percolation on the square lattice, with probability p_H(p_V) for the horizontal (vertical) bonds to be unbroken. For p_H=1- epsilon ( epsilon small) and p_V> sigma ( epsilon ) the percolating cluster is asymptotically within a cone phi ₊< phi < phi ₊. The author calculates phi _+or- and the fluctuations of the boundaries at phi = phi _+or- as power series in epsilon , up to terms approximately epsilon ², showing that the transverse spread of the percolating cluster is random walk-like. For any given p_H the percolation threshold p_V,c( epsilon ), defined by phi ₊= phi _- at p_V=p_V,c is also calculated.