On phase diagrams for directed percolation problems

Abstract

The authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which, differing from previous phenomenological or small-cell real space renormalisation groups, is sensitive to the existence of an infinite cluster along any lattice direction. This enables them to explore new aspects of the phase diagram for the problem, which were not accessible through existing treatments; in particular, the crossover exponent between one- and two-dimensional behaviour is shown to be exactly one for directed percolation. Numerical estimates are in agreement with known inequalities.