Infinite paths in randomly oriented lattices

The square lattice is used to generate an oriented graph in which a rightwa rd or upward arrow is present on each edge with probability a, and a leftwa rd or downward arrow with probability b. Independence between different edg es of the square lattice is assumed, but nothing is assumed concerning the dependence between the two possible orientations at any given edge. A prope rty of self-duality is exploited to show that, when a + b = 1, the process is, in a sense to be made precise, either critical or supercritical, but no t subcritical. This observation enables progress with the percolation probl em in which each horizontal edge is oriented rightward with probability p a nd otherwise leftward, and each vertical edge is oriented upward with proba bility p and otherwise downward. (C) 2001 John Wiley Bi Sons, Inc.