Directed Percolation in Two Dimensions: An Exact Solution

We consider a directed percolation process on an rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio at a value α_c(x, y) = [1 - y² - x(1 - y)²]/2y² where P(x, y) is discontinuous, and the critical exponent of the correlation length for α < α_c(x, y) is ν = 2.