Directed percolation in two dimensions: An exact solution

We consider a directed percolation process on an $\mathcal{M}$ x $\mathcal{N}$ 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 $a=\mathcal{M}\mathrm{/}\mathcal{N}$ at a value $a_c=[1-y^2-x(1-y{)}^2]\mathrm{/}2y^2$ where P(x,y) is discontinuous, and the critical exponent of the correlation length for $a<a_c$ is $\nu =2$.