Directed percolation in two dimensions: An exact solution

TL;DR

This paper derives an exact formula for the percolation probability in a two-dimensional directed percolation model on a rectangular lattice, revealing critical behavior and phase transition characteristics.

Contribution

It provides a closed-form expression for the percolation probability in a specific directed lattice model, identifying critical aspect ratios and exponents.

Findings

Percolation probability formula derived explicitly

Critical aspect ratio identified where phase transition occurs

Correlation length exponent found to be ν=2

Abstract

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