Directed Percolation in Two Dimensions: Numerical Analysis and an Exact Solution

Abstract

We study directed percolation in two dimensions. Critical properties, such as the threshold $p_c$ and the correlation length exponent $\nu$, depend on the direction $\varphi$ at which percolation is observed. We observe crossover from $\nu (\varphi =0)$ to $\nu \left(\varphi \ne 0\right)$; similar crossover is found when the probabilities assigned to vertical and horizontal bonds, $p_V$ and $p_H$, become unequal. For the case $p_H=1\mathrm{}$ we solved the problem exactly, obtaining $p_c\left(\varphi \right)={[1+cot(\frac{1}{4}\pi -\varphi \left)\right]}^{-1\mathrm{}}$, $\nu =2\mathrm{}$.