Oriented percolation in a random environment

Abstract

On the lattice ${\tilde{\mathbb{Z}}}_{+}^2:=\{(x,y)\in \mathbb{Z}\times {\mathbb{Z}}_{+}:x+y\text{is even}\}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:=\{(x,y)\in {\tilde{\mathbb{Z}}}_{+}^2:y=i\}$ are first declared to be bad or good with probabilities δ and $1-\mathit{\delta }$ respectively, independently of each other. Given the configuration of lines, sites on good lines are open with probability $p_G>p_c$, the critical probability for the standard oriented site percolation on ${\mathbb{Z}}_{+}\times {\mathbb{Z}}_{+}$, and sites on bad lines are open with probability $p_B$, some small positive number, independently of each other. We show that given any pair $p_G>p_c$ and $p_B>0$, there exists a $\mathit{\delta }(p_G,p_B)>0$ small enough, so that for $\mathit{\delta }\le \mathit{\delta }(p_G,p_B)$ there is a strictly positive probability of oriented percolation to infinity from the origin.