Oriented percolation in a random environment

Abstract: On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+ \colon y=i}$ are first declared to be bad or good with probabilities $\de$ and $1-\de$ 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 $\delta (p{G}, p{_B})>0$ small enough, so that for $\delta \le \delta(p_G,p_B)$ there is a strictly positive probability of oriented percolation to infinity from the origin.