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.