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 $\de$ 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.

arXiv admin note: substantial text overlap with arXiv:1204.3197