Oriented percolation in a random environment

TL;DR

This paper proves that in a randomly environment with good and bad lines, there is a positive probability of infinite oriented percolation if the probability on good lines exceeds the critical threshold and the bad lines are sufficiently rare.

Contribution

It establishes conditions under which oriented percolation persists in a random environment with mixed line qualities, extending percolation theory to more complex settings.

Findings

Percolation occurs with positive probability under certain conditions.

Existence of a threshold for the density of bad lines for percolation.

Percolation persists despite the presence of small-probability bad lines.

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.