Extinction of Contact and Percolation Processes in a Random Environment

Abstract

We consider the (inhomogeneous) percolation process on $\mathbf{Z}^d \times \mathbf{R}$ defined as follows: Along each vertical line $\{x\} \times \mathbf{R}$ we put cuts at times given by a Poisson point process with intensity $\delta(x)$, and between each pair of adjacent vertical lines $\{x\} \times \mathbf{R}$ and $\{y\} \times \mathbf{R}$ we place bridges at times given by a Poisson point process with intensity $\lambda(x, y)$. We say that $(x, t)$ and $(y, s)$ are connected (or in the same cluster) if there is a path from $(x, t)$ to $(y, s)$ made out of uncut segments of vertical lines and bridges. If we consider only oriented percolation, we have the graphical representation of the (inhomogeneous) $d$-dimensional contact process. We consider these percolation and contact processes in a random environment by taking $\delta = \{\delta(x); x \in \mathbf{Z}^d\}$ and $\lambda = \{\lambda(x,y); x,y \in \mathbf{Z}^d, \|x - y\|_2 = 1\}$ to be independent families of independent identically distributed strictly positive random variables; we use $\delta$ and $\lambda$ for representative random variables. We prove extinction (i.e., no percolation) of these percolation and contact processes, for almost every $\delta$ and $\lambda$, if $\delta$ and $\lambda$ satisfy $\mathbf{E}\{(\log(1 + \lambda))^\beta\} < \infty \text{and} \mathbf{E}\bigg\{\bigg(\log\big(1 + \frac{1}{\delta}\big)\bigg)^\beta\bigg\} < \infty$ for some $\beta > 2d^2\bigg(1 + \sqrt{1 + \frac{1}{d}} + \frac{1}{2d}\bigg),$ and if $\mathbf{E}\bigg\{\bigg(\log\big(1 + \frac{\lambda}{\delta}\big)\bigg)^\beta\bigg\}$ is sufficiently small.