Critical infection rates for contact processes on open clusters of oriented percolation in $Z^d$

TL;DR

This paper investigates contact processes on open clusters of oriented percolation in high-dimensional lattices, establishing the equality of critical infection rates in different cases and their asymptotic behavior as dimension increases.

Contribution

It proves the equality of quenched and annealed critical infection rates and determines their asymptotic value as the dimension tends to infinity.

Findings

Critical infection rates in quenched and annealed cases are almost surely equal.

Critical infection rates asymptotically approach (dp)^{-1} as dimension increases.

The results hold for open clusters of oriented percolation in Z^d.

Abstract

In this paper we are concerned with contact processes on open clusters of oriented percolation in $Z^d$, where the disease spreads along the direction of open edges. We show that the two critical infection rates in the quenched and annealed cases are equal with probability one and are asymptotically equal to $(dp)^{-1}$ as the dimension $d$ grows to infinity, where $p$ is the probability of edge `open'.