Phase Transition for the Large-Dimensional Contact Process with Random Recovery Rates on Open Clusters

Abstract

In this paper we are concerned with the contact process with random recovery rates on open clusters of bond percolation on $\mathbb {Z}^d$. Let $\xi$ be a random variable such that $P(\xi \ge 1)=1$, which ensures $\mathrm{E}\frac{1}{\xi }<+\infty$, then we assign i. i. d. copies of $\xi$ on the vertices as the random recovery rates. Assuming that each edge is open with probability p and the infection can only spread through the open edges, then we obtain that

$$\begin{aligned} \limsup _{d\rightarrow +\infty }\lambda _d\le \lambda _c=\frac{1}{p\mathrm{E}\frac{1}{\xi }}, \end{aligned}$$

where $\lambda _d$ is the critical value of the process on $\mathbb {Z}^d$, i.e., the maximum of the infection rates with which the infection dies out with probability one when only the origin is infected at $t=0$. To prove the above main result, we show that the following phase transition occurs. Assuming that $\lceil \log d\rceil$ vertices are infected at $t=0$, where these vertices can be located anywhere, then when the infection rate $\lambda >\lambda _c$, the process survives with high probability as $d\rightarrow +\infty$ while when $\lambda <\lambda _c$, the process dies out at time $O(\log d)$ with high probability.