Phase transition for large dimensional contact process with random recovery rates on open clusters

In this paper we are concerned with contact process with random recovery rates on open clusters of bond percolation on $\mathbb{Z}^d$. Let $ξ$ be a positive random variable, then we assigned i. i. d. copies of $ξ$ on the vertices as the random recovery rates. Assuming that each edge is open with probability $p$ and $\log d$ vertices are occupied at $t=0$, we prove that the following phase transition occurs. When the infection rate $λ<λ_c=1/({p{\rm E}\frac{1}ξ})$, then the process dies out at time $O(\log d)$ with high probability as $d\rightarrow+\infty$, while when $λ>λ_c$, the process survives with high probability.

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