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

Abstract: In this paper we are concerned with contact process with random recovery rates on open clusters of bond percolation on $\mathbb{Z}^d$. Let $\xi$ be a positive random variable, then we assigned i. i. d. copies of $\xi$ 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 $\lambda<\lambda_c=1/({p{\rm E}\frac{1}{\xi}})$, then the process dies out at time $O(\log d)$ with high probability as $d\rightarrow+\infty$, while when $\lambda>\lambda_c$, the process survives with high probability.