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

TL;DR

This paper studies the contact process with random recovery rates on high-dimensional open clusters, establishing a phase transition in survival probability depending on the infection rate and the distribution of recovery rates.

Contribution

It introduces a phase transition analysis for the contact process with random recovery rates on open clusters in high dimensions, deriving critical infection thresholds.

Findings

Process dies out quickly when infection rate is below critical value

Process survives with high probability above critical infection rate

Phase transition depends on the expected inverse of recovery rates

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.