Phase Transition for a Non-Attractive Infection Process in Heterogeneous Environment

#### M. Gottschau, M. Heydenreich, K. Matzke, C. Toninelli

2018, v.24, Issue 1, 39-56

ABSTRACT

We consider a non-attractive three state contact process on $\mathbb Z$ and prove that there exists a regime of survival as well as a regime of extinction. In more detail, the process can be regarded as an infection process in a dynamic environment, where non-infected sites are either healthy or passive. Infected sites can recover only if they have a healthy site nearby, whereas non-infected sites may become infected only if there is no healthy and at least one infected site nearby. The transition probabilities are governed by a global parameter $q$: for large $q$, the infection dies out, and for small enough $q$, we observe its survival. The result is obtained by a coupling to a discrete time Markov chain, using its drift properties in the respective regimes.

Keywords: contact process, phase transition, survival versus extinction