Multitype Contact Process on $\mathbb{Z}$: Extinction and Interface

Abstract

We consider a two-type contact process on $\mathbb{Z}$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty ,L)$ and $(L,\infty )$. We also show that, starting from the configuration in which all sites in $(-\infty ,0]$ are occupied by type 1 particles and all sites in $(0,\infty )$ are occupied by type 2 particles, the process ${\rho }_t$ defined by the size of the interface area between the two types at time $t$ is tight.