Multitype Contact Process on $\Z$: Extinction and Interface

TL;DR

This paper analyzes the behavior of a two-type contact process on the integer lattice, establishing conditions for extinction of a type and demonstrating the tightness of the interface size over time.

Contribution

It provides new criteria for extinction based on initial configurations and proves tightness of the interface size in a specific initial setup.

Findings

A type becomes extinct with probability 1 if initially confined to a finite interval and the other type occupies infinitely many sites outside.

The process's interface size remains tight over time starting from a step initial configuration.

Extinction occurs if and only if the initial configuration is confined and the other type is widespread.

Abstract

We consider a two-type contact process on $\Z$ 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.