Tightness for the interface of the one-dimensional contact process

Abstract

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(−∞, 0]$ have Infection 1 and sites in $[1, ∞)$ have Infection 2, then consider the process $ρ_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $ρ_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343–370].