Tightness for the interface of the one-dimensional contact process

TL;DR

This paper proves that the size of the interface between two competing infections in a one-dimensional contact process remains tight over time, confirming a longstanding conjecture by Cox and Durrett.

Contribution

It establishes the tightness of the interface size distribution in a symmetric, finite-range contact process with two infection types, resolving a conjecture from 1995.

Findings

Distribution of interface size is tight over time

Confirms Cox and Durrett's 1995 conjecture

Provides insight into the spatial structure of competing infections

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 $(-\infty,0]$ have Infection 1 and sites in $[1,\infty)$ have Infection 2, then consider the process $\rho_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $\rho_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343--370].