Convergence of the one-dimensional contact process with two types of particles and priority

TL;DR

This paper studies a two-type contact process on the integer line, proving convergence to specific invariant measures and revealing complex long-term behavior influenced by initial configurations.

Contribution

It introduces a model with two interacting particle types and demonstrates convergence to a unique invariant measure different from the classic contact process.

Findings

Convergence to a distinct invariant measure from the classic process.

Initial configurations determine the convex combination of invariant measures.

The process exhibits complex long-term behavior depending on initial states.

Abstract

We consider a symmetric finite-range contact process on $\mathbb{Z}$ with two types of particles (or infections), which propagate according to the same supercritical rate and die (or heal) at rate $1$. Particles of type $1$ can enter any site in $(-\infty,0]$ that is empty or occupied by a particle of type $2$ and, analogously, particles of type $2$ can enter any site in $[1,\infty)$ that is empty or occupied by a particle of type $1$. Also, almost one particle can occupy each site. We prove that the process beginning with all sites in $(-\infty,0]$ occupied by particles of type 1 and all sites in $[1,\infty)$ occupied by particles of type 2 converges in distribution to an invariant measure different from the nontrivial invariant measure of the classic contact process. In addition, we prove that for any initial configuration the process converges to a convex combination of four invariant measures.