Complete convergence theorem for the contact process with avoidance

Prove a complete convergence theorem for the contact process with avoidance on \(\mathbb{Z}^d\) that gives the limiting distribution from arbitrary initial configurations as a mixture of the extinction measure and an appropriate nontrivial invariant measure, analogous to the complete convergence theorem for the classical contact process.

Background

For the classical contact process, attractiveness and self-duality yield a complete convergence theorem describing the limiting law from any initial infected set. The contact process with avoidance is not known to possess these properties. The paper instead proves only existence of a nontrivial translation- and time-invariant measure in a sufficiently high-dimensional supercritical regime, without establishing uniqueness or convergence from arbitrary initial states.