On projections of the supercritical contact process: uniform mixing and cutoff phenomenon

TL;DR

This paper studies the supercritical contact process on infinite graphs, proving uniform mixing properties and the cutoff phenomenon in the supercritical regime, with implications for understanding infection spread and mixing times.

Contribution

It establishes uniform mixing (phi-mixing) for projections of the process and proves the cutoff phenomenon on  lattices, advancing understanding of supercritical contact processes.

Findings

Projection of the process is ta-mixing.

Cutoff phenomenon is proven for  lattices.

Large deviation estimates underpin the proofs.

Abstract

We consider the contact process on a countable-infinite and connected graph of bounded degree. For this process started from the upper invariant measure, we prove certain uniform mixing properties under the assumption that the infection parameter is sufficiently large. In particular, we show that the projection of such a process onto a finite subset forms a process which is $\phi$-mixing. The proof of this is based on large deviation estimates for the spread of an infection and general correlation inequalities. In the special case of the contact process on $\mathbb{Z}^d$, $d\geq1$, we furthermore prove the cutoff phenomenon, valid in the entire supercritical regime.