Stochastic Domination in Space-Time for the Contact Process

TL;DR

This paper extends Liggett and Steif's results by establishing space-time stochastic domination properties for the supercritical contact process on trees and lattices, revealing new mixing behaviors and correlation structures.

Contribution

It introduces space-time stochastic domination results for the contact process on trees and lattices, combining existing methods with new properties and correlation inequalities.

Findings

Existence of a subset of vertices on $T_d$ with positive fraction where the process dominates an independent spin-flip process.

Supercritical contact process on $ ext{Z}^d$ in space-time slabs dominates an i.i.d. Bernoulli measure.

Results imply strong mixing properties for the process in certain space-time regions.

Abstract

Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on $T_d$ with $d\geq2$. We show that, for large infection rate, there exists a subset $\Delta$ of the vertices of $T_d$, containing a "positive fraction" of all the vertices of $T_d$, such that the following holds: The contact process on $T_d$ observed on $\Delta$ stochastically dominates an independent spin-flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on $\mathbb{Z}^d$ observed on certain $d$-dimensional space-time slabs stochastically dominates an i.i.d.\ Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.