Stochastic Domination in Space-Time for the Contact Process

Abstract

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