Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree

Abstract

It is known that the contact process on a homogeneous tree of degree $d+1\geq3$ has a weak survival phase, in which the infection survives with positive probability but nevertheless eventually vacates every finite subset of the tree. It is shown in this paper that in the weak survival phase there exists a spherically symmetric invariant measure whose density decays exponentially at infinity, thus confirming a conjecture of Liggett. The proof is based on a study of the relationships between various thermodynamic parameters and functions associated with the contact process initiated by a single infected site. These include (1) the growth profile, which determines the exponential rate of growth in space-time on the event of survival, (2) the exponential rate $\beta$ of decay of the hitting probability function at infinity (also studied by the author) and (3) the exponential rate $\beta$ of decay in time $t$ of the probability that the initial infected site is infected at time $t$. It is shown that $\beta$ is a strictly increasing function of the infection rate $\lambda$ in the weak survival phase, and that $\beta = 1/\sqrt{d}$ at the upper critical point $\lambda_2$ demarcating the boundary between the weak 2 and strong survival phases. It is also shown that $\eta < 1$ except at $\lambda_2$, where $\eta =1$.