The Annals of Probability

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

Steven P. Lalley

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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 $\eta =1$.

Article information

Source
Ann. Probab., Volume 27, Number 1 (1999), 206-225.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677259

Digital Object Identifier
doi:10.1214/aop/1022677259

Mathematical Reviews number (MathSciNet)
MR1681122

Zentralblatt MATH identifier
0954.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Contact process homogeneous tree weak survival

Citation

Lalley, Steven P. Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree. Ann. Probab. 27 (1999), no. 1, 206--225. doi:10.1214/aop/1022677259. https://projecteuclid.org/euclid.aop/1022677259


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