Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree

• 成果类型: Article
• 署名作者: Lalley, SP
• 署名单位: Purdue University System; Purdue University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1999
• 页码: 206-225
• 关键词: intermediate phase; set

摘要

It is known that the contact process on a homogeneous tree of degree d + 1 greater than or equal to 3 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 eta 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/root d at the upper critical point lambda(2) demarcating the boundary between the weak and strong survival phases. It is also shown that eta <1 except at lambda(2), where eta = 1.