Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree
type: Item
instance of: scholarly article
title: Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree
zbMATH Open document ID: 0954.60090
author: Steven P. Lalley
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: Isotropic contact process on an infinite homogeneous tree of degree at least three is investigated. It is a continuous Markov process \(A_t\), where \(A_t\) are finite subsets of infected sites of the tree. The initial value is the root \(e\) of the tree. The switch from \(A_t\) to its complement happens at rate 1 and the switch from the complement of \(A_t\) to \(A_t\) at rate \(\lambda\) times the number of infected neighbours of \(t\). It is known that there are three phases under these circumstances, the extinction, weak survival, and strong survival. The paper examines the case, where for some interval \((\lambda_1,\lambda_2]\) of \(\lambda\)'s the weakly supercritical contact process appears (i.e. the weak survival occurs). It means more exactly that \(|A_t|\to\infty\) with positive probability and \(A_t\cap B\to\emptyset\) for finite subsets \(B\) of the tree almost surely. Two important quantities are the exponential rate \(\beta\) of decay of hitting probability function at \(\infty\), i.e. \(\beta=\lim_{n=d(x,e)\to\infty} (P\{(\exists t) x\in A_t\})^{{1}/{n}}\) and the growth profile \(V(s)=\log\lim_{n=d(x,e)\to\infty} (P\{x\in A_{ns}\})^{1/n}\). It is shown that \(\beta\) is strictly increasing for \(\lambda \in (\lambda_1,\lambda_2]\), that \(V(s)\) is concave, continuous, at most \(\log\beta\), and the asymptotic behaviour of \(V(s)\) at \(0_+\) and at \(\infty\) is described. Liggett's function \(\varphi\) that enables to recognize the existence of a spherically invariant measure for the contact process that satisfies certain exponential decay law is described in terms of \(V(s)\). Using the mentioned results and the result of \textit{S. P. Lalley} and \textit{T. Sellke} [ibid. 26, No.~2, 644-657 (1998; Zbl 0937.60093)] that \(\beta\leq {1}/{\sqrt{d}}\) for the weakly supercritical process, the conjecture that a spherically invariant measure with the exponential decay exists is verified if \(\lambda\in (\lambda_1,\lambda_2)\).
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1359655
zbMATH Keywords: contact process
zbMATH Keywords: homogeneous tree
zbMATH Keywords: weak survival
reviewed by: Petr Holický
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677259
OpenAlex ID: W2112695558
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DOI: 10.1214/AOP/1022677259
description: scientific article; zbMATH DE number 1359655