Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree
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 \(\lambda\in (\lambda_1,\lambda_2)\).
scientific article; zbMATH DE number 1359655