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Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree. (English) Zbl 0954.60090
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 S. P. Lalley and 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 $$\lambda\in (\lambda_1,\lambda_2)$$.
Reviewer: P.Holicky (Praha)

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
contact process; homogeneous tree; weak survival
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##### References:
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