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The complete convergence theorem of the contact process on trees. (English) Zbl 0876.60092
The contact processes \(\xi^A_t\) \((\xi^A_0=A\subset T)\) on a homogeneous tree \(T\) with degree \(d\geq 3\) and root 0 are studied. Let \(\lambda\cdot d\) be the rate of \(x\in T\) becoming occupied if vacant and let the rate of becoming vacant at an occupied \(x\) be 1. Theorem 1 shows that the distributions of \(\xi^A_t\) converge to the mixture of the Dirac measure concentrated at the empty configuration and of the “upper invariant measure”, i.e. the complete convergence theorem holds, if \(\lambda>\lambda_c=\inf\{\lambda; P(0\in \xi^0_t \text{ i.o.})>0\}\). For \(\lambda=\lambda_c\), by Theorem 2, \(P(0\in \xi^0_t \text{ i.o.})=0\). Theorem 3 says that for \(\lambda=\lambda_c\) there are infinitely many extremal stationary distributions of the contact process. Together with known results these ones complete the picture of stationary measures of the contact process on a homogeneous tree in the dependence on \(\lambda\). The author conjectures that the result analogous to Theorem 1 should hold for any homogeneous graph \(G\).
Reviewer: P.Holicky (Praha)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
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