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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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