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The contact process on trees. (English) Zbl 0762.60098
Consider a contact process on an \(n\)-ray tree with root \(\rho\). The infected vertices turn into healthy vertices at Poisson rate 1 and each infected vertex independently infects each of its healthy neighbors at Poisson rate \(\lambda\). Let \(\xi_ t\) denote the set of vertices infected at time \(t\). Define \[ \lambda_ 1=\inf\{\lambda: \mathbb{P}_ \rho[\xi_ t\neq\emptyset\text{ for all }t]>0\},\qquad\lambda_ 2=\inf\{\lambda:\varliminf_{t\to\infty}\mathbb{P}_ \rho[\rho\in \xi_ t]>0\}. \] The paper proves that \(0<\lambda_ 1<\lambda_ 2<\infty\) for all \(n>2\). This means that the contact process on a tree exhibits a multiple phase transition, a remarkable result which differs from the contact process on the lattice. Moreover, some concrete estimates of \(\lambda_ 1\) and \(\lambda_ 2\) are also presented: \(\lambda_ 1\approx 1/n\) and \(\lambda_ 2\approx 1/\sqrt n\) for large \(n\). Finally, the processes on nonhomogeneous trees are studied.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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