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The existence of an intermediate phase for the contact process on trees. (English) Zbl 0878.60061
Summary: Let \(\mathbb{T}_d\) be a homogeneous tree in which every vertex has \(d\) neighbors. A new proof is given that the contact process on \(\mathbb{T}_d\) exhibits two phase transitions when \(d\geq 3\), a behavior which distinguishes it from the contact process on \(\mathbb{Z}^n\). This is the first proof which does not involve calculation of bounds on critical values, and it is much shorter than the previous proof for the binary tree, \(\mathbb{T}_3\). The method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties, for which no such result was previously known.

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