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The contact process on periodic trees. (English) Zbl 1434.60295
Summary: A little over 25 years ago R. Pemantle [Ann. Probab. 20, No. 4, 2089–2116 (1992; Zbl 0762.60098)] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values \(\lambda_1\) and \(\lambda_2\) for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is \((n,a_1,\ldots, a_k)\) with \(\max_i a_i \le Cn^{1-\delta}\) and \(\log (a_1 \cdots a_k)/\log n \to b\) as \(n\to \infty \). We show that the critical value for local survival is asymptotically \(\sqrt{c (\log n)/n}\) where \(c=(k-b)/2\). This supports Pemantle’s claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

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