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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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
local survival; block construction; sharp asymptotics
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##### References:
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