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2020 The contact process on periodic trees
Xiangying Huang, Rick Durrett
Electron. Commun. Probab. 25: 1-12 (2020). DOI: 10.1214/20-ECP305


A little over 25 years ago Pemantle [6] 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.


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Xiangying Huang. Rick Durrett. "The contact process on periodic trees." Electron. Commun. Probab. 25 1 - 12, 2020.


Received: 22 September 2019; Accepted: 8 March 2020; Published: 2020
First available in Project Euclid: 25 March 2020

zbMATH: 1434.60295
MathSciNet: MR4089731
Digital Object Identifier: 10.1214/20-ECP305

Primary: 60K35


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