The contact process on periodic trees

Abstract

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.