The Contact Process on Periodic Trees

A little over 25 years ago Pemantle 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 generation is $(n,a_1,\dots ,a_k)$ 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.