The Contact Process on Periodic Trees

Abstract

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 ${\max }_i{a}_i\le C{n}^{1-\delta }$ 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.

12 pages, 1 figure