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 $(n,a_1,ldots, a_k)$ with $max_i a_i le Cn^{1-delta}$ and $log(a_1 cdots a_k)/log n to b$ as $ntoinfty$. We show that the critical value for local survival is asymptotically $sqrt{c (log n)/n}$ where $c=(k-b)/2$. This supports Pemantles 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.