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 the critical values $\lambda_1$ and $\lambda_2$ for global and local survival were different. Here, we will consider the case of trees in which the degrees of vertices are periodic. We will compute bounds on $\lambda_1$ and $\lambda_2$ and for the corresponding critical values $\lambda_g$ and $\lambda_\ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $\lambda_2$ of $(1,n)$ trees and generalize that result to the $(a_1,\ldots, a_k, n)$ tree when $\max_i a_i \le n^{1-\epsilon}$ and $a_1 \cdots a_k = n^b$. We also give results for $\lambda_g$ and $\lambda_\ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.