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 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 $(a_1,\dots ,a_k,n)$ 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 $abc$.

27 pages