The Contact Process on Periodic Trees

TL;DR

This paper analyzes the contact process on periodic trees, providing bounds and asymptotics for critical values related to survival, and extends known results to more complex tree structures with explicit formulas depending on vertex degrees.

Contribution

It introduces new sharp asymptotic results for the critical value on certain periodic trees and generalizes previous findings to more complex tree structures with explicit formulas.

Findings

Sharp asymptotics for $oldsymbol{ ext{(1,n)}}$ trees' critical value $oldsymbol{ ext{(}oldsymbol{ ext{lambda}}_2 ext{)}}$

Extension of results to $oldsymbol{ ext{(a}_1, ext{...,a}_k, ext{n)}}$ trees with bounds on degrees

Explicit formulas for critical values depending only on sums and products of degrees

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