The Contact Process on Periodic Trees

TL;DR

This paper analyzes the contact process on periodic trees with varying degrees, showing how the critical value for local survival depends on maximum degree and smaller degrees, extending Pemantle's earlier work.

Contribution

It provides an asymptotic formula for the critical value on periodic trees with degree sequences growing as specified, highlighting the influence of both maximum and smaller degrees.

Findings

Critical value for local survival is asymptotically rac{rac{c (rac{rac{rac{rac{rac{rac{c (rac{rac{rac{rac{rac{rac{c (rac{rac{rac{rac{rac{rac{log n)}{n}} where c=(k-b)/2.

The critical value depends largely on the maximum degree but smaller degrees also significantly influence the outcome.

The results extend Pemantle's findings to a broader class of 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 $(n,a_1,\ldots, a_k)$ with $\max_i a_i \le Cn^{1-\delta}$ 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.