Abstract
A little over 25 years ago Pemantle [6] 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 generations 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.
Highlights
The contact process can be defined on any graph as follows: occupied sites become vacant at rate 1, while vacant sites become occupied at rate λ times the number of occupied neighbors
Pemantle [7] began the study of contact processes on trees
On page 2103, Pemantle says that “for reasonably regular non-homogeneous trees the critical value is determined by M the maximum number of children and is at most rM −1/2 where r is a logarithmic measure of how far apart vertices with M children are from each other.”
Summary
The contact process can be defined on any graph as follows: occupied sites become vacant at rate 1, while vacant sites become occupied at rate λ times the number of occupied neighbors. Pemantle showed that λ1 < λ2 when d ≥ 3 by getting upper bounds on λ1 and lower bounds on λ2. As Pemantle notes on page 2103, the upper and lower bounds are different orders of magnitude. He continues with “Which of these asymptotics for λ2 is sharp if either? On the (1, n) tree, as n → ∞ the critical value λ2 ∼ c2(log n)/n where c2 = 1/2. On page 2103, Pemantle says that “for reasonably regular non-homogeneous trees the critical value is determined by M the maximum number of children and is at most rM −1/2 where r is a logarithmic measure of how far apart vertices with M children are from each other.”. Prove that λ1 < λ2 on the (2,3,4) tree
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