The contact process on periodic trees

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 ⅄1 and ⅄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, a1,..., ak) with maxiai ≤ Cn^1-δand log(a1· · · ak)= log n→b as n→ ꝏ. We show that the critical value for local survival is asymptotically √c(log)/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.

Duke Scholars

Author Richard Timothy Durrett Mathematics