Exponential growth and continuous phase transitions for the contact process on trees

Abstract

We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $\lambda _{1}$ for weak survival, and the survival probability $p(\lambda )$ is continuous with respect to the infection rate $\lambda $. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $\lambda _{1}<\lambda _{2}$, which confirms a conjecture of Stacey’s [12]. We also prove that if the contact process survives strongly at $\lambda $ then it survives strongly at a $\lambda '<\lambda $, which implies that the process does not survive strongly at the critical value $\lambda _{2}$ for strong survival.