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

TL;DR

This paper analyzes the contact process on trees, proving exponential growth in supercritical cases, continuity of survival probability at criticality, and confirming a conjecture about two phase transitions on periodic trees.

Contribution

It establishes the exponential growth rate, continuity of survival probability at critical points, and confirms the existence of two phase transitions on periodic trees.

Findings

Exponential growth of infected sites in supercritical contact process.

Continuity of survival probability at the critical infection rate.

Existence of two distinct phase transitions on periodic 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 \cite{Stacey}. 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.