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2020 Exponential growth and continuous phase transitions for the contact process on trees
Xiangying Huang
Electron. J. Probab. 25: 1-21 (2020). DOI: 10.1214/20-EJP483

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.

Citation

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Xiangying Huang. "Exponential growth and continuous phase transitions for the contact process on trees." Electron. J. Probab. 25 1 - 21, 2020. https://doi.org/10.1214/20-EJP483

Information

Received: 11 December 2019; Accepted: 8 June 2020; Published: 2020
First available in Project Euclid: 11 July 2020

zbMATH: 07252709
MathSciNet: MR4125782
Digital Object Identifier: 10.1214/20-EJP483

Subjects:
Primary: 60K35

JOURNAL ARTICLE
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