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arXiv 1911.03330math.PR
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树上接触过程的指数增长性和连续相变

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

Xiangying Huang

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摘要：

研究了Galton-Watson树和周期树上的超临界接触过程。证明了如果接触过程弱生存，则它支配超临界Crump-Mode-Jagers分支过程.因此，受感染网站的数量呈指数级快速增长。因此，我们得出结论，接触过程在弱存活的临界值$λ_1$处消亡，并且存活概率$p(λ)$相对于感染率$λ$是连续的。应用这一事实，我们证明了一般周期树上的接触过程经历了$λ_1<λ_2$意义上的两个相变，从而证实了Stacey的\cite{Stacey}的一个猜想.我们还证明，如果接触过程强烈生存在$λ$，那么它强烈生存在一个$λ'<λ$，这意味着该过程不强烈生存在强生存的临界值$λ_2$。

英文摘要：

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 $λ_1$ for weak survival, and the survival probability $p(λ)$ is continuous with respect to the infection rate $λ$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $λ_1<λ_2$, which confirms a conjecture of Stacey's \cite{Stacey}. We also prove that if the contact process survives strongly at $λ$ then it survives strongly at a $λ'<λ$, which implies that the process does not survive strongly at the critical value $λ_2$ for strong survival.

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