随机图上阈值- $θ\geq 2$接触过程的一个不连续相变
A discontinuous phase transition in the threshold-$θ\geq 2$ contact process on random graphs
摘要：
本文研究了一般度随机图上的离散阈值$θ\geq 2$接触过程。对于给定度分布$μ$的随机图，我们证明了如果$μ$是$θ+2$下界的，并且对所有的$k>0$都有有限的$k$阶矩，那么随机图上的离散时间阈值- $θ$接触过程在亚稳出现时表现出不连续的相变，从而回答了Chatterjee和Durrett \cite{cd13}的一个问题.具体地说，我们建立了：（i）对于任何足够大的感染概率$p>p_1$，从全感染状态w. h. p.开始的过程在$e^{Θ(n)}$ -时间内存活，保持大的感染密度;（ii）对于任何$p<1$，如果初始密度小于$\varepsilon(p)>0$，则它在$O(\log n)$ -时间w. h. p.中消亡。我们还解释了一些扩展到更一般的随机图，包括Erdens-Rényi图。证明了随机$(θ+1)$ -正则图上的阈值$θ$接触过程在时间$n^{O(1)}$ W. H. P.中消亡.
英文摘要：
We study the discrete-time threshold-$θ\geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $μ$, we show that if $μ$ is lower bounded by $θ+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$θ$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w.h.p. survives for $e^{Θ(n)}$-time, maintaining a large density of infection; (ii) for any $p<1$, if the initial density is smaller than $\varepsilon(p)>0$, then it dies out in $O(\log n)$-time w.h.p.. We also explain some extensions to more general random graphs, including the Erdős-Rényi graphs. Moreover, we prove that the threshold-$θ$ contact process on a random $(θ+1)$-regular graph dies out in time $n^{O(1)}$ w.h.p..
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