First passage percolation in hostile environment is not monotone
First passage percolation in hostile environment is not monotone
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恶劣环境中的第一代渗透并不单调
DOI：
10.1214/24-ejp1145
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发表时间：
2021
影响因子：
1.4
通讯作者：
Alexandre O. Stauffer
中科院分区：
文献类型：
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作者：
Elisabetta Candellero;Alexandre O. Stauffer
我们研究了一个具有竞争的自然增长过程，该过程由两个首次通过的渗流过程--$fpp_1$和$fpp_{\lambda}$--在图上展开。$fpp_1$从原点开始，以$1$的速率扩散，而$fpp_{\lambda}$从根据参数$\u\in(0，1)$的Bernoulli渗流分布的一组随机顶点开始(以延迟的方式)，并以某个固定的速率$\lambda>；0$扩散。在以前的作品中(参见[SS19，CS，FS])已经证明，当$\MU$足够小时，则存在$\lambda$的非空值范围，使得最终被$FPP_1$感染的簇可以是具有正概率的无穷大。但是，如果$\MU$足够大，则发生此事件的概率为零。获得无限$fpp_1$簇的概率是$\MU$的单调函数，这似乎很直观。在这项工作中，我们通过构造一个可以找到两个值$0<；\MU_1&Lt；\MU_2&Lt；1$的图证明了这一结论是错误的，使得对于足够小的$\lambda$，如果$\MU=\MU_1$，则所寻求的概率为零，并且如果$\MU=\MU_2$，则该概率有界于零。
We study a natural growth process with competition, modeled by two first passage percolation processes, $FPP_1$ and $FPP_{\lambda}$, spreading on a graph. $FPP_1$ starts at the origin and spreads at rate $1$, whereas $FPP_{\lambda}$ starts (in a"delayed"manner) from a random set of vertices distributed according to Bernoulli percolation of parameter $\mu\in (0,1)$, and spreads at some fixed rate $\lambda>0$. In previous works (cf. [SS19, CS, FS]) it has been shown that when $\mu$ is small enough then there is a non-empty range of values for $\lambda$ such that the cluster eventually infected by $FPP_1$ can be infinite with positive probability. However the probability of this event is zero if $\mu$ is large enough. It might seem intuitive that the probability of obtaining an infinite $FPP_1$ cluster is a monotone function of $\mu$. In this work, we prove that, in general, this claim is false by constructing a graph for which one can find two values $0<\mu_1<\mu_2<1$ such that for all $\lambda$ small enough, if $\mu=\mu_1$ then the sought probability is zero, and if $\mu=\mu_2$ then this probability is bounded away from zero.
DOI：
10.1214/20-aihp1134
发表时间：
2021
期刊：
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
影响因子：
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作者：
Candellero E
通讯作者：
Candellero E