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arXiv 0901.0661math.PR
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无限长程渗流团簇的生长

The growth of the infinite long-range percolation cluster

Pieter Trapman

备注：
Published in at http://dx.doi.org/10.1214/09-AOP517 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Probability 2010, Vol. 38, No. 4, 1583-1608
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摘要：

我们考虑$\mathbb{Z}^d$上的长程渗流，其中距离$r$处的两个顶点由边连接的概率由$p(r)=1-\exp[-λ(r)]\in(0,1)$给出，并且不同边的存在或不存在是独立的。这里，$λ(r)$是一个严格正的、非增的、规则变化的函数。我们研究了$k$ -球在原点周围的大小的渐近增长，$|\mathcal{B}_k|$，也就是说，对于$k\to\infty$，对于不同的$λ(r)$，在原点的图形距离$k$内的顶点数。我们表明，条件的起源是在（唯一的）无限集群，nonempty类的非增加定期变化的$λ(r)$存在，其中，分别为：$\bullet$ $|\mathcal{B}_k|^{1/k}\to\infty$几乎肯定; $\bullet$存在$1<a_1<a_2<\infty$，使$\lim_{k\to \infty}\mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}<a_2)=1$ ; $\bullet$ $|\mathcal{B}_k|^{1/k}\to1$几乎肯定。这一结果可以应用于空间SIR流行病。特别是，制度确定的基本繁殖数，$R_0$，这是一个重要的数量流行病在非结构化人群中，有一个有用的对应空间流行病。

英文摘要：

We consider long-range percolation on $\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\exp[-λ(r)]\in(0,1)$ and the presence or absence of different edges are independent. Here, $λ(r)$ is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the $k$-ball around the origin, $|\mathcal{B}_k|$, that is, the number of vertices that are within graph-distance $k$ of the origin, for $k\to\infty$, for different $λ(r)$. We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying $λ(r)$ exist, for which, respectively: $\bullet$ $|\mathcal{B}_k|^{1/k}\to\infty$ almost surely; $\bullet$ there exist $1<a_1<a_2<\infty$ such that $\lim_{k\to \infty}\mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}<a_2)=1$; $\bullet$ $|\mathcal{B}_k|^{1/k}\to1$ almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, $R_0$, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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