Asymptotics of the critical time in Wiener sausage percolation with a small radius
Asymptotics of the critical time in Wiener sausage percolation with a small radius
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小半径维纳香肠渗滤临界时间的渐近
DOI：
10.30757/alea.v13-17
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Poisat
中科院分区：
文献类型：
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作者：
D. Erhard;J. Poisat
本文考虑$\R^d$上的连续渗流模型，其中$d\geq 4 $，其占据集由半径为r$的独立Wiener香肠的并给出，其初始点按齐次Poisson点过程分布.它是由埃哈德在以前的工作中建立的，Martnez和Poisat引用EMP 13的结论：（1）如果r足够小，在临界时刻t c（r）t中存在一个非平凡的渗流转变;（2）在超临界状态下，无界团簇是唯一的.本文研究了当半径r收敛到0时临界时间的渐近性质。后者似乎不能从简单的尺度参数推导出来。我们证明了对于$d\geq 4$，存在一个正常数$c$使得当$d=4$时$c^{-1}\sqrt{\log（1/r）}\leq t\_c（r）\leq c\sqrt{\log（1/r）}$，当$d\geq 5$时$c^{-1}r^{（4-d）/2}\leq t\_c（r）\leq c\ r^{（4-d）/2}$，当$r$收敛到0 $。我们推导出沿着的Wiener香肠，这可能是独立的兴趣的能力的方式矩估计。
We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t\_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t\_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.