文章基本信息

标题：On the easiest way to connect k points in the Random Interlacements process
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作者：Hubert Lacoin ; Johan Tykesson
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2013
卷号：X
页码：505-524
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider the random interlacements process with intensity u on Zd,d ≥ 5 (call it Iu), built from a Poisson point process on the space of doubly infinitenearest neighbor trajectories on Zd. For k ≥ 3 we want to determine the minimalnumber of trajectories from the point process that is needed to link together kpoints in Iu. Letn(k, d) := ⌈d2(k − 1)⌉ − (k − 2).We prove that almost surely given any k points x1, ..., xk ∈ Iu, there is a sequenceof n(k, d) trajectories γ1, ..., γn(k,d) from the underlying Poisson point process suchthat the union of their tracesSn(k,d)i=1 Tr(γi) is a connected set containing x1, . . . , xk.Moreover we show that this result is sharp, i.e. that a.s. one can find x1, ..., xk ∈ Iuthat cannot be linked together by n(k, d) − 1 trajectories.
关键词：Random Interlacement; Connectivity; Percolation; Random Walk.