2012/06/19 by Hubert Lacoin, Lacoin, Hubert, Johan Tykesson +1
Mathematics · Physics and Astronomy · #60K35 #82B41 #82D30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82B41 #msc:82D30
paper · pdf · doi:10.48550/arxiv.1206.4216
16 pages, 2 figures. The section where notation is introduced has been modified to avoid text overlap with another paper on the subject
arxiv created 2012/07/04 · arxiv updated 2012/07/05
We consider the random interlacements process with intensity u on \mathbb Zd, d≥ 5 (call it Iu), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on \mathbb Zd. For k≥ 3 we want to determine the minimal number of trajectories from the point process that is needed to link together k points in \mathcal Iu. Let n(k,d):=\lceil \frac d 2 (k-1) \rceil - (k-2). We prove that almost surely given any k points x1,...,xk∈ \mathcal Iu, there is a sequence ofof n(k,d) trajectories γ1,...,γn(k,d) from the underlying Poisson point process such that the union of their traces \bigcupi=1n(k,d)\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 in Iu that cannot be linked together by n(k,d)-1 trajectories.