dc.contributor.author Tykesson, Johan dc.contributor.author Lacoin, Hubert dc.date.accessioned 2014-01-10T16:30:52Z dc.date.available 2014-01-10T16:30:52Z dc.date.issued 2013 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/12405 dc.language.iso en en dc.subject Random Walk en dc.subject Percolation en dc.subject Connectivity en dc.subject Random Interlacement en dc.subject.ddc 519 en dc.title On the easiest way to connect $k$ points in the Random Interlacements process en dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Department of Computer Science and Applied Mathematics http://www.wisdom.weizmann.ac.il/ Weizmann Institute of Science;Israël dc.description.abstracten We consider the random interlacements process with intensity $u$ on ${\mathbb Z}^d$, $d\ge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${\mathbb Z}^d$. For $k\ge 3$ we want to determine the minimal number of trajectories from the point process that is needed to link together $k$ points in $\mathcal I^u$. Let $$n(k,d):=\lceil \frac d 2 (k-1) \rceil - (k-2).$$ We prove that almost surely given any $k$ points $x_1,...,x_k\in \mathcal I^u$, there is a sequence ofof $n(k,d)$ trajectories $\gamma^1,...,\gamma^{n(k,d)}$ from the underlying Poisson point process such that the union of their traces $\bigcup_{i=1}^{n(k,d)}\tr(\gamma^{i})$ is a connected set containing $x_1,...,x_k$. Moreover we show that this result is sharp, i.e. that a.s. one can find $x_1,...,x_k in I^u$ that cannot be linked together by $n(k,d)-1$ trajectories. en dc.relation.isversionofjnlname Alea dc.relation.isversionofjnlvol 10 en dc.relation.isversionofjnldate 2013 dc.relation.isversionofjnlpages 505-524 en dc.identifier.urlsite http://fr.arxiv.org/abs/1206.4216 en dc.relation.isversionofjnlpublisher Instituto nacional de matemática pura e aplicada en dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.relation.forthcoming non en dc.relation.forthcomingprint non en
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