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On the easiest way to connect $k$ points in the Random Interlacements process

dc.contributor.authorTykesson, Johan
dc.contributor.authorLacoin, Hubert
dc.date.accessioned2014-01-10T16:30:52Z
dc.date.available2014-01-10T16:30:52Z
dc.date.issued2013
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/12405
dc.language.isoenen
dc.subjectRandom Walken
dc.subjectPercolationen
dc.subjectConnectivityen
dc.subjectRandom Interlacementen
dc.subject.ddc519en
dc.titleOn the easiest way to connect $k$ points in the Random Interlacements processen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherDepartment of Computer Science and Applied Mathematics http://www.wisdom.weizmann.ac.il/ Weizmann Institute of Science;Israël
dc.description.abstractenWe 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.isversionofjnlnameAlea
dc.relation.isversionofjnlvol10en
dc.relation.isversionofjnldate2013
dc.relation.isversionofjnlpages505-524en
dc.identifier.urlsitehttp://fr.arxiv.org/abs/1206.4216en
dc.relation.isversionofjnlpublisherInstituto nacional de matemática pura e aplicadaen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen


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