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Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster.

dc.contributor.authorLacoin, Hubert
dc.date.accessioned2014-07-22T13:23:21Z
dc.date.available2014-07-22T13:23:21Z
dc.date.issued2014
dc.identifier.issn0178-8051
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/13768
dc.language.isoenen
dc.subjectRandom media
dc.subjectPolymers
dc.subjectPercolation
dc.subjectSelf-avoiding walk
dc.subjectDisorder relevance
dc.subject.ddc519en
dc.titleNon-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster.
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenIn this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $$\mathbb{Z }^d$$ . More precisely, we count $$Z_N$$ , the number of self-avoiding paths of length $$N$$ on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of $$Z_N$$ , $$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$ , which we call the connective constant of the dilute lattice. After proving that this connective constant is a.s. non-random, we focus on the two-dimensional case and show that for every percolation parameter $$p\in (1/2,1)$$ , almost surely, $$Z_N$$ grows exponentially slower than its expected value. In other words, we prove that $$\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}$$ , where the expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on $$\mathbb{Z }^2$$ , so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.
dc.relation.isversionofjnlnameProbability Theory and Related Fields
dc.relation.isversionofjnlvol159
dc.relation.isversionofjnlissue3-4
dc.relation.isversionofjnldate2014
dc.relation.isversionofjnlpages777-808
dc.relation.isversionofdoihttp://dx.doi.org/10.1007/s00440-013-0520-1
dc.identifier.urlsitehttps://arxiv.org/abs/1203.6051v3
dc.relation.isversionofjnlpublisherSpringer
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenon
dc.description.halcandidateoui
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2017-03-13T13:12:21Z


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