dc.contributor.author Lacoin, Hubert dc.date.accessioned 2014-07-22T13:23:21Z dc.date.available 2014-07-22T13:23:21Z dc.date.issued 2014 dc.identifier.issn 0178-8051 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/13768 dc.language.iso en en dc.subject Random media dc.subject Polymers dc.subject Percolation dc.subject Self-avoiding walk dc.subject Disorder relevance dc.subject.ddc 519 en dc.title Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster. dc.type Article accepté pour publication ou publié dc.description.abstracten In 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.isversionofjnlname Probability Theory and Related Fields dc.relation.isversionofjnlvol 159 dc.relation.isversionofjnlissue 3-4 dc.relation.isversionofjnldate 2014 dc.relation.isversionofjnlpages 777-808 dc.relation.isversionofdoi http://dx.doi.org/10.1007/s00440-013-0520-1 dc.identifier.urlsite https://arxiv.org/abs/1203.6051v3 dc.relation.isversionofjnlpublisher Springer dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.relation.forthcoming non en dc.relation.forthcomingprint non en dc.description.ssrncandidate non dc.description.halcandidate oui dc.description.readership recherche dc.description.audience International dc.relation.Isversionofjnlpeerreviewed oui dc.date.updated 2017-03-13T13:12:21Z
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