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A limit theorem for the survival probability of a simple random walk among power-law renewal traps

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorPoisat, Julien
HAL ID: 6620
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorSimenhaus, François
dc.date.accessioned2019-02-25T12:48:24Z
dc.date.available2019-02-25T12:48:24Z
dc.date.issued2018
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/18486
dc.language.isoenen
dc.subjectRandom walks in random trapsen
dc.subjectpolymers in random environmentsen
dc.subjectparabolicAnderson modelen
dc.subjectsurvival probabilityen
dc.subjectFKG inequalitiesen
dc.subjectRay-Knight theoremsen
dc.subject.ddc519en
dc.titleA limit theorem for the survival probability of a simple random walk among power-law renewal trapsen
dc.typeDocument de travail / Working paper
dc.description.abstractenWe consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1−e −β , where β is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent γ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is γ/(γ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter β that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case β corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5.en
dc.identifier.citationpages36en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris-Dauphineen
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-01878052en
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2018-09
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2019-02-25T12:40:13Z
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