hal.structure.identifier CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] dc.contributor.author Poisat, Julien HAL ID: 6620 hal.structure.identifier CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] dc.contributor.author Simenhaus, François dc.date.accessioned 2022-02-25T15:11:47Z dc.date.available 2022-02-25T15:11:47Z dc.date.issued 2022 dc.identifier.uri https://basepub.dauphine.psl.eu/handle/123456789/22775 dc.language.iso en en dc.subject Random walks in random obstacles en dc.subject polymers in random environments en dc.subject parabolic Anderson model en dc.subject survival probability en dc.subject localization en dc.subject one-city theorem en dc.subject Markov renewal processes en dc.subject.ddc 519 en dc.title Localization of a one-dimensional simple random walk among power-law renewal obstacles en dc.type Document de travail / Working paper dc.description.abstracten We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces. en dc.publisher.city Paris en dc.identifier.citationpages 55 en dc.relation.ispartofseriestitle Cahier de recherche CEREMADE, Université Paris Dauphine-PSL en dc.identifier.urlsite https://hal.archives-ouvertes.fr/hal-03526023 en dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.identifier.citationdate 2022 dc.description.ssrncandidate non dc.description.halcandidate non en dc.description.readership recherche en dc.description.audience International en dc.date.updated 2022-02-25T15:09:34Z hal.author.function aut hal.author.function aut
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