Asymptotic direction in random walks in random environment revisited

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dc.contributor.authorDrewitz, Alexander
dc.contributor.authorRamirez, Alejandro F.
dc.date.accessioned2024-01-10T13:11:19Z
dc.date.available2024-01-10T13:11:19Z
dc.date.issued2010
dc.description.abstractConsider a random walk {X(n) : n >= 0} in an elliptic i.i.d. environment in dimensions d >= 2 and call P(0) its averaged law starting from 0. Given a direction I is an element of S(d-1), A(l) = {lim(n ->infinity) Xn . l = infinity} is called the event that the random walk is transient in the direction I. Recently Simenhaus proved that the following are equivalent: the random walk is transient in the neighborhood of a given direction; P(0)-a.s. there exists a deterministic asymptotic direction; the random walk is transient in any direction contained in the open half space defined by this asymptotic direction. Here we prove that the following are equivalent: P(0)(A(l) boolean OR A(-l)) = 1 in the neighborhood of a given direction; there exists an asymptotic direction v such that P(0) (A(upsilon) boolean OR A(-upsilon)) = 1 and P(0)-a.s we have lim(n ->infinity) X(n)/vertical bar X(n)vertical bar = 1(A upsilon)upsilon - 1(A-upsilon)upsilon; P(0) (A(l) boolean OR A(-l)) = 1 if and only if l . upsilon not equal 0. Furthermore, we give a review of some open problems.
dc.description.funderInternational Research Training Group
dc.description.funderBerlin Mathematical School
dc.description.funderFondo Nacional de Desarrollo Cientifico y Tecnologico
dc.description.funderIniciativa Cientifica Milenio
dc.fechaingreso.objetodigital2024-05-15
dc.format.extent14 páginas
dc.fuente.origenWOS
dc.identifier.doi10.1214/09-BJPS028
dc.identifier.issn0103-0752
dc.identifier.urihttps://doi.org/10.1214/09-BJPS028
dc.identifier.urihttps://repositorio.uc.cl/handle/11534/78032
dc.identifier.wosidWOS:000296129700005
dc.information.autorucMatemática;Ramirez A.;S/I;70094
dc.issue.numero2
dc.language.isoen
dc.nota.accesocontenido completo
dc.pagina.final225
dc.pagina.inicio212
dc.publisherBRAZILIAN STATISTICAL ASSOCIATION
dc.revistaBRAZILIAN JOURNAL OF PROBABILITY AND STATISTICS
dc.rightsacceso abierto
dc.subjectRandom walk in random environment
dc.subjectrenewal times
dc.subjectasymptotic directions
dc.subjectPLANAR RANDOM-WALKS
dc.subjectZERO-ONE LAW
dc.titleAsymptotic direction in random walks in random environment revisited
dc.typeartículo
dc.volumen24
sipa.codpersvinculados70094
sipa.indexWOS
sipa.trazabilidadCarga SIPA;09-01-2024
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