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Localization crossover for the continuous Anderson Hamiltonian in 1-d

hal.structure.identifierDépartement de Mathématiques et Applications - ENS Paris [DMA]
dc.contributor.authorDumaz, Laure
HAL ID: 739617
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorLabbé, Cyril
HAL ID: 9675
dc.date.accessioned2022-02-28T10:35:26Z
dc.date.available2022-02-28T10:35:26Z
dc.date.issued2021
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22792
dc.language.isoenen
dc.subjectAnderson Hamiltonianen
dc.subjectHill’s operatoren
dc.subjectlocalizationen
dc.subjectdiffusionen
dc.subjectPoisson statisticsen
dc.subjecthypocoercivityen
dc.subjectMalliavin calculusen
dc.subject.ddc519en
dc.titleLocalization crossover for the continuous Anderson Hamiltonian in 1-den
dc.typeDocument de travail / Working paper
dc.description.abstractenWe investigate the behavior of the spectrum of the continuous Anderson Hamiltonian HL, with white noise potential, on a segment whose size L is sent to infinity. We zoom around energy levels E either of order 1 (Bulk regime) or of order 1≪E≪L (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of HL. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.en
dc.publisher.cityParisen
dc.identifier.citationpages63en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris Dauphine-PSLen
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-03436108en
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2021
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2022-02-28T10:32:52Z
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