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Author: Biskup, Marek × Subject: Anderson localisation ×

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    Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Biskup, Marek; König, Wolfgang
    We consider random Schrödinger operators of the form Delta+zeta , where D is the lattice Laplacian on Zd and Delta is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for sigma with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where zeta takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.
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    Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Biskup, Marek; König, Wolfgang; Santos, Renato Soares dos
    We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schrödinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.