Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails

Abstract

We consider random Schrödinger operators of the form $\Delta +\xi$, where $\Delta$ is the lattice Laplacian on ${\mathbb{Z}}^d$ and $\xi$ 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 ${\mathbb{Z}}^d$. We show that for $\xi$ 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 $\xi$ 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.

36 pages

References in corpus (3)

• A two cities theorem for the parabolic Anderson model
• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials
• Localisation and ageing in the parabolic Anderson model with Weibull potential

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• Spectral Analysis of the Quantum Random Energy Model
• Eigenvalue vs perimeter in a shape theorem for self-interacting random walks
• On the spectral gap in the Kac-Luttinger model and Bose-Einstein condensation
• The Parabolic Anderson Model on a Galton-Watson tree revisited
• Deviation of top eigenvalue for some tridiagonal matrices under various moment assumptions