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

We consider random Schrödinger operators of the form $Δ+ξ$, where $Δ$ is the lattice Laplacian on $\mathbb Z^d$ and $ξ$ 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 $ξ$ 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 $ξ$ 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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