Eigenvalue Order Statistics for Random Schrödinger Operators with Doubly-Exponential Tails

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 Dirichlet 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, and the corresponding eigenfunctions are exponentially localized in regions where $\xi$ takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue/eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass