Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
- Biskup, Marek
- König, Wolfgang
2010 Mathematics Subject Classification
- 60F05 60G55 70H40 70B80
- parabolic Anderson model, random Schrödinger operator, eigenvalue order statistics, Poisson point process convergence, Anderson localisation
We consider random Schrödinger operators of the form Δ + ξ, where $Delta; is the lattice Laplacian on Zd 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 Zd. 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.
- Comm. Math. Phys., 341 (2016) pp. 179--218.