Local eigenvalue statistics for higher-rank Anderson models after Dietlein–Elgart

Abstract

We use the method of eigenvalue level spacing developed by Dietlein and Elgart [Level spacing and Poisson statistics for continuum random Schrödinger operators, J. Eur. Math. Soc. (JEMS) 23(4) (2021) 1257–1293] to prove that the local eigenvalue statistics (LES) for the Anderson model on [Formula: see text], with uniform higher-rank [Formula: see text], single-site perturbations, is given by a Poisson point process with intensity measure [Formula: see text], where [Formula: see text] is the density of states at energy [Formula: see text] in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna [Eigenvalue statistics for random Schrödinger operators with non-rank one perturbations, Comm. Math. Phys. 340(1) (2015) 125–143], who proved that the LES is a compound Poisson process with Lévy measure supported on the set [Formula: see text]. Our proofs are an application of the ideas of Dietlein and Elgart to these higher-rank lattice models with two spectral band edges, and illustrate, in a simpler setting, the key steps of the proof of Dietlein and Elgart.