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.

We use the method of eigenvalue level spacing developed by Dietlein and Elgart [7] to prove that the local eigenvalue statistics (LES) for the Anderson model on Z d , with uniform higher-rank m 2, single-site perturbations, is given by a Poisson point process with intensity measure n(E0) ds, where n(E0) is the density of states at energy E0 in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna [13], who proved that the LES is a compound Poisson process with Lévy measure supported on the set {1, 2, . . . , m}. Our proofs are an application of the ideas of Dieltein 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 Dieltein and Elgart. Contents 1. Statement of the problem 1.1. The higher-rank Anderson model 1.2. Results for higher-rank Anderson models 1.3. Simplifications for the higher-rank Anderson model 1.4. Some open problems 1.5. Contents of the paper 2. Background: Wegner and generalized Minami estimates for higher-rank Anderson models 2.1. Spectral averaging and the Wegner estimate 2.2. The generalized Minami estimate 3. Removal of eigenvalue degeneracies 3.1. Eigenvalue splitting: Removal of one degeneracy in I 3.2. Eigenvalue splitting: Induction to all eigenvalues in I 4. Eigenvalue level spacing estimate 5. The weak Minami estimate 6. Simplicity of the eigenvalues in the localization regime 7. The LES is a Poisson point process A. One-parameter perturbations B. A Cartan-type lemma and the size of bad configurations C. Discrete Laplacians: Boundary conditions, eigenvalues, and eigenvectors References PDH is partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 843327. This article is partially based on the doctoral dissertation of SH submitted in partial fulfillment of the PhD degree in mathematics at the University of Kentucky.