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

2022/08/07 by Herschenfeld, Samuel, Hislop, Peter D.
#35J10 #35P20 #81Q10 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2208.03598

Abstract

We use the method of eigenvalue level spacing developed by Dietlein and Elgart (arXiv:1712.03925) 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 (arXiv:1809.01236), 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.

Related

• On the local eigenvalue spacings for certain Anderson-Bernoulli Hamiltonians
• On Eigenvalue spacings for the 1-D Anderson model with singular site distribution
• Eigenvalue Statistics for higher rank Anderson model over Canopy tree
• Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges
• Poisson Statistics in the Non-Homogeneous Hierarchical Anderson Model