Correlations Estimates in the Lattice Anderson Model

Abstract

We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schrödinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new $n$-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least $n$ eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.

16 pages

References in corpus (3)

• Finite-Volume Fractional-Moment Criteria for Anderson Localization
• Simplicity of eigenvalues in the Anderson model
• The repulsion between localization centers in the Anderson model

Cited by in corpus (22)

• Eigenvector localization for random band matrices with power law band width
• Generalized eigenvalue-counting estimates for the Anderson model
• Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method
• Poisson statistics of eigenvalues in the hierarchical Anderson model
• The scaling limit of the critical one-dimensional random Schrodinger operator
• Minami's estimate: beyond rank one perturbation and monotonicity
• Simplicity of eigenvalues in Anderson-type models
• Decorrelation estimates for the eigenlevels of the discrete Anderson model in the localized regime
• Matrix regularizing effects of Gaussian perturbations
• Localization via fractional moments for models on $\mathbb{Z}$ with single-site potentials of finite support
• On the Joint Distribution of Energy Levels of Random Schroedinger Operators
• Conductivity and the current-current correlation measure
• A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian
• Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model
• Level Spacing for Non-Monotone Anderson Models
• Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials
• Spectral statistics for weakly correlated random potentials
• Poisson eigenvalue statistics for random Schrödinger operators on regular graphs
• Grand Ensembles of deterministic operators. II. Localization for generic `haarsh' potentials
• Decorrelation estimates for some continuous and discrete random Schr{ö}dinger operators in dimension one, without covering condition
• Poisson Statistics in the Non-Homogeneous Hierarchical Anderson Model
• Eigenvalue Correlations in Continuum one-dimensional Anderson Models