Discrete Alloy-Type Models: Regularity of Distributions and Recent Results

#### M. Tautenhahn, I. Veselic

2015, v.21, №3, 823-846

ABSTRACT

We consider discrete random Schr\"odinger operators on $\ell^2 (\ZZ^d)$ with a potential of discrete alloy-type structure.
That is, the potential at lattice site $x \in \ZZ^d$ is given by a linear combination of independent identically distributed random variables, possibly
with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly $\tau$-H\"older continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.

Keywords: discrete alloy type model, correlated Anderson model, density of states, Wegner estimate, conditional density, non-monotone randomness, Poisson eigenvalue statistics