74 research outputs found
Discrete alloy-type models: Regularity of distributions and recent results
We consider discrete random Schr\"odinger operators on with a potential of discrete alloy-type structure. That is, the
potential at lattice site 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 -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.Comment: 20 pages, 0 figure
Lipschitz-continuity of the integrated density of states for Gaussian random potentials
The integrated density of states of a Schroedinger operator with random
potential given by a homogeneous Gaussian field whose covariance function is
continuous, compactly supported and has positive mean, is locally uniformly
Lipschitz-continuous. This is proven using a Wegner estimate
Uniform existence of the integrated density of states for models on \ZZ^d
We provide an ergodic theorem for certain Banach-space valued functions on
structures over \ZZ^d, which allow for existence of frequencies of finite
patterns. As an application we obtain existence of the integrated density of
states for associated finite-range operators in the sense of convergence of the
distributions with respect to the supremum norm. These results apply to various
examples including periodic operators, percolation models and nearest-neighbour
hopping on the set of visible points. Our method gives explicit bounds on the
speed of convergence in terms of the speed of convergence of the underlying
frequencies. It uses neither von Neumann algebras nor a framework of random
operators on a probability space.Comment: 15 page
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