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Positivity

, Volume 12, Issue 4, pp 571–589 | Cite as

Uniform existence of the integrated density of states for models on \({\mathbb{Z}}^d\)

  • Daniel Lenz
  • Peter Müller
  • Ivan Veselić
Article

Abstract

We provide an ergodic theorem for certain Banach-space valued functions on structures over \({\mathbb{Z}}^d\), which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated discrete 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.

Keywords

Random Schrödinger operator integrated density of states uniform ergodic theorem 

Mathematics Subject Classification (2000)

Primary 37A30, 81Q10 Secondary 34P05, 47B80, 47N50 

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Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Fakultät für MathematikChemnitzGermany
  2. 2.Institut für Theoretische PhysikGöttingenGermany
  3. 3.Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft & Fakultät für MathematikTU ChemnitzGermany

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