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

Uniform existence of the integrated density of states for models on $${\mathbb{Z}}^d$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2008 by Springer Science + Business Media B.V.
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2238-3
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: May 27, 2008

References

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