Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d

Details Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d

2007-12-11

arxiv.org/abs/0712.1740v2

Proc. Symp. Pure Math. vol. 77 (2008), 87-108

Physics

Mathematical Physics

22 pages; minor extensions and updates

Scientific paper

We give an overview and extension of recent results on ergodic random Schr\"odinger operators for models on $\mathbb{Z}^d$. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed Shubin-Pastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.

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