An Application of Singular Traces to Crystals and Percolation

Abstract

For a certain class of discrete metric spaces, we provide a formula for the density of states. This formula involves Dixmier traces and is proven using recent advances in operator theory. Various examples are given of metric spaces for which this formula holds, including crystals, quasicrystals and the infinite cluster resulting from super-critical bond percolation on ${\mathbb{Z}}^d$.

arXiv comment28 pages, no figures

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Cited by in corpus (3)

• A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
• A General Dixmier Trace Formula for the Density of States on Open Manifolds
• Cyclic cocycles and quantized pairings in materials science