An Application of Singular Traces to Crystals and Percolation

TL;DR

This paper develops a formula for the density of states in specific discrete metric spaces using Dixmier traces, with applications to crystals, quasicrystals, and percolation clusters.

Contribution

It introduces a novel application of singular traces to compute the density of states in complex discrete spaces, extending operator theory methods.

Findings

Formula applies to crystals, quasicrystals, and percolation clusters.

Demonstrates the use of Dixmier traces in spectral analysis.

Provides examples validating the formula's applicability.

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 $\mathbb{Z}^d$.