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Aperiodic substitution tilings provide popular models for quasicrystals,materials exhibiting aperiodic order. We study the graph Laplacian associatedwith four tilings from the mutual local derivability class of the Penrosetiling, as well as the Ammann--Beenker tiling. In each case we exhibitlocally-supported eigenfunctions, which necessarily cause jump discontinuitiesin the integrated density of states for these models. By bounding themultiplicities of these locally-supported modes, in several cases we provideconcrete lower bounds on this jump. These results suggest a host of questionsabout spectral properties of the Laplacian on aperiodic tilings, which wecollect at the end of the paper.

Discontinuities of the Integrated Density of States for Laplacians Associated with Penrose and Ammann-Beenker Tilings