Continuity of the integrated density of states on random length metric graphs

Abstract

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge length are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.

31 pages, 2 figures; introduction extended, references updated

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• Some abstract Wegner estimates with applications
• Uniform existence of the integrated density of states on metric Cayley graphs
• Lifshitz asymptotics and localization for random breather models