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

TL;DR

This paper studies the properties of the integrated density of states in random quantum graphs, establishing formulas, regularity results, and characterizing eigenfunctions, with applications to periodic and random edge length models.

Contribution

It introduces a comprehensive analysis of the integrated density of states on random metric graphs, including a trace formula, Wegner estimate, and characterization of eigenfunctions.

Findings

Trace per unit volume formula established

Wegner estimate proved for random edge lengths

Discontinuities of the IDS characterized in periodic 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.