Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Zd

We consider ergodic random Schrödinger operators on the metric graph Zd with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.