Density of states of randomly sprinkled graphs

Independently at every vertex $x$ of a large core graph $\mathbb G$, we draw a random graph and connect $x$ to a subset of its vertices. This model of a randomly sprinkled graph captures qualitative features of commonly observed graphs; it can also be regarded as a model of quantum disorder, where disorder arises from local perturbations to the graph geometry. We show that it is naturally connected both to the Anderson model and to site percolation on $\mathbb G$ by deriving an Anderson-percolation representation for its spectrum. We characterize the support of the spectrum and derive quantitative bounds on the integrated density of states. We also analyse in detail the regime of sparse sprinkling, in particular investigating the behaviour of the density of states in the vicinity of atoms.