Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

In this paper we study the spectrum of long-range percolation graphs. The underlying geometry is given in terms of a finitely generated amenable group. We prove that the integrated density of states (IDS) or spectral distribution function can be approximated uniformly in the energy variable. This result is already new for percolation on ℤd. Using this, we are able to characterize the set of discontinuities of the IDS.