Georgia Institute of Technology College of Sciences

The spectrum of the discrete Laplacian on a infinite graph, or of its random perturbations such as the Anderson tight-binding model, encodes a rich information about the structure of that space. While natural questions abound (nature of the spectrum, localization of eigenfunctions, behavior of the spectral measure), few admit complete answers outside of very specific cases. In this talk, we will briefly survey some of the main open questions in the area. We will then present an elementary geometric criterion that provides control over the  regularity of the spectral measure.