Sparse graphs and their Benjamini-Schramm limits: a spectral tour

Abstract: Sparse graphs with bounded average degree form a rich class of discrete structures where local geometry strongly influences global behavior. The Benjamini-Schramm (BS) convergence offers a natural framework to describe their asymptotic local structure. In this note, we survey spectral aspects of BS convergence and their applications, with a focus on random Schreier graphs and covering graphs. We review some recent progress on the spectral decomposition of the local operators on graphs. We discuss the behavior of extreme eigenvalues and the growing role of strong convergence in distribution, which rules out spectral outliers. We also give a new application of strong convergence to the typical graph distance between vertices in Schreier graphs