The sound of random graphs and Cayley graphs

Infinite random graphs, such as Galton-Watson trees and percolation clusters, may have real numbers that are eigenvalues with probability one, providing a consistent "sound". These numbers correspond to atoms in their density-of-states measure.

When does the sound exist? When is the measure purely atomic? I will review many examples and show some elementary techniques developed in joint works with Charles Bordenave and Arnab Sen.

Remarkably, understanding the spectra of random graphs also yields results about deterministic Cayley graphs of lamplighter groups. In joint work with L. Grabowski we answer an old question of W. Luck.