Localization for Trees with Random Geometry

Abstract: We will discuss metric tree graphs with random geometry in the sense that number of edges attached to each vertex and the length of each edge are random variables. We will examine the particular case of radial symmetry, that is, the case in which these random variables depend solely on the combinatorial distance to the root of the tree. Our main result is that the Kirchhoff Laplacian on such graphs exhibits spectral and dynamical localization almost surely. Because the branching number only takes integral values, one must be able to handle singular distributions (e.g. Benoulli), which typically present the greatest challenges in the study of random operators.