Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)

Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from the known Poisson distribution of cycle counts in regular graphs, in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely. The result is analogous to the so called "diagonal approximation" in Quantum Chaos. We also show that by assuming that the spectral correlations are given by RMT to all orders, we can compute the leading deviations from the Poisson distribution for cycle counts. We provide numerical evidence which supports this conjecture.Comment: 15 pages, 5 figure

Spectral cross correlations of magnetic edge states

We observe strong, non-trivial cross-correlations between the edge states found in the interior and the exterior of magnetic quantum billiards. Our analysis is based on a novel definition of the edge state spectral density which is rigorous, practical and semiclassically accessible.Comment: 4 pages, 3 figures (high quality version available at http://www.klaus-hornberger.de