Localization transition on a Cayley tree via spectral statistics

The spectral statistics of a Cayley tree are numerically studied. The statistics are nonuniversal due to the high ratio of boundary sites. Once the boundary sites are connected to each other in a way that preserves the local structure of the tree, the universal statistics of the spectra is recovered. A clear localization transition is observed as function of on-site disorder strength, with a critical disorder W c = 11.44′0.08 and critical index v=0.51′0.05. The value of v fits nicely to its mean-field value, while the value of W c is puzzling.