Anderson localisation on a Cayley tree: a new model with a simple solution

A new model is introduced for wave propagation on a disordered Cayley tree. The model has sufficient simplifying features (related to the absence of time-reversal symmetry and to phase randomisation) that a straightforward study is possible of the probability distribution for eigenstate amplitudes. It is shown that the model supports extended eigenstates at weak scattering and that eigenstates are exponentially localised at strong scattering. Two further distinctions are also shown to be important. The localised phase is insulating if the exponential decay with distance of eigenstate amplitudes is faster than the exponential growth in number of sites with distance from an origin; otherwise eigenfunctions are not square-integrable and the phase is conducting. Correspondingly, in the extended phase there are large amplitude fluctuations unless the correlation length is smaller than the exponential growth rate of site number with distance.