The problem of Anderson localization on hierarchical lattices has attracted attention recently in the context of many-body localization ptoblem. We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice and Random Regular Graphs.
For Bethe lattice, we show that the wave function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally tree-like structure without a boundary (random regular graph) where extended wavefunctions are ergodic.
For RRG, we focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that presence of even very large loops (with size comparable with the diameter of the structure) modifies the properties of delocalized phase a lot. We show that the numerics can be interpreted in terms of the finite-size crossover from small to large system, separated by the correlation volume, diverging exponentially at the localization transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wavefunction statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis non-trivial.