Delocalization and ergodicity of the Anderson model on Bethe lattices

We review the state of the art on the delocalized non-ergodic regime of the Anderson model on Bethe lattices. We also present new results using Belief Propagation, which consists in solving the self-consistent recursion relations for the Green's functions directly on a given sample. This allows us to numerically study very large system sizes and to directly access observables related to the eigenfunctions and energy level statistics. In agreement with recent works, we establish the existence of a delocalized non-ergodic phase on Cayley trees. On random regular graphs instead our results indicate that ergodicity is recovered when the system size is larger than a cross-over scale $N_c (W)$, which diverges exponentially fast approaching the localization transition. This scale corresponds to the size at which the mean-level spacing becomes smaller than the Thouless energy $E_{Th} (W)$. Such energy scale, which vanishes exponentially fast approaching the localization transition, is the one below which ergodicity in the level statistics is restored in the thermodynamic limit. Remarkably, the behavior of random regular graphs below $N_c (W)$ coincides with the one found close to the root of loop-less infinite Cayley trees, {\it i.e.} only above $N_c (W)$ the effects of loops emerge and random regular graphs behave differently from Cayley trees. Our results indicate that ergodicity is recovered in the thermodynamic limit on random regular graph. However, all observables probing volumes smaller than $N_c(W)$ and times smaller than $\hbar/E_{Th} (W)$ are expected to behave as if there were an intermediate phase. Given the very fast divergence of $N_c(W)$ and $\hbar/E_{Th} (W)$ these non-ergodic effects are very pronounced in a large region preceding the localization transition, and they can be related to the intermediate phase present on Cayley trees.