  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.
