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

Delocalization and ergodicity of the Anderson model on Bethe lattices