  We prove a result of delocalization for the Anderson model on the regular
tree (Bethe lattice). When the disorder is weak, it is known that large parts
of the spectrum are a.s. purely absolutely continuous, and that the dynamical
transport is ballistic. In this work, we prove that in such AC regime, the
eigenfunctions are also delocalized in space, in the sense that if we consider
a sequence of regular graphs converging to the regular tree, then the
eigenfunctions become asymptotically uniformly distributed. The precise result
is a quantum ergodicity theorem.
