  We give three different proofs of the main result of Anantharaman-Le Masson,
establishing quantum ergodicity -- a form of delocalization --for
eigenfunctions of the laplacian on large regular graphs of fixed degree. These
three proofs are much shorter than the original one, quite different from one
another, and we feelthat each of the four proofs sheds a different light on the
problem. The goal of this exploration is to find a proof that could be adapted
for othermodels of interest in mathematical physics, such as the Anderson model
on large regular graphs, regular graphs with weighted edges, or possibly
certain models of non-regular graphs.A source of optimism in this direction is
that we are able to extend the last proof to the case of anisotropic random
walks on large regular graphs.
