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Quantum ergodicity on large graphs

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.