Graph Eigenfunctions and Quantum Unique Ergodicity

We apply the techniques of our previous paper to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of $\mathbb{H}\times\mathbb{H}$. In both cases, we show that quantum limit measures of such sequences of eigenfunctions carry positive entropy on almost every ergodic component. Together with prior work of the second named author, this implies Quantum Unique Ergodicity for such functions.Comment: 8 page

Recent results of quantum ergodicity on graphs and further investigation

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.Comment: To appear in "Annales de la facult\'e des Sciences de Toulouse