Quantum mixing is a stronger version of quantum ergodicity introduced by Zelditch and Sunada, where the aim is to compare the asymptotics of observable averages of the form
with uniform averages of the form delta_{j,k} _j, where psi_j are the eigenfunctions of the underlying Laplacian. This gives information on both the asymptotic density |psi_j(x)|^2 and the eigenfunction correlations psi_j(x)psi_k(y).
I will discuss a new approach to this problem that we developed with Charles Bordenave and Cyril Letrouit, for sequences of Schreier graphs converging in the sense of Benjamini-Schramm to an infinite Cayley graph that has absolutely continuous spetrum. In contrast to earlier results of quantum ergodicity, we do not need the limiting graph to be a tree or Z^d-periodic. In the event that the sequence converges strongly, we obtain stronger conclusions.
If time allows I will discuss more briefly some results with Kiran Kumar on the ergodicity of quantum walks. The frameworks are distinct, but will be presented together as different ways of understanding and possibly characterizing quantum ergodicity (eigenvectors vs dynamics vs spectral type).