The quantum ergodicity theorem states that on compact hyperbolic surfaces, most eigenfunctions of the Laplacian equidistribute spatially in the large eigenvalue limit. We will present an alternative equidistribution theorem for eigenfunctions where the eigenvalues stay bounded and we take instead sequences of compact hyperbolic surfaces converging to the plane in the sense of Benjamini and Schramm. This approach is motivated by joint works with Anantharaman, Brooks and Lindenstrauss on eigenvectors of the discrete Laplacian on regular graphs. The proof uses an ergodic theorem of Nevo.
Joint work with Tuomas Sahlsten.