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 quantum ergodicity on regular graphs, and
equidistribution results for holomorphic forms in the level aspect by
Nelson et al. The proof presented is more elementary and geometric than
the usual proof of quantum ergodicity and uses an ergodic theorem of Nevo.
Joint work with Tuomas Sahlsten.