  We present a quantum ergodicity theorem for fixed spectral window and
sequences of compact hyperbolic surfaces converging to the hyperbolic plane in
the sense of Benjamini and Schramm. This addresses a question posed by Colin de
Verdi\`{e}re. Our theorem is inspired by results for eigenfunctions on large
regular graphs by Anantharaman and the first-named author. It applies in
particular to eigenfunctions on compact arithmetic surfaces in the level
aspect, which connects it to a question of Nelson on Maass forms. The proof is
based on a wave propagation approach recently considered by Brooks,
Lindenstrauss and the first-named author on discrete graphs. It does not use
any microlocal analysis, making it quite different from the usual proof of
quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave
propagator with renormalised averaging operators over discs, which simplifies
the analysis and allows us to make use of a general ergodic theorem of Nevo. As
a consequence of this approach, we require little regularity on the
observables.
