# Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

# Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

Please note special day (Tuesday). Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière is an equidistribution result of eigenfunctions of the Laplacian in large frequency limit on a Riemannian manifold with an ergodic geodesic flow. We complement this work by introducing a Quantum Ergodicity theorem on hyperbolic surfaces, where instead of taking high frequency limits, we fix an interval of frequencies and vary the geometric parameters of the surface such as volume, injectivity radius and genus. In particular, we are interested of such results under Benjamini-Schramm convergence of hyperbolic surfaces. This work is inspired by analogous results for holomorphic cusp forms and eigenfunctions for large regular graphs. Unlike in the QE theorem, our methods do not rely on pseudodifferential calculus but instead on a wave propagation approach, which was recently considered by Brooks, Le Masson and Lindenstrauss in the graph theoretic setting. We still employ ergodic theory in the form of exponential decay of correlations for the geodesic flow on hyperbolic surfaces. Joint work with Etienne Le Masson (Bristol).