Tuomas Sahlsten (Manchester)

Delocalisation of waves under scaling limits

Abstract:
We establish quantitative quantum ergodicity type delocalisation theorem for waves on hyperbolic surfaces of large genus. In the compact setting our assumptions hold for random surfaces in the sense of Weil-Petersson volume in the Teichm├╝ller space due to the work of Mirzakhani and in non-compact setting for arithmetic surfaces coming from congruence covers of the modular surface. The methods are based on Benjamini-Schramm scaling limits of metric measure spaces and Stein type harmonic analysis ergodic theorems, and are inspired by similar results on graphs. We plan to give a gentle introduction to the field before going to our results. Joint work with Etienne Le Masson (Cergy-Pontoise University, France).