I will discuss some of the recent progress on the delocalisation of eigenstates of Schrödinger operators on random models of hyperbolic surfaces of large genus. We review how these relates to the parallel new developments in optimal spectral gap, spectral statistics and on large (random) graphs, and then discuss some of the main challenges differentiating this study to the graph case. Joint work with K. Hippi, F. Lequen, S. Mikkelsen and H. Ueberschär.
