Delocalisation of Laplacian Eigenfunctions on Large Genus Random Surfaces

Abstract

In this thesis, we seek to understand delocalisation properties exhibited by Laplacian eigenfunctions on closed hyperbolic surfaces of large genus. In particular, we study the shape and spread of the eigenfunctions over the surface. We then exhibit stronger results that hold with high probability for surfaces in the Weil-Petersson random surface model. The first contribution presented here is a study of the Lp norms of the eigenfunctions. Understanding the magnitude of these norms offers insight into the shape of the eigenfunctions; for example, how large they can be at any point. Our results show that these norms decay with respect to a parameter involving geodesic loops on the surface. We then study this parameter probabilistically, leading to decay rates on the Lp norms logarithmic in the surface genus. Next, we study the geometry of the surfaces themselves more precisely by introducing the tangle-free parameter of a surface. This looks at what types of subsurfaces can be embedded inside a surface. We demonstrate that knowledge of the size of this parameter translates to information on the structure of geodesics in the surface whose lengths are of a similar size. We then study the parameter probabilistically, showing that the local geometry of these surfaces is similar to that of regular graphs. Using this tangle-free framework, we then study the extent to which eigenfunctions can concentrate on subsets of the surface. In particular, we show near full concentration can only happen on subsets of size at least exponential in the tangle-free parameter, or probabilistically, at least the genus to some power.

Date of Award	31 Dec 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Charles Walkden (Supervisor)