Delocalisation of eigenfunctions on large genus random surfaces

We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an L2 normalised eigenfunction restricted to a measurable subset of the surface has squared L2-norm ε > 0, only if the set has a relatively large size—exponential in the geometric parameter. For random surfaces with respect to the Weil—Petersson probability measure, we then show, with high probability as g → ∞, that the size of the set must be at least the genus of the surface to some power dependent upon the eigenvalue and ε