Delocalisation of eigenfunctions on large genus random surfaces

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