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 $L^2$ normalised eigenfunction restricted to a measurable subset of the surface has squared $L^2$-norm $\epsilon >0$, 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 $g\to \mathrm{\infty }$, that the size of the set must be at least the genus of the surface to some power dependent upon the eigenvalue and $\epsilon$.