Short geodesic loops and Lp norms of eigenfunctions on large genus random surfaces

We give upper bounds for L^p norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus g→+∞, we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have almost surely at most one such loop of length less than clogg for small enough c>0. This allows us to deduce that the L^p norms of L² normalised eigenfunctions on X are bounded by 1/√logg almost surely in the large genus limit for any p>2+ε for ε>0 depending on the spectral gap λ₁(X) of X.