Short geodesic loops and $$L^p$$ norms of eigenfunctions on large genus random surfaces

Abstract

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 $$g \rightarrow +\infty $$ , we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $$c \log g$$ for small enough $$c > 0$$ . This allows us to deduce that the $$L^p$$ norms of $$L^2$$ normalised eigenfunctions on X are $$O(1/\sqrt{\log g})$$ with high probability in the large genus limit for any $$p > 2 + \varepsilon $$ for $$\varepsilon > 0$$ depending on the spectral gap $$\lambda _1(X)$$ of X, with an implied constant depending on the eigenvalue and the injectivity radius.