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 $g\to +\infty$, 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 a $O(1/\sqrt{\log g})$ with high probability in the large genus limit for any $p>2+\epsilon$ for $\epsilon >0$ depending on the spectral gap $\hat{I}{»}_1(X)$ of $X$, with an implied constant depending on the eigenvalue and the injectivity radius.

37 pages, 1 figure, v3: Many updates and improvements in the proof of the geometric side. To appear in GAFA