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 WeilPetersson 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 $p > 2 + \varepsilon$ 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.09961
 Bibcode:
 2019arXiv191209961G
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Mathematics  Dynamical Systems;
 Mathematics  Probability;
 37D40;
 11F72
 EPrint:
 37 pages, 1 figure, v3: Many updates and improvements in the proof of the geometric side. To appear in GAFA