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

TL;DR

This paper establishes bounds on the $L^p$ norms of Laplacian eigenfunctions on large genus hyperbolic surfaces, showing that these norms tend to decrease as the genus increases, with high probability.

Contribution

It introduces new bounds for eigenfunction norms based on short geodesic loops and demonstrates that random large genus surfaces typically have at most one such loop, leading to asymptotic norm estimates.

Findings

$L^p$ norms are $O(1/\sqrt{\log g})$ for large genus $g$

Most random hyperbolic surfaces have at most one short geodesic loop

Norm bounds depend on spectral gap and injectivity radius

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 $p > 2 + \varepsilon$ 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.