Mean of the $L^{\infty }$-norm for $L^2$-normalized random waves on compact aperiodic Riemannian manifolds

Abstract

This article concerns upper bounds for $L^{\infty }$-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold $(M,g).$ We study $f_{\hat{I}}»$ chosen uniformly at random from the space of $L^2$-normalized linear combinations of Laplace eigenfunctions with eigenvalues in the interval $(λ^2, \lr{λ+1}^2].$ Our main result is that the expected value of $\norm{f_λ}_\infty$ grows at most like $C\sqrt{\log \hat{I}»}$ as $\hat{I}»\to \infty$, where $C$ is an explicit constant depending only on the dimension and volume of $(M,g).$ In addition, we obtain concentration of the $L^{\infty }$-norm around its mean and median and study the analogous problems for Gaussian random waves on $(M,g).$

Withdrawn due to significant overlap with the work of Burq-Lebeau (arXiv:1111.7310). The authors have uploaded a new version of the withdrawn paper, "Fixed Frequency Eigenfunction Immersions and Supremum Norms of Random Waves," which acknowledges the preceding work and gives a significantly simper argument that is more geometric and substantively different from the technique of Burq-Lebeau