  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 $f_{\lambda}$ chosen uniformly at random from the
space of $L^2$-normalized linear combinations of Laplace eigenfunctions with
eigenvalues in the interval $(\lambda^2, \lr{\lambda+1}^2].$ Our main result is
that the expected value of $\norm{f_\lambda}_\infty$ grows at most like $C
\sqrt{\log \lambda}$ as $\lambda \to \infty$, 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 $(M,g).$
