  It has been known since the work of Avakumov\'ic, H\"ormander and Levitan
that, on any compact smooth Riemannian manifold, if $-\Delta_g \psi_\lambda =
\lambda \psi_\lambda$, then $\|\psi_\lambda\|_{L^\infty} \leq C
\lambda^{\frac{d-1}{4}} \|\psi_\lambda\|_{L^2}$. It is believed that, on
manifolds of negative curvature, such a bound can be largely improved; however,
only logarithmic improvements in $\lambda$ have been obtained so far. In the
present paper, we obtain polynomial improvements over the previous bound in a
generic setting, by adding a small random pseudodifferential perturbation to
the Laplace-Beltrami operator.
