  Let $X$ be a finite-area non-compact hyperbolic surface. We study the
spectrum of the Laplacian on random covering surfaces of X and on random
unitary bundles over X. We show that there is a constant $c > 0$ such that,
with probability tending to 1 as $n \to \infty$, a uniformly random degree-$n$
Riemannian covering surface $X_n$ of $X$ has no Laplacian eigenvalues below
$\frac{1}{4}-c\frac{(\log\log\log n)^2}{\log \log n}$ other than those of $X$
and with the same multiplicities. We also show that with probability tending to
1 as $n\to \infty$, a random unitary bundle $E_{\phi}$ over $X$ of rank $n$ has
no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log n)^2}{\log n}$.
