  Let $X$ be a compact connected hyperbolic surface, that is, a closed
connected orientable smooth surface with a Riemannian metric of constant
curvature -1. For each $n\in\mathbf{N}$, let $X_{n}$ be a random degree-$n$
cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces
of $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associated
Laplacian operator $\Delta_{X}$ or $\Delta_{X_{n}}$. We say that an eigenvalue
of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove
that for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$,
there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\varepsilon$. We
conjecture that the same result holds with $\frac{3}{16}$ replaced by
$\frac{1}{4}$.
