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Let $X$ be a compact connected hyperbolic surface, that is, a closedconnected orientable smooth surface with a Riemannian metric of constantcurvature -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 spacesof $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associatedLaplacian operator $\Delta_{X}$ or $\Delta_{X_{n}}$. We say that an eigenvalueof $X_n$ is new if it occurs with greater multiplicity than in $X$. We provethat for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$,there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\varepsilon$. Weconjecture that the same result holds with $\frac{3}{16}$ replaced by$\frac{1}{4}$.

A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\varepsilon$