Effective lower bounds for spectra of random covers and random unitary bundles

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 $n\to \mathrm{\infty }$, 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 $n\to \mathrm{\infty }$, a random unitary bundle $E_{\phi }$ over $X$ of rank $n$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log \log n{)}^2}{\log n}$.