Polynomial bounds for eigenfunctions and eigenvalues on random covers of hyperbolic surfaces

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ cover, taken uniformly at random. We show that, with high probability, the $L^\infty$ norm of every Laplace eigenfunction on $X_n$ with bounded eigenvalue decays polynomially in $n$. This gives a polynomial de