Eigenvalue rigidity of hyperbolic surfaces in the random cover model

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. We also obtain a polynomial improvement on the $L^{\infty}$ bound of the eigenfunctions. Our proof relies on the Selberg trace formula and a variant of the polynomial method.