Let $X$ be a compact 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. We also obtain an improved $L^{\infty}$ bound on the eigenfunctions. Our proof relies on the Selberg (pre-)trace formula and a variant of the polynomial method.
This is joint work with Zhongkai Tao.
