Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus

We investigate the spectral distribution of the twisted Laplacian associated with uniform square-integrable bounded harmonic 1-form on typical hyperbolic surfaces of high genus. First, we estimate the spectral distribution by the supremum norm of the corresponding harmonic form. Subsequently, we show that the square-integrable bounded harmonic form exhibits a small supremum norm for typical hyperbolic surfaces of high genus. Based on these findings, we prove a uniform Weyl law for the distribution of real parts of the spectrum on typical hyperbolic surfaces.