arXiv2025
The core focus of this series of two articles is the study of the distribution of the length spectrum of closed hyperbolic surfaces of genus $g$, sampled randomly with respect to the Weil-Petersson probability measure. In the first article, we introduced a notion of local topological type $T$, and established the existence of a density function $V_g^T(l)$ describing the distribution of the lengths of all closed geodesics of type $T$ in a genus $g$ hyperbolic surface. We proved that $V_g^{T}(l)$ admits an asymptotic expansion in powers of $1/g$. We introduced a new class of functions, called Friedman-Ramanujan functions, and related it to the study of the spectral gap $λ_1$ of the Laplacian. In this second part, we provide a variety of new tools allowing to compute and estimate the volume functions $V_g^{T}(l)$. Notably, we construct new sets of coordinates on Teichmüller spaces, distinct from Fenchel-Nielsen coordinates, in which the Weil-Petersson volume has a simple form. These coordinates are tailored to the geodesics we study, and we can therefore prove nice formulae for their lengths. We use these new ideas, together with a notion of pseudo-convolutions, to prove that the coefficients of the expansion of $V_g^{T}(l)$ in powers of $1/g$ are Friedman-Ramanujan functions, for any local topological type $T$. We then exploit this result to prove that, for any $ε>0$, $λ_1 \geq \frac14 - ε$ with probability going to one as $g \rightarrow + \infty$, or, in other words, typical hyperbolic surfaces have an asymptotically optimal spectral gap.