Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II

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 $\lambda_1$ 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 Teichm\"uller 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 $\epsilon>0$, $\lambda_1 \geq \frac14 - \epsilon$ with probability going to one as $g \rightarrow + \infty$, or, in other words, typical hyperbolic surfaces have an asymptotically optimal spectral gap.