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

Understand

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.