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

Understand

In this series of articles, we analyse the level-sets of length functions on the moduli space of compact hyperbolic surfaces of fixed genus.

• This work ultimately culminates in a proof that typical hyperbolic surfaces have an optimal spectral gap.
• In this first article, we introduce new volume functions $V_g^T(l)$, counting the expected number of closed geodesics of type $T$ and length $l$ on a random hyperbolic surface of genus $g$.
• So far, this function has only been considered in the case where the type is simple, in which case it can be expressed as a combination of Weil-Petersson volumes polynomials, as proven by Mirzakhani.