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This paper studies the Weil-Petersson measure on the moduli space of compacthyperbolic surfaces of genus $g$. We define "volume functions" associated witharbitrary topological types of closed geodesics, generalising the "volumepolynomials" studied by M. Mirzakhani for simple closed geodesics. Ourprogramme is to study the structure of these functions, focusing on theirbehaviour in the limit $g \to +\infty$. Motivated by J. Friedman's work onrandom graphs, we prove that volume functions admit asymptotic expansions toany order in powers of $1/g$, and claim that the coefficients in theseexpansions belong to a newly-introduced class of functions called"Friedman-Ramanujan functions". We prove the claim for closed geodesics fillinga surface of Euler characteristic 0 and -1. This result is then applied toprove that a random hyperbolic surface has spectral gap $> 2/9 - \epsilon$ withhigh probability as $g \to +\infty$, using the trace method and cancellationproperties of Friedman-Ramanujan functions.

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