Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps (2304.02678v3)

Abstract: This two-part paper studies the Weil-Petersson measure on the moduli space of compact hyperbolic surfaces of genus $g$. In this first part, we define "volume functions" $V_g^{T}(l)$ associated with arbitrary topological types $T$ of closed geodesics, generalising the "volume polynomials" studied by M. Mirzakhani for simple closed geodesics. Our programme is to study the structure of these functions, focusing on their behaviour as a function of $l$ in the limit $g\rightarrow +\infty$. In this first article, motivated by J. Friedman's work on random graphs, we prove that volume functions admit asymptotic expansions to any order in powers of $1/g$, and claim that the coefficients in these expansions belong to a newly-introduced class of functions called "Friedman-Ramanujan functions". We prove the claim for closed geodesics filling a surface of Euler characteristic $0$ and $-1$. This result is then applied to prove that a random hyperbolic surface has spectral gap $\geq 2/9-\epsilon$ with high probability as $g \rightarrow +\infty$, using the trace method and cancellation properties of Friedman--Ramanujan functions.