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

TL;DR

This paper introduces new volume functions for counting geodesics on random hyperbolic surfaces, proves their asymptotic expansion in genus, and relates these functions to spectral gaps, revealing new mathematical structures called Friedman-Ramanujan functions.

Contribution

It develops an integral expression for volume functions of geodesics of any type and shows their coefficients form Friedman-Ramanujan functions, advancing understanding of spectral gaps in hyperbolic geometry.

Findings

Established asymptotic expansion of volume functions in powers of 1/g

Identified coefficients as Friedman-Ramanujan functions

Linked these functions to spectral gap analysis

Abstract

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. We provide an integral expression for $V_g^T(l)$ for any prescribed type $T$, which we use to prove that $V_g^T(l)$ admits a full asymptotic expansion in powers of $1/g$. We then claim that the coefficients in this expansion, as a function of the length variable $l$, belong to a newly-introduced class of functions called "Friedman-Ramanujan functions". We relate this claim to the study of the spectral gap of the Laplace-Beltrami operator, and prove it when $T$ fills a surface of Euler characteristic $0$ or $-1$, providing a method to explicitly compute all coefficients in the second-order expansion. We conclude by displaying how the presence of tangles (which is an event of vanishing probability) prevents the sum over all types to satisfy the Friedman-Ramanujan property at the second order.