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

Abstract: 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 $λ_1$ of the Laplacian. In this second part, we provide a variety of new tools allowing to compute and estimate the volume functions $V_g^{T}(l)$. Notably, we construct new sets of coordinates on Teichmüller spaces, distinct from Fenchel-Nielsen coordinates, in which the Weil-Petersson volume has a simple form. These coordinates are tailored to the geodesics we study, and we can therefore prove nice formulae for their lengths. We use these new ideas, together with a notion of pseudo-convolutions, to prove that the coefficients of the expansion of $V_g^{T}(l)$ in powers of $1/g$ are Friedman-Ramanujan functions, for any local topological type $T$. We then exploit this result to prove that, for any $ε>0$, $λ_1 \geq \frac14 - ε$ with probability going to one as $g \rightarrow + \infty$, or, in other words, typical hyperbolic surfaces have an asymptotically optimal spectral gap.

• The paper establishes Friedman–Ramanujan functions as key descriptors for the asymptotic behavior of Weil–Petersson volumes on high-genus hyperbolic surfaces.
• It introduces new Teichmüller coordinate systems and a diagrammatic decomposition to effectively analyze geodesic loops with complex self-intersections.
• The study demonstrates stability of pseudo-convolution structures under a differentiation-like operator, advancing the analysis of spectral gaps.

Summary

The paper “Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II” (2502.12268) develops a comprehensive framework for analyzing the asymptotic behavior of Weil–Petersson volume functions associated with closed geodesic length spectra on random hyperbolic surfaces. Building on techniques introduced in the first part, the authors introduce a new class of functions—termed Friedman–Ramanujan functions—that arise naturally in the asymptotic expansion of volume functions when the genus tends to infinity. In addition, the paper develops new coordinate systems on Teichmüller spaces, which are distinct from the classical Fenchel–Nielsen coordinates, and are adapted to the study of geodesic loops with complex self-intersections. Through a careful diagrammatic decomposition of multi-loops (including the analysis of so-called generalized eights), the authors are able to derive pseudo-convolution structures that capture the interaction of the lengths attributed to bridges and simple portions within these diagrams. By establishing stability properties of these pseudo-convolutions under an operator analogous to the differentiation operator ℒ, the paper proves that the asymptotic coefficients of the expansion satisfy the Friedman–Ramanujan property in a weak sense.

New Coordinate Systems and Diagram Decomposition

A central technical innovation is the introduction of alternative coordinate systems on the moduli space of hyperbolic structures, which are defined via a detailed decomposition of a given multi-loop into “diagrams.” In these diagrams the originally intersecting geodesic representatives are “opened” through a choice of surgery (either the first or second kind), leading to a description in terms of a simple (or nearly simple) multi-loop β together with a collection of “bars” whose endpoints subdivide β into segments. The authors define a natural labelling and sign convention on the set Θ (comprising pairs (bar index, ±)) and further develop notions such as unshielded simple portions and crossing indices. This decomposition enables the authors to express the length of a closed geodesic in terms of a cyclic product involving lengths of bars and the “cell” lengths (denoted by τ). An explicit formula is derived via matrix representations in PSL(2, ℝ), where the product of matrices of the form aᵗ = diag(e^t/2, e^–t/2) and wᵗ = [cosh t sinh t; sinh t cosh t] can be manipulated to yield a combinatorial expression ℓ(γ) = Qₘ(𝑳,𝒕)