The tight length spectrum of large-genus random hyperbolic surfaces with many cusps

TL;DR

The paper analyzes the primitive tight length spectrum of random hyperbolic surfaces of large genus with many cusps, proving a Poisson point process limit under a regime where the cusp count $n_g$ grows faster than a cubic in the genus. The authors extend Mirzakhani–Petri results to the tight geometry setting by employing a tight Weil–Petersson recursion due to Budd–Zonneveld and a tight-version Mirzakhani integration formula, enabling precise volume and intersection-number estimates. In the high-genus, many-cusps regime, after suitable normalization—scaling by $(oldmu_c-oldmu_g)^{1/4}$ or by $(n_g/g)^{1/4}$—the tight length spectrum converges to the same Poisson intensity $(rac{ ext{cosh}(t)-1}{t})dt$, with explicit constants controlling the scale. The work also yields corollaries about tight systoles and conjectures a Brownian-surface analogue for large genus, highlighting a deep link between random hyperbolic geometry with cusps, tight-geodesic combinatorics, and universal Poisson statistics in length spectra.

Abstract

Since the work of Mirzakhani and Petri on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number of which grows with the genus. We prove that if the number of cusps grows fast enough and we restrict attention to special geodesics that are tight, we recover upon proper normalization the same Poisson point process in the large genus limit for the length statistics. The proof relies on a recursion formula for tight Weil-Petersson volumes obtained recently by Budd and Zonneveld and on a generalization of Mirzakhani's integration formula to the tight setting.

Figures (9)

• Figure 1: A hyperbolic surface of genus $8$ with many cusps. The blue curves are the $4$ shortest tight geodesics on the surface.
• Figure 2: On the left a surface chosen under $\mathbb{P}^{\mathrm{WP}}_{8}$ and on the right a surface chosen under $\mathbb{P}^{\mathrm{WP}}_{8,74}$. On the right, the two geodesics $\gamma_1$ and $\gamma_2$ are very close in length, but not homotopic due to the cusps that separate them. The presence of a large number of cusps entails a fractal structure for the right surface which is not the case on the left.
• Figure 3: On the left the surface $\Sigma_{g,n}$. On the right, two examples of markings which give the same hyperbolic surface $X$, however this gives two different points in $\mathcal{T}_{g,n}(\mathbf{L})$. Indeed to obtain the bottom marking we have first applied a full twist $\tau$ along the geodesic $\gamma_1$ in $\Sigma_{g,n}$.
• Figure 4: We take $g=3, n=2, p=1$ and we consider a marking $(X,\phi)$. The geodesic $\phi(\gamma_1)$ is not tight since $\phi(\gamma_2)$ has a smaller length and $\gamma_1$ and $\gamma_2$ are homotopic in $\Sigma_{3,2,1}$.
• Figure 5: In this example, with $n=0$, the curves $\gamma_3$ and $\gamma_4$ are part of the same extended homotopy class, thus none of them is tight. The curve $\gamma_2$ is the tight representative of $\gamma_1$.

Theorems & Definitions (49)

• Theorem 1.1: Theorem 4.1 of Mirzakhani_petri_2019
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 1.5
• Conjecture 1.7
• Theorem 2.1: budd2023topological, Theorem $4$
• Proposition 3.1
• proof
• Proposition 3.2