Short geodesics and small eigenvalues on random hyperbolic punctured spheres

2022/09/30 by Will Hide, Joe Thomas, Hide, Will +1 · 2 citations
Mathematics · #32G15 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2209.15568

openalex publication_date 2022/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with n cusps in the regime n→∞. Inspired by work of Mirzakhani and Petri \citeMi.Pe19, we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with n cusps has at least k=o(n) arbitrarily small eigenvalues tends to 1 as n→∞.

Cited by

• A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries
• Random graphs, expanding families and the construction of noncompact hyperbolic surfaces with uniform spectral gaps

Related

• The tight length spectrum of large-genus random hyperbolic surfaces with many cusps
• Large-n asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces
• Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps
• Asymptotics of shortest filling closed multi-geodesics
• Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus