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Short geodesics and small eigenvalues on random hyperbolic punctured spheres

Will Hide, Joe Thomas

TL;DR

The paper proves that for random genus-zero hyperbolic punctured spheres with $n$ cusps under the Weil-Petersson measure, the lengths of short closed geodesics, rescaled by $\sqrt{n}$, converge to a Poisson point process with an explicit intensity constant $\alpha$ tied to Bessel function zeros. This Poisson statistics framework extends to counts of short geodesics that separate off a fixed number of cusps, yielding independent Poisson limits with computable means; these results hinge on Mirzakhani’s integration formula and precise volume asymptotics. A key geometric consequence is that the systole is typically of order $n^{-1/2}$, with topological-type constraints implying that, with positive probability, the systole can separate off at least $k$ cusps for any fixed $k$, and analytic consequences show that the random surface carries many small eigenvalues: for any $k=o(n)$, with high probability $\lambda_{k}(X)$ is below a constant multiple of $\sqrt{k/n}$. Collectively, the work connects geometric/topological data (short geodesics and cusp-separating curves) with spectral properties, advancing understanding of random hyperbolic surfaces in the cusp-dominated regime and offering explicit constants via volume asymptotics and Bessel-function identities.

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 $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.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 $n\to\infty$.

Short geodesics and small eigenvalues on random hyperbolic punctured spheres

TL;DR

The paper proves that for random genus-zero hyperbolic punctured spheres with cusps under the Weil-Petersson measure, the lengths of short closed geodesics, rescaled by , converge to a Poisson point process with an explicit intensity constant tied to Bessel function zeros. This Poisson statistics framework extends to counts of short geodesics that separate off a fixed number of cusps, yielding independent Poisson limits with computable means; these results hinge on Mirzakhani’s integration formula and precise volume asymptotics. A key geometric consequence is that the systole is typically of order , with topological-type constraints implying that, with positive probability, the systole can separate off at least cusps for any fixed , and analytic consequences show that the random surface carries many small eigenvalues: for any , with high probability is below a constant multiple of . Collectively, the work connects geometric/topological data (short geodesics and cusp-separating curves) with spectral properties, advancing understanding of random hyperbolic surfaces in the cusp-dominated regime and offering explicit constants via volume asymptotics and Bessel-function identities.

Abstract

We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with cusps in the regime . Inspired by work of Mirzakhani and Petri \cite{Mi.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 cusps has at least arbitrarily small eigenvalues tends to as .
Paper Structure (15 sections, 23 theorems, 152 equations)

This paper contains 15 sections, 23 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Overview of main results
  3. Relations to existing work
  4. Background
  5. Moduli space
  6. Weil-Petersson metric
  7. Mirzakhani's integration formula
  8. Volume bounds
  9. Poisson Statistics
  10. Contribution of nested multicurves
  11. Contribution of unnested curves
  12. Proof of Theorem \ref{['thm:main-thm']}
  13. Proof of Theorem \ref{['thm:main-thm-2']}
  14. Systole
  15. Small eigenvalues

Key Result

Theorem 1.1

Let $\ell\in\mathbb{N}$ and suppose that $0\leq a_{i}<b_{i}$ are real numbers for $i=1,\ldots,\ell$ such that the intervals $[a_{i},b_{i}]$ are pairwise disjoint. Then, the sequence of random vectors converges in distribution as $n\to\infty$ to a vector of independent Poisson distributed random variables with means where $J_{\alpha}$ are Bessel functions of the first kind and $j_{0}$ is the firs

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 40 more