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Length spectrum of large genus random metric maps

Simon Barazer, Alessandro Giacchetto, Mingkun Liu

TL;DR

This work analyzes the length spectrum of short cycles on uniform random metric ribbon graphs in the large-genus limit. Leveraging a Teichmüller-theory framework combined with Aggarwal's large-genus intersection-number asymptotics, the authors prove that the cycle lengths converge in distribution to a Poisson point process with explicit intensity $\lambda_{\mu}(\ell)=(\cosh(\ell/\mu)-1)/\ell$ under the boundary-scaling $|L|\sim \mu\,12\,g$, extending previous results from unicellular maps to multi-faced cases. The girth is shown to follow a non-homogeneous exponential distribution with rate $\lambda_{\mu}$, and the analysis reveals precise asymptotics for Kontsevich volumes via saddle-point methods and integration formulas. The paper also discusses fundamental limitations for closed curves and provides numerical evidence supporting the PPP description, highlighting the deep link between combinatorial moduli, intersection theory, and random geometry.

Abstract

We study the length of short cycles on uniformly random metric maps (also known as ribbon graphs) of large genus using a Teichmüller theory approach. We establish that, as the genus tends to infinity, the length spectrum converges to a Poisson point process with an explicit intensity. This result extends the work of Janson and Louf to the multi-faced case.

Length spectrum of large genus random metric maps

TL;DR

This work analyzes the length spectrum of short cycles on uniform random metric ribbon graphs in the large-genus limit. Leveraging a Teichmüller-theory framework combined with Aggarwal's large-genus intersection-number asymptotics, the authors prove that the cycle lengths converge in distribution to a Poisson point process with explicit intensity under the boundary-scaling , extending previous results from unicellular maps to multi-faced cases. The girth is shown to follow a non-homogeneous exponential distribution with rate , and the analysis reveals precise asymptotics for Kontsevich volumes via saddle-point methods and integration formulas. The paper also discusses fundamental limitations for closed curves and provides numerical evidence supporting the PPP description, highlighting the deep link between combinatorial moduli, intersection theory, and random geometry.

Abstract

We study the length of short cycles on uniformly random metric maps (also known as ribbon graphs) of large genus using a Teichmüller theory approach. We establish that, as the genus tends to infinity, the length spectrum converges to a Poisson point process with an explicit intensity. This result extends the work of Janson and Louf to the multi-faced case.
Paper Structure (19 sections, 21 theorems, 89 equations, 9 figures)

This paper contains 19 sections, 21 theorems, 89 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The results
  3. Related works and proof strategy
  4. Outlook
  5. Background
  6. Combinatorial Teichmüller and moduli spaces
  7. The symplectic structure, the length function, and the integration formula
  8. Connection with intersection numbers and Aggarwal's asymptotic formula
  9. Random metric ribbon graphs and the method of moments
  10. The combinatorial length spectrum
  11. Setup
  12. Volume estimates
  13. Simple, distinct, non-separating closed curves
  14. The negligible terms
  15. Proof of the main estimate
  16. ...and 4 more sections

Key Result

Theorem A

For any fixed $n$, the random multiset $\Lambda(\bm{G}_{g,L})$, viewed as a point process on $\mathbb{R}_{\geq 0}$, converges in distribution as $g \to \infty$ to a Poisson point process of intensity $\lambda$ defined by

Figures (9)

  • Figure 1: A ribbon graph of genus $0$ with $3$ faces (left) and a ribbon graph of genus $1$ with $1$ face (right).
  • Figure 2: In blue, the cycle length statistics of random unicellular metric maps of genus $g = 2$, $8$, and $64$, sampled over $10^3$ units and properly rescaled. The predicted intensity $\lambda$ is depicted in lime.
  • Figure 3: The combinatorial Teichmüller space of a one-holed torus (left), and the corresponding moduli space.
  • Figure 4: The stable graph corresponding to an ordered tuple of curves.
  • Figure 5: Example (left) and non-example (right) of a cycle on an embedded metric ribbon graph of type $(0,4)$.
  • ...and 4 more figures

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem D
  • Theorem 2.1: Combinatorial Wolpert's formula ABCGLW
  • Theorem 2.2: Integration formula ABCGLW
  • Theorem 2.3: Symplectic volumes as intersection numbers Kon92
  • Theorem 2.4: Large genus asymptotic of intersection numbers Agg21
  • Theorem 2.5: Uniform bound on intersection numbers Agg21
  • Theorem 2.6: Method of moments
  • ...and 24 more