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The separating systole and the genus ratio of high genus triangulations

Baptiste Louf

TL;DR

This work analyzes random triangulations of increasing size and genus, proving the separating systole grows only logarithmically with the triangulation size and establishing a convergence result for the genus ratio, complementing prior size-ratio findings. It develops a robust enumerative framework based on the Goulden–Jackson recurrence, boundary decompositions, and first-moment bounds, then couples these with geometric constructions—such as nice cycles and fat/thin eights—to translate global topological constraints into tractable counting problems. The main contributions are a precise asymptotic bound for sepsys and an explicit form for the genus-ratio limit function ψ(θ) in terms of λ(θ) and a generating-series correction, together with rigorous bounds ruling out short simple, fat, and thin eights whp. The paper concludes with conjectures on simple SNCC existence, sepsys concentration, and deeper connections to high-genus geometric models, guiding future research in random maps and enumerative topology.

Abstract

We show that the separating systole of high genus triangulations is of logarithmic order (in the size of the triangulation). Our methods also allow us to show an enumerative result, i.e. the convergence of the "genus ratio" for high genus triangulations. This complements the convergence of the "size ratio" that was proven in previous work with Budzinski.

The separating systole and the genus ratio of high genus triangulations

TL;DR

This work analyzes random triangulations of increasing size and genus, proving the separating systole grows only logarithmically with the triangulation size and establishing a convergence result for the genus ratio, complementing prior size-ratio findings. It develops a robust enumerative framework based on the Goulden–Jackson recurrence, boundary decompositions, and first-moment bounds, then couples these with geometric constructions—such as nice cycles and fat/thin eights—to translate global topological constraints into tractable counting problems. The main contributions are a precise asymptotic bound for sepsys and an explicit form for the genus-ratio limit function ψ(θ) in terms of λ(θ) and a generating-series correction, together with rigorous bounds ruling out short simple, fat, and thin eights whp. The paper concludes with conjectures on simple SNCC existence, sepsys concentration, and deeper connections to high-genus geometric models, guiding future research in random maps and enumerative topology.

Abstract

We show that the separating systole of high genus triangulations is of logarithmic order (in the size of the triangulation). Our methods also allow us to show an enumerative result, i.e. the convergence of the "genus ratio" for high genus triangulations. This complements the convergence of the "size ratio" that was proven in previous work with Budzinski.

Paper Structure

This paper contains 24 sections, 21 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Definitions and notations
  4. Maps
  5. Paths and cycles
  6. Homotopy and homology
  7. Preliminaries: enumeration
  8. Previous results
  9. Additional results
  10. Case 1:
  11. Case 2:
  12. Case 3:
  13. Case 4:
  14. The genus ratio: proof of \ref{['thm_ratio']}
  15. Preliminaries: geometry
  16. ...and 9 more sections

Key Result

Theorem 1

With high probabilitythat is, with probability tending to $1$ as $n \rightarrow \infty$, abbreviated later as whp., the separating systole of $\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\theta$) such that whp

Figures (5)

  • Figure 1: Left: a fat eight, right: a thin eight
  • Figure 2: The paths and cycles introduced in \ref{['lem_te']}
  • Figure 3: The paths introduced in \ref{['lem_fe']}
  • Figure 4: Cutting along a fat eight.
  • Figure 5: Cutting along a thin eight, the two cases.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Goulden--Jackson formula
  • Lemma 4
  • Theorem 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 26 more