Separating Systole in High Genus Triangulations

• The paper shows that the separating systole in high-genus triangulations grows logarithmically with the number of faces, establishing tight bounds through probabilistic and combinatorial arguments.
• It employs a decomposition of SNCCs into canonical patterns and uses union-bound arguments alongside precise enumerative estimates to derive the logarithmic bounds.
• The study introduces a new genus ratio limit and highlights analogies with hyperbolic surfaces, advancing our understanding of random maps and asymptotic surface theory.

The separating systole of high genus triangulations concerns the asymptotic behavior, enumeration, and combinatorics of shortest separating non-contractible cycles (SNCCs) in random triangulations of large genus. These cycles are central to understanding the global geometric and topological properties of random maps embedded in surfaces and have far-reaching implications for probabilistic, enumerative, and geometric models of random high-genus surfaces.

1. Background and Definitions

A map $m$ is a finite, connected multigraph (with loops and multiple edges allowed) embedded without overlap in a compact oriented surface $\Sigma$, up to homeomorphism. The genus $g(m)$ of $m$ is that of $\Sigma$. A triangulation is a rooted map all of whose faces are triangles (degree 3). For $2n$ faces and genus $g$, denote the set of such triangulations as

$T(2n, g),\quad \tau(n, g) = |T(2n,g)|.$

A cycle in a map is a closed walk; it is simple if no vertex is revisited except at the start/end. A cycle is contractible (null-homotopic) if it can be continuously shrunk to a point. It is separating (null-homologous) if its complement comprises two regions of positive genus, equivalently if its class in the homology group vanishes. A separating non-contractible cycle (SNCC) is a separating cycle not null-homotopic. The separating systole of a triangulation $T$ is

$$\mathrm{sepsys}(T) = \min\{ \ell(\gamma) \mid \gamma \subset T \text{ is a SNCC} \}.$$

2. Logarithmic Growth of the Separating Systole

The primary enumerative and probabilistic result is that for a uniform triangulation $\mathbf{T}$ of genus $g_n$ with $2n$ faces and $g_n/n \rightarrow \theta \in (0, \frac{1}{2})$, the separating systole is logarithmic in $n$ with high probability (whp). Specifically, there exist constants $0 < A(\theta) < B(\theta) < \infty$ such that

$$A\log n \leq \mathrm{sepsys}(\mathbf{T}) \leq B\log n$$

with probability tending to 1 as $n \to \infty$. This establishes $\mathrm{sepsys}(\mathbf{T}) = \Theta(\log n)$ whp (Louf, 4 Dec 2025).

• The upper bound is a consequence of a deterministic result of Costantini–Viviani–Hutchinson–Mazzola, which gives an $O(\log n)$ bound in the regime $g \asymp \theta n$.
• The lower bound is established by a union-bound argument: For $L = o(\log n)$, the expected number of SNCCs of length $\leq L$ vanishes as $n \to \infty$.

This demonstrates that, in contrast to the systole of certain compact hyperbolic surfaces, random triangulations generally do not admit separating cycles of length smaller than logarithmic order as the genus grows proportionally with the size.

3. Combinatorial and Enumerative Techniques

The lower bound is established by a decomposition of SNCCs into three canonical combinatorial patterns:

• Simple separating cycles: A single simple non-contractible separating cycle.
• Thin eights: Two non-separating cycles joined by a path.
• Fat eights: A single non-separating cycle with an attaching path forming a figure-eight.

For each pattern of length $\ell$, one marks the corresponding structure in the triangulation, cuts along it, and uses precise enumerative results about triangulations with boundaries to bound the number of such marked maps. The heart of the approach leverages asymptotic enumerative estimates for triangulations in the high-genus regime: $$\tau(n, g) = n^{2g} \exp\left( n f\left( \frac{g}{n} \right) + o(n) \right)$$ with $$\frac{\tau(n-1,g)}{\tau(n,g)} = \lambda(g/n) + o(1)$$, where $f$ is a concave function and $\lambda$ is strictly decreasing (Louf, 4 Dec 2025).

Key technical lemmas include:

• Filling boundaries: The number of triangulations with boundary cycles of total perimeter $p$ and $m$ interior faces is bounded by: $$|T_{p_1, \dots, p_k}(m, g)| \leq (3m+3p)^{k-1} \, \tau\left(\frac{m+p}{2}, g\right)$$
• General ratio bounds: For $g_n/n \ge \theta$,

$$\frac{\tau(n,g_n)}{\tau(n-m, g_n-h)} \geq a_\theta^h \left( \frac{n}{n-m} \right)^{2(g_n-h) - 2g_n} \exp\left(n f(g_n/n) - (n-m) f((g_n-h)/(n-m)) + o(m)\right)$$

These results allow for the first-moment method to show scarcity of short SNCCs.

4. The Genus Ratio and High-Genus Asymptotics

Beyond establishing the growth of the separating systole, the analysis produces a new enumerative limit: $$\frac{n^2\, \tau(n, g_n-1)}{\tau(n, g_n)} \rightarrow \psi(\theta)$$ for an explicit analytic function $\psi(\theta)$ as $n \to \infty$, with $g_n/n \to \theta$ in $(0, \frac{1}{2})$. This "genus ratio" limit complements the previously known "size ratio" results and is obtained by isolating the dominant asymptotic terms in the Goulden–Jackson recurrence for face-and-genus enumeration (Louf, 4 Dec 2025).

The Goulden–Jackson recurrence provides the fundamental recursive structure for $\tau(n,g)$, allowing extraction of these asymptotic growth rates and ratio limits.

5. Broader Context: Random Surfaces and Metric Invariants

The logarithmic scaling of the separating systole aligns with broader phenomena in both random map and hyperbolic geometry frameworks:

• In random hyperbolic surfaces (e.g., in the Weil–Petersson model), shortest non-contractible geodesics also exhibit logarithmic growth in genus.
• The separating systole, as a global metric invariant, complements other investigations of geometric properties such as diameter, injectivity radius, and general systolic geometry for random maps and surfaces.

These results emphasize that random large-genus combinatorial maps display metric properties similar to those of classical random hyperbolic surfaces, reinforcing deep analogies between these models. The genus-ratio limit also provides a foundational easing for probabilistic and asymptotic enumeration analyses in high-genus regimes.

6. Connections to Hyperbolic Surfaces and Algorithmic Aspects

For explicit Riemann surfaces with maximal symmetry—such as the generalized Bolza surfaces $\mathbb{M}_g$ constructed as $\mathbb{H}^2/\Gamma_g$, where $\Gamma_g$ is generated by isometries acting on the Poincaré disk—a closed-form formula for the systole exists: $\mathrm{sys}(\mathbb{M}_g) = 2 \arcosh(1 + 2 \cos(\pi/(2g)))$ (Ebbens et al., 2021). This geometric systole is distinct from the combinatorial setting of random triangulations, yet the interplay between hyperbolic geometry, combinatorial topology, and efficient algorithmic detection of systolic cycles informs the paper of separating systoles in both random and explicit surface settings.

Algorithmic approaches based on Delaunay triangulations, as described for generalized Bolza surfaces, allow efficient identification of systolic loops by maintaining triangulation graphs and tracking the homotopy of shortest cycles. Although these approaches target the geometric systole, the conceptual framework underscores the combinatorial-geometric nature of SNCCs and their detection in large genus settings.

7. Implications and Future Perspectives

The convergence results for the separating systole and genus ratio for high-genus triangulations provide new tools for enumerative and probabilistic surface theory. These findings clarify the typical landscape of global invariants for random maps, with strong connections to random geometry and moduli theory.

The methodology—combining combinatorial reductions, precise asymptotic enumeration, and structural decomposition of separating cycles—is anticipated to be broadly applicable to the paper of random maps, possibly extending to other types of embedded graphs and surfaces, or to higher-dimensional analogues. The analogies with random hyperbolic surfaces suggest promising directions for the synthesis of geometric and combinatorial random surface theories (Louf, 4 Dec 2025, Ebbens et al., 2021).