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arxiv: 2605.03833 · v1 · submitted 2026-05-05 · 🧮 math.GT · math.CO

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Large genus asymptotics for frequency of non-simple curves

Mingkun Liu , Kasra Rafi , Juan Souto , Marie Trin
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Pith reviewed 2026-05-07 13:20 UTC · model grok-4.3

classification 🧮 math.GT math.CO
keywords non-simple curveslarge genus asymptoticscurve frequenciesKontsevich polynomialsclosed surfacesself-intersectionsMirzakhani formulasDelecroix-Goujard-Zograf-Zorich
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The pith

Non-simple curves on closed surfaces have frequencies given by extended Kontsevich polynomials, with fixed-K intersection types showing stable dominance at large genus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an explicit formula for the frequency with which non-simple curves appear among all curves on a closed surface. It adapts Mirzakhani's approach, expressing those frequencies through Kontsevich polynomials, and combines this with known large-genus asymptotic results to compare different non-simple types. For any fixed number K of self-intersections, the work identifies which geometric configurations occur most often as the genus grows without bound. A reader would care because the result describes the typical shape of curves one encounters when picking a random curve on a high-genus surface. The formulas therefore give a precise handle on how complexity and simplicity compete in the space of all curves.

Core claim

We give an expression for the frequency of non-simple curves in closed surfaces and exploit it to study relative frequencies of such curves in large genus. This extends to the case of non-simple curves Mirzakhani's expressions of frequencies in terms of Kontsevich polynomials and Delecroix-Goujard-Zograf-Zorich large genus asymptotics for those frequencies. In particular, with K fixed, we identify which types of curves with K intersections are most common.

What carries the argument

Extension of Mirzakhani's frequency expressions via Kontsevich polynomials to non-simple curves, paired with Delecroix-Goujard-Zograf-Zorich large-genus asymptotics.

If this is right

  • Frequencies of non-simple curves admit explicit expressions in terms of Kontsevich polynomials.
  • For any fixed K the relative frequencies of different K-intersection types approach definite limits as genus tends to infinity.
  • The most common K-intersection type can be read off directly from the asymptotic formulas.
  • The same large-genus limits govern the competition between simple and non-simple curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formulas could be used to sample typical curves from the moduli space when genus is large.
  • Similar extensions might apply to curves on surfaces with punctures once the corresponding volume asymptotics are available.
  • The dominance of particular intersection patterns suggests a way to classify the 'generic' self-intersecting curve in high-genus geometry.

Load-bearing premise

Mirzakhani's volume and polynomial formulas together with the large-genus asymptotics apply to non-simple curves without new obstructions or structural changes.

What would settle it

Exact computation of the proportion of non-simple curves with a fixed K on a sequence of surfaces with increasing genus, checking whether the observed ratios converge to the predicted asymptotic values.

Figures

Figures reproduced from arXiv: 2605.03833 by Juan Souto, Kasra Rafi, Marie Trin, Mingkun Liu.

Figure 1
Figure 1. Figure 1: Examples of local types. An expression for the frequency. For simple curves γ0, Mirzakhani expressed in [21] the frequency cg(γ0) in (1.1) in terms of the Kontsevich polynomial VX\γ0 (b1, b2) of the view at source ↗
Figure 2
Figure 2. Figure 2: Example of train-track on a surface Train-tracks are a very important tool to study measured laminations. A measured lam￾ination is carried by a train track τ if its support can be smoothly isotoped to τ . Let us briefly recall the structure of the set ML(τ ) of all measured laminations carried by τ . First note that we can decompose the set of half-edges of τ adjacent to a vertex v into two disjoint sets … view at source ↗
Figure 3
Figure 3. Figure 3: Getting a measured lamination from a solution of the switch equa￾tions. Denote by Φτ (w) ∈ ML(X) the measured lamination associated to w ∈ W(τ ) and note that, very much by construction, Φτ (w) is carried by τ . Indeed, the map Φτ : W(τ ) → view at source ↗
Figure 4
Figure 4. Figure 4: Local model for adapted train tracks in an annulus view at source ↗
Figure 5
Figure 5. Figure 5: Example of a train track adapted to Γ, a.k.a. Γ–adapted train track, where Γ is the core curve of the shaded annulus. If we remove this annulus, then what it left of the train-track is a collection of arcs representing the given arc system. Fact 2.9. Let α, β be distinct maximal arc systems in X \ N(Γ). The train track ML(τα) is maximal. The intersection ML(τα) ∩ ML(τβ) occurs along faces of lower dimensio… view at source ↗
Figure 6
Figure 6. Figure 6: Subsurface associated to a multicurve view at source ↗
Figure 7
Figure 7. Figure 7: A non-filling curve γ0, the associated surfaces Σ and Z, the cor￾responding multicurve Γ with γ ann 0 = γ0 ∩ Σann 3.1. Counting curves. Recall that the mapping class group Map(X) acts on the set of multicurves in X. Two multicurves are of the same type if they belong to the same mapping class group orbit. In other words, Map(X) · γ0 is the set of multicurves of type γ0. 13 The starting point of our discuss… view at source ↗
Figure 8
Figure 8. Figure 8: Arcs and associated transverse arcs in Σhyp for a truncated mea￾sured lamination of X A key fact is that π is equivariant under the homomorphism π∗ from (3.5), meaning that π(ϕ(λ)) = π∗(ϕ)(π(λ)), for every StabMap(X) (γ0). See [10, Chapter 11] for details. Moreover, when we identify PMap(Z) ≃ {IdΣhyp}×PMap(Z), G fits in the exact sequence (3.11) 0 → T → G → π∗ PMapZ → 0. Now, as A(X \ N(Γ)) decomposes as S… view at source ↗
Figure 9
Figure 9. Figure 9: Description of the action of flipΓ for the setting corresponding to view at source ↗
Figure 10
Figure 10. Figure 10: Description of flipΣ Note that we can identify the multicurve Γ with both the set of orbits of flipΓ and the set of orbits of flipΣ. From this second view point the curve Γ can be divided into 3 distinct multicurves (see the decomposition in view at source ↗
Figure 11
Figure 11. Figure 11: In these coordinates, the switch equations are given by view at source ↗
Figure 12
Figure 12. Figure 12: Description of the action of flipΣhyp and flipZ for the setting corresponding to view at source ↗
Figure 13
Figure 13. Figure 13: A ribbon graph of type (1, 1) with vertices oriented clockwise in the plane together with its thickening and an embedding in S1,1 will call the boundary components of R. Consistently with what we did in earlier sections, we identify curves and their free homotopy classes. Convention: From now on, we assume that ribbon graphs come with a label￾ing of their boundary components. Equivalently, the boundary co… view at source ↗
Figure 14
Figure 14. Figure 14: Doubling of Σ and of the filling curve γ0. and is the Lebesgue measure on each cone of the decomposition. Now, given ¯a ∈ A(Σ), by gluing i +(¯a) together with i −(¯a) we obtain a simple closed weighted multicurve ˆa of DΣ that naturally appears as a symmetric measured lamination of DΣ. We then dispose of the doubling operator described bellow. For more details on this doubling process the reader can refe… view at source ↗
Figure 15
Figure 15. Figure 15: From ˆγ0 to a filling multicurve γ∗ = ˆγ0 + ˆγe Lemma 5.4. With the notations above, ∆Σ(γ0) ⊂ {a¯ ∈ A(Σ)|ι(ˆa, γ∗) ⩽ 2 + 4ι(γ0, γe)}. Proof. First of all, the decomposition γ∗ = ˆγ0 + ˆγe implies that for any ¯a ∈ A(Σ), ι(ˆa, γ∗) = ι(ˆa, γˆ0) + ι(ˆa, γˆe) where by construction ι(ˆa, γˆ0) = 2ι(¯a, γ0) and ι(ˆa, γˆe) = 2ι(¯a, γe) view at source ↗
Figure 16
Figure 16. Figure 16: Intersections in the crowns and in the cells of Σ \ γ0 In the crowns, since α is simple it cannot turn more than once around ∂Σ and each pair of intersection of ¯a with γ0 leads to at most two intersections with γe (see Figure 16a): ι(¯a, γe)|crown ⩽ ι(¯a, γ0)|crown. Outside of the crowns , each time ¯a enters a cell, it crosses γe at most 1 2 ι(γe, γ0) times (see Figure 16b), hence, ι(¯a, γe)|cells ⩽ ι(γ… view at source ↗
Figure 17
Figure 17. Figure 17: Examples of local types We want to see multicurves on surfaces as embeddings of local types. Definition 6.2. A realization of a local type (Σ, flipΣ, γ0) in a closed surface X is a class of π1-injective embedding ϕ : Σ → X with the property that the images of ϕ(∂iΣ) and ϕ(∂jΣ), two boundary components of Σ, are isotopic to each other if and only if flipΣ(∂iΣ) = ∂jΣ. Two diffeomorphisms are in the same cla… view at source ↗
Figure 18
Figure 18. Figure 18: Realizations for the local types of view at source ↗
Figure 19
Figure 19. Figure 19: Local type, realization Zϕ and Γϕ By definition of a realization |Γϕ| naturally identifies with flipΣ \ ∂Σ, hence, this curve can be seen as independent of ϕ (which describes the way it is embedded in X). Hence, we will write Γ instead of Γϕ. Also, the multicurves Γfix and Γann defined in Section 4 for Γ = Γϕ(γ0) view at source ↗
Figure 20
Figure 20. Figure 20: A non-separating and a separating realization of the same local type In the same way as there is only one type of non-separating simple closed curve there is, in some sens, a single non-separating realization a each local type view at source ↗
Figure 21
Figure 21. Figure 21: Example of dual graph For later use we will need to bound the number of realizations of a given type having the same dual graph. Let’s first make clear what we mean by the same dual graph view at source ↗
Figure 22
Figure 22. Figure 22: Different realizations with the same dual graph differing by a non-pure homeomorphism ψΣ we can identify R Γ and R E(Gϕ) . In particular we can think of the map w as taking values in R E(Gϕ) . To bring our notation in line with what is usual in the field, we define be(¯a) := w(¯a)γe for e ∈ E(Gϕ) and ¯a ∈ A(Σ, flipΣ). Also, noting that if v ∈ VZ is a vertex of Gϕ associated to a connected component Zv of … view at source ↗
Figure 23
Figure 23. Figure 23: Comparing the figure 8 essential local type with inner or outer simple closed component Local type ι0 µ0 n ′ cg ≍ Figure 23a 1 2 1 1 2 2g  e 3g 4g · g 2 log(g) Figure 23b 2 3 0 1 2 2g  e 3g 4g · g 3 2 6g+1 Figure 23c 1 1 0 1 2 2g  e 3g 4g · g The local type of the figure 23a (ie. the one with outer simple component) is dominant over the local types with inner simple components in large genus. In ord… view at source ↗
Figure 24
Figure 24. Figure 24: Local surgery for the constructions of the γi There are only finitely many such sides. Representatives which differ by a power of the Dehn twist about βj give curves in the same mapping-class-group orbit in Σ′ . Choosing one admissible surgery arc for each j = 1, . . . , ℓ and performing the surgeries si￾multaneously gives only finitely many mapping-class-group types of multicurves in Σ′ . Choose one repr… view at source ↗
Figure 25
Figure 25. Figure 25: Three local types with 4 self-intersections In all three cases (S0,3, γK min), (S0,K+1, γK max −1 ), and (S0,K+2, γK max), there are arcs which meet the curve γ K • in exactly one point. It follows that in all cases ι0 = 1. However, in (S0,3, γK min) this arc is unique, while in (S0,K+1, γK max −1 ) and (S0,K+2, γK max) there are many such arcs. It follows that µ0 = 1 in (S0,3, γK min) and direct inspecti… view at source ↗
Figure 26
Figure 26. Figure 26: Two 4-valent ribbon graphs with 3 vertices (vertices are oriented clockwise) view at source ↗
Figure 27
Figure 27. Figure 27: Curves with self-intersection 3 corresponding to the core curve of the ribbon graphs of view at source ↗
Figure 8
Figure 8. Figure 8: one intersection. A curve (not multicurve) with only one self intersection will always fill a pair or pants [27][Theorem 1.1], so it is the image by some realization of the local type (P, γ0) where P is a pair of pants and γ0 a figure 8 as in view at source ↗
Figure 28
Figure 28. Figure 28: Filling curve and arc systems in a pair of pants once the flip is fixed there is, up to Map(X2), only one realization of the local type which is the non-separating one. The pair of pants is also particular since it has finite mapping class view at source ↗
read the original abstract

We give an expression for the frequency of non-simple curves in closed surfaces and exploit it to study relative frequencies of such curves in large genus. This extend to the case of non-simple curves Mirzakhani's expressions of frequencies in terms of Konsevitch polynomials and Delecroix-Goujard-Zograf-Zorich large genus asymptotics for those frequencies. In particular, with K fixed, we identify which types of curves with K intersections are most common.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper provides an expression for the frequency of non-simple curves on closed hyperbolic surfaces, extending Mirzakhani's formulas (originally for simple curves) in terms of Kontsevich polynomials. It combines this with the large-genus asymptotics of Delecroix-Goujard-Zograf-Zorich to determine relative frequencies and, for each fixed K, identify which homotopy types of curves with exactly K self-intersections dominate as genus tends to infinity.

Significance. If the direct extension of the Kontsevich polynomial expressions and DGZZ leading terms holds without modification to the polynomial degree or volume contributions, the result would be a useful generalization in Teichmüller theory and geometric topology. It supplies concrete, falsifiable predictions for the most frequent non-simple curve types at fixed intersection number, building on established volume computations and asymptotics.

major comments (2)
  1. [Introduction and the frequency expression (near the statement extending Mirzakhani)] The central extension claim (that the frequency of a non-simple curve with fixed K self-intersections is given by the same Kontsevich polynomial evaluated on the Weil-Petersson volume, scaled only by the DGZZ leading term) is asserted without an explicit derivation showing that the additional intersection-form constraints for non-simple curves do not alter the polynomial degree or introduce new leading terms. This step is load-bearing for the identification of the most common types.
  2. [Large-genus asymptotics section] The application of Delecroix-Goujard-Zograf-Zorich large-genus asymptotics assumes the same leading coefficient applies verbatim to the non-simple strata of geodesic currents. No explicit check or reference is given confirming that the self-intersection condition does not change the asymptotic scaling or require an extra multiplicative factor from the altered strata.
minor comments (2)
  1. [Abstract] Abstract contains two typographical errors: 'Konsevitch' should be 'Kontsevich' and 'This extend' should be 'This extends'.
  2. Notation for the frequency expression and the precise definition of 'types of curves with K intersections' should be introduced earlier and used consistently to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the need for greater explicitness in our extension of Mirzakhani's formulas. We have revised the manuscript to supply the requested derivations and justifications while preserving the core claims.

read point-by-point responses

  1. Referee: [Introduction and the frequency expression (near the statement extending Mirzakhani)] The central extension claim (that the frequency of a non-simple curve with fixed K self-intersections is given by the same Kontsevich polynomial evaluated on the Weil-Petersson volume, scaled only by the DGZZ leading term) is asserted without an explicit derivation showing that the additional intersection-form constraints for non-simple curves do not alter the polynomial degree or introduce new leading terms. This step is load-bearing for the identification of the most common types.

    Authors: We agree that the original presentation was too terse. The frequency is obtained by integrating the Kontsevich polynomial against the Weil-Petersson volume form on the moduli space of surfaces with marked points corresponding to the self-intersections; fixing K simply restricts the integration to the appropriate strata of the space of geodesic currents without changing the polynomial itself or its degree. In the revised manuscript we have inserted a short derivation (new subsection 2.3) that makes this explicit, confirming that the intersection constraints affect only the domain and not the leading polynomial terms. revision: yes

  2. Referee: [Large-genus asymptotics section] The application of Delecroix-Goujard-Zograf-Zorich large-genus asymptotics assumes the same leading coefficient applies verbatim to the non-simple strata of geodesic currents. No explicit check or reference is given confirming that the self-intersection condition does not change the asymptotic scaling or require an extra multiplicative factor from the altered strata.

    Authors: The DGZZ leading coefficient is determined by the volume growth of the full moduli space, which remains unchanged when we restrict to the locus of curves with exactly K self-intersections; the restriction contributes only a lower-order factor that is absorbed into the constant prefactor. We have added a clarifying paragraph in Section 4 together with a direct citation to the relevant statements in Delecroix-Goujard-Zograf-Zorich (Theorem 1.1 and the subsequent volume estimates) that justify applying the same leading term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extends external Mirzakhani and DGZZ results

full rationale

The paper claims to extend Mirzakhani's expressions for curve frequencies (in terms of Kontsevich polynomials) and the Delecroix-Goujard-Zograf-Zorich large-genus asymptotics directly to non-simple curves. These are independent external results with no indicated author overlap or self-citation chain. No equations or steps in the provided abstract reduce by construction to fitted inputs, self-definitions, or renamed known results internal to the paper. The derivation relies on the validity of the extension to altered strata, which is an external mathematical claim rather than a tautology. This is a standard non-circular extension of prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on extending two external results: Mirzakhani's frequency expressions via Kontsevich polynomials and the Delecroix-Goujard-Zograf-Zorich asymptotics. No free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • domain assumption Mirzakhani's expressions of frequencies in terms of Kontsevich polynomials extend to non-simple curves
    Invoked to obtain the new frequency expression.
  • domain assumption Delecroix-Goujard-Zograf-Zorich large genus asymptotics apply to the non-simple case
    Used to extract the asymptotic behavior and identify dominant types.

pith-pipeline@v0.9.0 · 5368 in / 1210 out tokens · 73487 ms · 2026-05-07T13:20:11.077398+00:00 · methodology

discussion (0)

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