Integrals of general geometric random variables on the moduli space of hyperbolic surfaces

Victor Le Guilloux (IRMA)

arxiv: 2605.23626 · v1 · pith:TJEWHUELnew · submitted 2026-05-22 · 🧮 math.GT

Integrals of general geometric random variables on the moduli space of hyperbolic surfaces

Victor Le Guilloux (IRMA) This is my paper

Pith reviewed 2026-05-25 02:44 UTC · model grok-4.3

classification 🧮 math.GT

keywords moduli spacehyperbolic surfacesWeil-Petersson measureclosed geodesicsmapping class groupFenchel-Nielsen coordinatesasymptoticsgeometric random variables

0 comments

The pith

An integration formula reduces integrals of lengths from any mapping class orbit to one-dimensional Lebesgue densities, improving the asymptotic for E[N_γ(a)] as a tends to infinity.

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

The paper constructs an integration formula that applies to random variables on the moduli space of hyperbolic surfaces depending on lengths of closed geodesics in any fixed mapping class group orbit. This generalizes prior formulas restricted to simple geodesics or those with exactly one self-intersection. The authors also give an explicit expression for the length of an arbitrary closed loop in Fenchel-Nielsen coordinates. Substituting this expression shows that the integral equals an integral over the real line against a density with respect to Lebesgue measure. Asymptotic analysis of the density at fixed genus and boundary number then refines Mirzakhani's equivalent for the Weil-Petersson expectation of the number of geodesics in the orbit of a given loop γ having length at most a.

Core claim

Using the integration formula together with the general expression of the length function in Fenchel-Nielsen coordinates, the integral of a geometric random variable reduces to an integral over R against a measure with density with respect to Lebesgue measure. Studying the asymptotic behavior of this density at fixed genus and number of boundaries then yields an improvement of Mirzakhani's asymptotic equivalent of the Weil-Petersson expectation E[N_γ(a)] as a tends to infinity, for an arbitrary closed loop γ. The result also recovers the earlier conclusions for eight-shaped geodesics as a special case.

What carries the argument

The integration formula that converts moduli-space integrals involving lengths from a fixed mapping class orbit into one-dimensional integrals against a Lebesgue density, using the explicit length expression in Fenchel-Nielsen coordinates.

If this is right

The expected number E[N_γ(a)] of geodesics in any fixed mapping class orbit admits a refined asymptotic expansion as a grows.
Any geometric random variable built from lengths in an arbitrary orbit becomes expressible as a one-dimensional integral.
The density function governing the reduced integral can be analyzed asymptotically while keeping genus and boundary number fixed.
Results previously obtained only for simple geodesics or eight-shaped geodesics now follow as special cases of the general formula.

Where Pith is reading between the lines

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

The same density reduction may permit explicit or numerical evaluation of the integral for particular low-complexity orbits.
Higher moments or the full distribution of the geodesic count could be accessible by applying the formula to suitable powers or generating functions.
The technique suggests a route to treating random variables that depend on lengths from several distinct orbits simultaneously.

Load-bearing premise

The length of an arbitrary closed loop admits an explicit expression in Fenchel-Nielsen coordinates that allows the full moduli-space integral to reduce to a single real-line integral against a density.

What would settle it

A numerical computation, for a concrete low-genus surface and a specific loop γ, of the actual Weil-Petersson expectation E[N_γ(a)] at successively larger values of a that deviates from the claimed improved asymptotic would disprove the reduction and the resulting asymptotic statement.

Figures

Figures reproduced from arXiv: 2605.23626 by Victor Le Guilloux (IRMA).

**Figure 1.** Figure 1: The resolutions of a loop at the self-intersection point p. Since the resolutions of a loop γ at a self-intersecting point can be taken in an arbitrarily small neighborhood of γ, its resolutions are homotopic to loops on the surface S(γ) filled by γ. Lemma 3.2 ([RY14]). Let γ be a closed loop self-intersecting transversally at p. One way of resolving the self-intersection at p splits γ into two loops α, α′… view at source ↗

**Figure 2.** Figure 2: The resolutions of an eight-shaped loop. The curves α and α ′ are the separating resolutions and β is the non-separating one. Definition 3.3 ([RY14]). Let γ be a loop, p be a self-intersection point of γ and let t1, t2 ∈ (0, 1), t1 < t2, such that γ(t1) = γ(t2) = p. If the subloop γ|[t1,t2] is homotopically trivial, we call it a curl of γ and p is its vertex. A Reidemeister move I on γ at p consists in re… view at source ↗

**Figure 3.** Figure 3: A Reidemeister move I removing a curl of vertex p. The length of a non-simple geodesic on a hyperbolic surface is closely related to the length of its resolutions. We can therefore describe an algorithm consisting in successively resolving a closed geodesic at its self-intersecting points in order to find an expression of its length in terms of lengths of simple closed geodesics. Lemma 3.4 ([RY14]). Let γ … view at source ↗

**Figure 4.** Figure 4: The only possible case for the curling number of the non-separating resolution of γ at p to be 1. We shall denote by ch and sh the hyperbolic cosinus and sinus instead of the usual notations cosh and sinh. This slight gain of space will become convenient for our further calculations in Paragraph 3.3. Lemma 3.5 ([RY14]). Let γ be a non-simple closed geodesic on a hyperbolic surface X and let us denote by α,… view at source ↗

**Figure 5.** Figure 5: Resolutions of γ at self-intersecting points near the curve δ. However it is possible that the non-separating resolution at one of the self-intersecting points pi has curling number 1. In this case we denote by p ′ i the simple vertex of the contractible triangle bounded by γ with double vertex pi . Let αi , α′ i be the separating resolutions of γ at pi and βi be the non-separating resolution (with c(βi) =… view at source ↗

**Figure 6.** Figure 6: Resolutions at a point where the non-separating resolution has curling number 1. □ Remark. In general the curves α1, . . . , αk are not disjoint [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Non-disjoint simple resolutions of a loop filling a four-holed sphere. A first consequence of Lemma 3.6 is that we can deduce from it a monotony result for the length functions of loops with respect to the lengths of the boundaries of Sg,n. Indeed, while proving the main result of [Par05], Parlier also proved the following monotony property. Theorem 3.7 ([Par05]). Let P a pants decomposition of Sg,n and γ … view at source ↗

**Figure 8.** Figure 8: Change of Fenchel-Nielsen coordinates in a once-holed torus. Using this method, Mirzakhani obtained appropriate C 0 estimates on the length functions of arbitrary closed loops. However our method requires C 1 estimates of these functions. Unfortunately we cannot control the simple curves appearing in Lemma 3.6, hence we could have to compose several different changes of coordinates provided by Theorem 3.12… view at source ↗

**Figure 9.** Figure 9: Definitions of the lengths in the pair of pants bounded by α0, β0 and β1 with ℓα0 = x, ℓβ0 = y and ℓβ1 = z. Let P ⊂ S(γ) be a pair of pants in the complement of P and γeP be an incursion of γe in P. Let P0 be the pair of pants from which γeP enters in P and P1 be the pair of pants in which γeP enters after exiting P. We note α0, α1 ∈ P the boundaries of P such that the starting point of γeP is on α0 and th… view at source ↗

read the original abstract

In this article we provide an integration formula making us able to integrate random variables defined on the moduli space of hyperbolic surfaces which involve the lengths of closed geodesics belonging to a fixed arbitrary mapping class group orbit. This generalizes Mirzakhani's formula for simple geodesics and the integration formula of our previous paper on geodesics with exactly one self-intersection. We then compute the general expression of the length function of an arbitrary closed loop in Fenchel-Nielsen coordinates. Using this expression together with our integration formula, we prove that the integral of a geometric random variable can be expressed as an integral over R for a measure with density with respect to the Lebesgue measure. By studying the asymptotic behavior of this density function (at fixed genus and number of boundaries on the base surface), given an arbitrary closed loop $\gamma$, we obtain an improvement of Mirzakhani's asymptotic equivalent of the Weil-Petersson expectation E[N$\gamma$(a)], when a $\rightarrow$ $\infty$, of the number of geodesics in the same mapping class group orbit as $\gamma$ of length at most a. This also generalizes the conclusions of our previous article on eight-shaped geodesics.

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.

Desk Editor's Note private letter to a colleague

The paper gives a length formula in FN coordinates for arbitrary closed loops and claims this plus a new integration formula reduces moduli integrals of geometric RVs to 1D Lebesgue, improving Mirzakhani asymptotics for general MCG orbits.

read the letter

The core advance is an explicit length expression for any closed geodesic in Fenchel-Nielsen coordinates, paired with an integration formula that supposedly collapses the Weil-Petersson integral over the moduli space to a single integral against a density on R. This extends the author's earlier work on one-self-intersection curves and Mirzakhani's simple-curve case to arbitrary orbits, and it produces a sharper asymptotic for the expected count E[N_gamma(a)] as a goes to infinity at fixed genus and boundaries. If the reduction step holds, it would let people compute more general geodesic statistics without summing over the full mapping class group action each time. The length formula itself looks like standard trace identities applied to a pants decomposition, which is the right starting point. The soft spot is exactly the reduction claim: for a general orbit the length is a nonlinear function of several length and twist parameters, and nothing shown so far confirms that the volume form factors so cleanly that all but one coordinate integrate out in closed form, leaving a usable density whose asymptotics can be read off. The abstract states the result but does not display the intermediate expressions or error controls, so it is impossible to judge whether the improvement over Mirzakhani is genuine or formal. This is aimed at people already working on Weil-Petersson volumes and random hyperbolic surfaces; they will want to see the explicit density and the asymptotic derivation before citing. It is worth sending to referees so the derivations can be checked in detail.

Referee Report

1 major / 1 minor

Summary. The paper develops an integration formula for geometric random variables on the moduli space of hyperbolic surfaces involving lengths of closed geodesics from a fixed mapping class group orbit. This generalizes Mirzakhani's formula for simple geodesics and the author's prior work on one-self-intersection geodesics. The authors derive the general length function of an arbitrary closed loop in Fenchel-Nielsen coordinates, combine it with the integration formula to reduce the moduli-space integral to a one-dimensional integral over R against a Lebesgue density, and analyze the density's asymptotics (at fixed genus and boundaries) to obtain an improved asymptotic for the Weil-Petersson expectation E[N_γ(a)] as a → ∞.

Significance. If the claimed reduction holds, the result would extend Mirzakhani's techniques to arbitrary MCG orbits, offering a systematic way to evaluate such integrals and sharper asymptotics for geodesic counting functions. The explicit length expression in FN coordinates and the density analysis constitute concrete technical progress if the integration steps are fully rigorous.

major comments (1)

[The computation of the general expression of the length function in Fenchel-Nielsen coordinates (paragraph beginning 'We] The section deriving the general length expression in Fenchel-Nielsen coordinates and the subsequent reduction claim: for arbitrary MCG orbits the length is a nonlinear function of multiple length/twist parameters (via trace identities or cosh formulas). The manuscript must explicitly show how the Weil-Petersson volume form factors so that all but one coordinate integrate out in closed form, leaving a density w.r.t. Lebesgue measure; without this explicit integration the reduction to a 1D integral is not yet verified and is load-bearing for the asymptotic improvement.

minor comments (1)

Notation for the density function and the measure should be introduced with an explicit formula immediately after the reduction is stated, rather than left implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying the need to strengthen the explicit verification of the reduction step, which is indeed central to the paper's claims. We address the major comment point by point below and will revise the manuscript to incorporate additional details.

read point-by-point responses

Referee: The section deriving the general length expression in Fenchel-Nielsen coordinates and the subsequent reduction claim: for arbitrary MCG orbits the length is a nonlinear function of multiple length/twist parameters (via trace identities or cosh formulas). The manuscript must explicitly show how the Weil-Petersson volume form factors so that all but one coordinate integrate out in closed form, leaving a density w.r.t. Lebesgue measure; without this explicit integration the reduction to a 1D integral is not yet verified and is load-bearing for the asymptotic improvement.

Authors: We agree that the reduction to a one-dimensional integral is load-bearing and that the current presentation would benefit from greater explicitness. The manuscript already derives the general length expression in Fenchel-Nielsen coordinates via trace identities (expressing the length of an arbitrary closed curve as a nonlinear function of the relevant length and twist parameters) and invokes the integration formula to assert that all but one coordinate integrate out. However, the factoring of the Weil-Petersson volume form is not written out coordinate-by-coordinate for the general case. In the revised version we will add a dedicated subsection that performs this explicit integration: starting from the WP volume form in Fenchel-Nielsen coordinates, we will show term-by-term how the integrals over the unused length and twist parameters evaluate in closed form (using the structure of the integration formula), leaving a density with respect to Lebesgue measure on the single remaining variable. This will make the reduction fully rigorous and verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new general formulas and length expression provide independent content

full rationale

The derivation introduces a new integration formula generalizing Mirzakhani's (external) and the author's prior specific-case work, explicitly computes the general length function of an arbitrary closed loop in Fenchel-Nielsen coordinates, and uses these to reduce the moduli-space integral to a 1D Lebesgue integral whose density asymptotics improve an external result. No step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the central claims rest on explicit new expressions rather than tautological renaming or load-bearing self-reference. The paper is self-contained against external benchmarks like Mirzakhani's asymptotics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard structures from Teichmuller theory (Weil-Petersson measure, Fenchel-Nielsen coordinates, mapping class group action) already present in the cited Mirzakhani and prior self-work; no new free parameters or invented entities are mentioned.

axioms (2)

domain assumption The Weil-Petersson measure on the moduli space is well-defined and mapping-class-group invariant.
Used to define the expectation E[Nγ(a)] and the integration setting.
domain assumption Fenchel-Nielsen coordinates give a global parametrization of the hyperbolic structures.
Invoked to express the length function of an arbitrary closed loop.

pith-pipeline@v0.9.0 · 5739 in / 1506 out tokens · 31298 ms · 2026-05-25T02:44:41.959686+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We then compute the general expression of the length function of an arbitrary closed loop in Fenchel-Nielsen coordinates... prove that the integral of a geometric random variable can be expressed as an integral over R for a measure with density with respect to the Lebesgue measure.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ch(ℓγ/2) = 1/∏ sh(ℓα/2)^dα ⋅ ∑ aiT eT(...) (Corollary 3.20); length-type functions and simplified length-type functions (Definition 3.21, 3.25)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

Friedman-Ramanujan functions in random hyper- bolic geometry and application to spectral gaps II

doi: https://doi.org/10. 48550/arXiv.2304.02678. [AM25] Nalini Anantharaman and Laura Monk. “Friedman-Ramanujan functions in random hyper- bolic geometry and application to spectral gaps II”. Preprint

work page arXiv
[2]

Counting curves, and the stable length of currents

doi: https://doi.org/ 10.48550/arXiv.2502.12268. [EPS20] Viveka Erlandsson, Hugo Parlier, and Juan Souto. “Counting curves, and the stable length of currents”. In: Journal of the European Mathematical Society 22.6 (Feb. 2020), pp. 1675–1702. doi: https://doi.org/10.4171/JEMS/953. REFERENCES 47 [ES16] Viveka Erlandsson and Juan Souto. “Counting curves in h...

work page doi:10.48550/arxiv.2502.12268 2020
[3]

A proof of Alon’s second eigenvalue conjecture

[Fri03] Joel Friedman. “A proof of Alon’s second eigenvalue conjecture”. In: Proceedings of the thirty- fifth annual ACM symposium on Theory of computing (2003), pp. 720–724. [HS99] Joel Hass and Peter Scott. “Configurations of curves and geodesics on surfaces”. In: Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest (1999), pp. 201–213....

work page doi:10.2307/2007076 2003
[4]

Weil-Petersson volumes and intersection theory on the moduli space of curves

doi: https: //doi.org/10.1007/s10711-025-01043-0 . [Mir07a] Maryam Mirzakhani. “Weil-Petersson volumes and intersection theory on the moduli space of curves”. In: Journal of the American Mathematical Society 20.1 (Jan. 2007), pp. 1–23. doi: https://doi.org/10.1090/S0894-0347-06-00526-1 . [Mir07b] Maryam Mirzakhani. “Simple geodesics and Weil-Petersson vol...

work page doi:10.1007/s10711-025-01043-0 2007
[5]

Counting Mapping Class group orbits on hyperbolic surfaces

doi: https://doi.org/10.48550/arXiv.1601.03342. [Oka93] Takayuki Okai. “Effects of a change of pants decompositions on their Fenchel-Nielsen coordi- nates”. In: Kobe Journal of Mathematics 10 (1993), pp. 215–223. [Par05] Hugo Parlier. “Lengths of geodesics on Riemann surfaces with boundary”. In: Annales Academiæ Scientiarum Fennicæ 30 (2005), pp. 227–236....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1601.03342 1993

[1] [1]

Friedman-Ramanujan functions in random hyper- bolic geometry and application to spectral gaps II

doi: https://doi.org/10. 48550/arXiv.2304.02678. [AM25] Nalini Anantharaman and Laura Monk. “Friedman-Ramanujan functions in random hyper- bolic geometry and application to spectral gaps II”. Preprint

work page arXiv

[2] [2]

Counting curves, and the stable length of currents

doi: https://doi.org/ 10.48550/arXiv.2502.12268. [EPS20] Viveka Erlandsson, Hugo Parlier, and Juan Souto. “Counting curves, and the stable length of currents”. In: Journal of the European Mathematical Society 22.6 (Feb. 2020), pp. 1675–1702. doi: https://doi.org/10.4171/JEMS/953. REFERENCES 47 [ES16] Viveka Erlandsson and Juan Souto. “Counting curves in h...

work page doi:10.48550/arxiv.2502.12268 2020

[3] [3]

A proof of Alon’s second eigenvalue conjecture

[Fri03] Joel Friedman. “A proof of Alon’s second eigenvalue conjecture”. In: Proceedings of the thirty- fifth annual ACM symposium on Theory of computing (2003), pp. 720–724. [HS99] Joel Hass and Peter Scott. “Configurations of curves and geodesics on surfaces”. In: Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest (1999), pp. 201–213....

work page doi:10.2307/2007076 2003

[4] [4]

Weil-Petersson volumes and intersection theory on the moduli space of curves

doi: https: //doi.org/10.1007/s10711-025-01043-0 . [Mir07a] Maryam Mirzakhani. “Weil-Petersson volumes and intersection theory on the moduli space of curves”. In: Journal of the American Mathematical Society 20.1 (Jan. 2007), pp. 1–23. doi: https://doi.org/10.1090/S0894-0347-06-00526-1 . [Mir07b] Maryam Mirzakhani. “Simple geodesics and Weil-Petersson vol...

work page doi:10.1007/s10711-025-01043-0 2007

[5] [5]

Counting Mapping Class group orbits on hyperbolic surfaces

doi: https://doi.org/10.48550/arXiv.1601.03342. [Oka93] Takayuki Okai. “Effects of a change of pants decompositions on their Fenchel-Nielsen coordi- nates”. In: Kobe Journal of Mathematics 10 (1993), pp. 215–223. [Par05] Hugo Parlier. “Lengths of geodesics on Riemann surfaces with boundary”. In: Annales Academiæ Scientiarum Fennicæ 30 (2005), pp. 227–236....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1601.03342 1993