Weil–Petersson Random Hyperbolic Surface

• Weil–Petersson random hyperbolic surfaces are defined on moduli spaces using the normalized Weil–Petersson measure, offering a framework to model typical high-genus hyperbolic geometry.
• Key analyses reveal asymptotic behavior in length spectra, systole measurements, and pants decomposition lengths, which elucidate the complex geometry and topology of these surfaces.
• Advanced methods, including tangle-freeness and combinatorial tree bijections, establish convergence results, spectral expansions, and scaling limits in both high-genus and planar regimes.

A Weil–Petersson random hyperbolic surface is a hyperbolic surface sampled from a moduli space with respect to the normalized Weil–Petersson probability measure. In the closed case, one works on the moduli space $\mathcal{M}_g$ of genus-$g$ hyperbolic surfaces; in bordered or cusped settings, one uses the corresponding moduli spaces $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$ or $\mathcal{M}_{g,n}$ with fixed boundary lengths or cusps. This probabilistic model has become the standard framework for describing “typical” large-genus or large-complexity hyperbolic geometry, especially in work influenced by Mirzakhani’s integration formulas, Weil–Petersson volume asymptotics, and later developments on spectra, tangles, ribbon graphs, and scaling limits (Mirzakhani, 2010).

1. Probabilistic model and Weil–Petersson geometry

For a closed surface, the normalized Weil–Petersson measure is defined by

$$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$

where $V_g$ is the total Weil–Petersson volume of $\mathcal{M}_g$. In the notation used for expectations, if $f:\mathcal{M}_g\to\mathbb{R}$, then

$$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$

For bordered surfaces one analogously uses the normalized measures

$d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$

on moduli spaces $g$0 of genus-$g$1, $g$2-boundary hyperbolic surfaces with prescribed boundary lengths (He et al., 2023).

The Weil–Petersson measure is induced by a symplectic form on Teichmüller space and descends to a finite measure on moduli space. In Fenchel–Nielsen coordinates associated to a pants decomposition, the Weil–Petersson volume form is essentially Lebesgue measure: $g$3 This coordinate description underlies a large part of the probabilistic analysis of random surfaces, because it converts geometric constraints into measurable regions in length–twist space (Guth et al., 2010).

Large-scale asymptotics of Weil–Petersson volumes are central. For fixed $g$4 and $g$5, Mirzakhani proved recurrence relations such as

$g$6

together with bounds of the form

$g$7

for some $g$8. These asymptotics make it possible to compare the volume of geometrically constrained loci to the total volume of moduli space, which is the basic mechanism behind “with high probability” statements for random hyperbolic geometry (Mirzakhani, 2010).

2. Length spectrum, systoles, and typical large-genus geometry

One of the earliest statistical results for the closed model concerns the bottom of the length spectrum. If $g$9 denotes the number of primitive closed geodesics of $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$0 with lengths in $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$1, then for disjoint intervals $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$2 the vector

$\mathcal{M}_{g,n}(L_1,\ldots,L_n)$3

converges in distribution, as $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$4, to a vector of independent Poisson random variables with means

$\mathcal{M}_{g,n}(L_1,\ldots,L_n)$5

As an application, the expected systole has the large-genus limit

$\mathcal{M}_{g,n}(L_1,\ldots,L_n)$6

(Mirzakhani et al., 2017).

This constant-scale behavior for the simple systole contrasts sharply with other distinguished geodesic lengths. For separating simple closed geodesics, if $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$7 denotes the separating systole, then for any function $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$8 with $\mathcal{M}_{g,n}(L_1,\ldots,L_n)$9 and $\mathcal{M}_{g,n}$0,

$\mathcal{M}_{g,n}$1

with probability tending to $\mathcal{M}_{g,n}$2, and generically the minimizing curve separates $\mathcal{M}_{g,n}$3 into $\mathcal{M}_{g,n}$4. Moreover, there is a half-collar around such a curve of width

$\mathcal{M}_{g,n}$5

with probability tending to $\mathcal{M}_{g,n}$6 (Nie et al., 2020).

For non-simple closed geodesics, the relevant invariant is the non-simple systole

$\mathcal{M}_{g,n}$7

It is always realized by a figure-eight geodesic. If $\mathcal{M}_{g,n}$8 is sampled from $\mathcal{M}_{g,n}$9 with respect to the normalized Weil–Petersson measure, then

$$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$0

More precisely, for any $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$1 with $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$2,

$$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$3

and consequently $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$4 (He et al., 2023).

Global decomposition complexity also grows. A pants decomposition of a closed genus-$$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$5 surface uses $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$6 disjoint simple closed curves, and its total pants length is the sum of their lengths. For any $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$7, a Weil–Petersson random hyperbolic surface of genus $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$8 has total pants length at least $$\mathbb{P}_{g,\mathrm{WP}}(A)=\frac{\operatorname{Vol}_{\mathrm{WP}}(A)}{\operatorname{Vol}_{\mathrm{WP}}(\mathcal{M}_g)} =\frac{1}{V_g}\int_A d\mu_{WP},$$9 with probability tending to $V_g$0. The argument compares the volume of the region in moduli space admitting a pants decomposition of total length $V_g$1,

$V_g$2

to the total moduli space volume $V_g$3 (Guth et al., 2010).

A common misconception is that “random” in this model means uniformly thick at every scale. The large-genus results show a more structured picture: very short simple non-separating geodesics persist at constant scale, while separating and non-simple minimizers typically occur at logarithmic scale, and the total cost of a disjoint pants decomposition is polynomially larger than either phenomenon suggests.

3. Tangle-free surfaces and control of local topology

A major structural tool is the tangle-free hypothesis. A compact hyperbolic surface $V_g$4 is $V_g$5-tangle-free if all embedded pairs of pants and one-holed tori in $V_g$6 have total boundary length greater than $V_g$7. If such a subsurface exists with total boundary length $V_g$8, then $V_g$9 is $\mathcal{M}_g$0-tangled. This notion is motivated by the graph-theoretic notion of tangle-freeness and has precise geometric consequences (Monk et al., 2020).

For any $\mathcal{M}_g$1, a Weil–Petersson random closed surface of genus $\mathcal{M}_g$2 is $\mathcal{M}_g$3-tangle-free with high probability: $\mathcal{M}_g$4 This is described as almost optimal, since any surface is $\mathcal{M}_g$5-tangled. On an $\mathcal{M}_g$6-tangle-free surface, every primitive closed geodesic of length $\mathcal{M}_g$7 is simple, any two closed geodesics with total length $\mathcal{M}_g$8 are disjoint, and any closed geodesic of length $\mathcal{M}_g$9 lies in an embedded hyperbolic cylinder of width at least $f:\mathcal{M}_g\to\mathbb{R}$0. In addition, every ball of radius $f:\mathcal{M}_g\to\mathbb{R}$1 is isometric either to a ball in the hyperbolic plane or to a ball in a cylinder (Monk et al., 2020).

Conditioning on tangle-freeness is analytically difficult because multiplying the Weil–Petersson density by an indicator function destroys the smooth structure needed for Mirzakhani-type integration formulas. To address this, a Möbius inversion formula was introduced for $f:\mathcal{M}_g\to\mathbb{R}$2-tangles, where tangles are either simple closed geodesics of length $f:\mathcal{M}_g\to\mathbb{R}$3 or hyperbolic pairs of pants with all boundary lengths $f:\mathcal{M}_g\to\mathbb{R}$4. There exists a unique function $f:\mathcal{M}_g\to\mathbb{R}$5 such that

$f:\mathcal{M}_g\to\mathbb{R}$6

for every $f:\mathcal{M}_g\to\mathbb{R}$7 filled by tangles, and consequently

$f:\mathcal{M}_g\to\mathbb{R}$8

The function $f:\mathcal{M}_g\to\mathbb{R}$9 is multiplicative over components, and the inversion reorganizes conditioning by the filled subsurface rather than by an inclusion–exclusion over individual bad patterns (Anantharaman et al., 2024).

The same work proves that after excluding tangles, the number of local topological types of periodic geodesics of length $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$0 grows only polynomially in $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$1: $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$2 On tangled surfaces, by contrast, the growth is exponential. This sharply reduces combinatorial proliferation in trace-formula and counting arguments (Anantharaman et al., 2024).

4. Spectral theory, Benjamini–Schramm convergence, and spectral gap regimes

For closed surfaces of large genus, the local geometry of a typical Weil–Petersson random surface converges to that of $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$3 in the Benjamini–Schramm sense. More precisely, for suitable good sets of moduli space of probability tending to $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$4, one obtains quantitative control on the volume of the thin part, and from this one derives convergence of spectral measures to the Plancherel measure of $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$5, a uniform Weyl law, and bounds on multiplicities and small eigenvalues (Monk, 2020).

At the level of the first non-zero Laplace eigenvalue $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$6, the large-genus closed model exhibits near-optimal spectral expansion. A program initiated by Wu–Xue, Lipnowski–Wright, and then Anantharaman–Monk shows that for arbitrarily small $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$7,

$$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$8

so with probability tending to $$\mathbb{E}_{WP}[f]=\frac{1}{V_g}\int_{\mathcal{M}_g} f(X)\,dX.$$9, the spectral gap is arbitrarily close to $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$0 (Anantharaman et al., 2024). This was strengthened to a polynomial-rate statement: there exists $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$1 such that a Weil–Petersson random closed surface satisfies

$d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$2

with probability tending to $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$3 as $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$4. The proof adapts the polynomial method for strong convergence of random matrices, together with Selberg trace formula estimates and new effective expansions for ratios of Weil–Petersson volumes (Hide et al., 20 Aug 2025).

The cusp and many-cusp regimes behave differently. If $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$5 with $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$6, then for any $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$7, a generic finite-area genus-$d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$8 hyperbolic surface with $d\mu_{WP}=\frac{dV_{WP}}{V_{g,n}(L_1,\ldots,L_n)}$9 cusps has no non-zero eigenvalue of the Laplacian below

$g$00

as $g$01 (Hide, 2021). By contrast, for fixed genus $g$02 and $g$03, a random surface in $g$04 has a linear number of small Laplacian eigenvalues with high probability, and this is described as optimal up to a multiplicative constant by the theorem of Otal and Rosas (Hide et al., 2023).

These statements are not contradictory; they concern different asymptotic regimes. A plausible implication is that the phrase “typical spectral behavior” in the Weil–Petersson model is meaningful only after specifying whether complexity is driven by genus, by cusp number, or by long boundary lengths.

5. Long boundaries, asymptotic Weil–Petersson form, and ribbon graph models

For moduli spaces with geodesic boundary, the long-boundary regime has a distinct asymptotic description. If $g$05 and $g$06, then a hyperbolic surface with large boundary lengths resembles a graph after appropriate rescaling. More precisely, for a metric ribbon graph $g$07, the associated hyperbolic surface $g$08 satisfies

$g$09

in the Gromov–Hausdorff sense, and the normalized Weil–Petersson form converges to Kontsevich’s piecewise linear form: $g$10 on the combinatorial moduli space of metric ribbon graphs (Do, 2010).

This correspondence is realized through the Bowditch–Epstein spine construction. In recent work on hyperbolic surfaces with long boundaries, the spine map $g$11 is used to compare the critical exponent random variable $g$12 on $g$13 with the graph-theoretic critical exponent $g$14 on the moduli space of metric ribbon graphs equipped with the normalized Kontsevich measure $g$15. In the regime where all boundary lengths are uniformly comparable and tend to infinity, one has convergence in mean and in Wasserstein distance: $g$16 On the “great” locus where the spine is trivalent and all edge-lengths are large, the Radon–Nikodym derivative satisfies

$g$17

with $g$18 the minimal edge length (Talbott, 14 Jan 2025).

The many-cusp regime admits a different asymptotic formula. For fixed genus $g$19 and fixed number $g$20 of geodesic boundaries, as $g$21,

$g$22

where $g$23 is the modified Bessel function and $g$24 is the first positive zero of $g$25. This Bessel-function asymptotic governs expected counts of short separating curves and underlies the spectral and geodesic-statistical results for surfaces with many cusps (Hide et al., 2023).

6. Genus-zero tree bijections, punctured spheres, and scaling limits

In genus $g$26, Weil–Petersson random surfaces admit explicit combinatorial encodings by trees. For random punctured spheres $g$27, sampled according to the normalized Weil–Petersson measure, one has

$g$28

by Gauss–Bonnet. The Penner–Epstein–Bowditch decomposition yields a bijective encoding by a plane cubic tree with continuous angle labels, and the Weil–Petersson measure pushes forward to Lebesgue measure on the corresponding labeled-tree space (Budd et al., 26 Aug 2025).

This encoding leads to two distinct asymptotic geometries. Without rescaling, $g$29 converges in distribution in the local Benjamini–Schramm sense to a random infinite-volume hyperbolic surface with countably many punctures, homeomorphic to $g$30. After rescaling distances by $g$31, the compact core converges in the Gromov–Hausdorff topology to the Brownian sphere: $g$32 The proof mirrors techniques from random planar maps and reduces, via a sequence of transformations, to a model of single-type Galton–Watson trees (Budd et al., 26 Aug 2025).

Related tree-bijection results provide exact formulas for Weil–Petersson volumes and metric statistics in planar moduli spaces. For genus-$g$33 surfaces with one distinguished cusp and geodesic boundaries, a spine-tree bijection gives a combinatorial proof that the Weil–Petersson volume is a polynomial in the boundary lengths and yields an explicit distance-dependent three-point function: $g$34 For cusp-less planar hyperbolic surfaces, the bijection extends by decomposing a general surface into two half-tight cylinders and replacing cusp-distance by the Busemann function

$g$35

The stated applications include computation of Weil–Petersson volumes, access to distance statistics on random hyperbolic surfaces, and scaling limits when the number of boundaries becomes large (Budd et al., 10 Dec 2025, Zonneveld, 11 Dec 2025).

Taken together, these genus-$g$36 results show that the Weil–Petersson random-surface formalism is not restricted to asymptotic counting in high genus. It also supports exact bijective descriptions, explicit metric observables, and continuum scaling limits closely parallel to those of random planar maps.