WP Random Hyperbolic Surfaces

• Weil–Petersson random hyperbolic surfaces are closed or bordered oriented hyperbolic surfaces sampled from moduli space using the WP measure.
• They provide a rigorous probabilistic framework to analyze asymptotic geometric, spectral, and topological invariants, linking moduli space, intersection theory, and combinatorial models.
• The study employs tools like the Selberg trace formula, asymptotic WP volume growth, and random matrix techniques to reveal universal properties in high-genus limits.

Weil–Petersson random hyperbolic surfaces are closed or bordered oriented hyperbolic surfaces sampled from moduli space with respect to the probability measure arising from the Weil–Petersson (WP) metric. This framework places the detailed geometry, topology, and spectral theory of “typical” high-genus hyperbolic surfaces on rigorous probabilistic footing, facilitating asymptotic, average-case, and extremal analysis of their geometric invariants. The structure of the WP measure and its interplay with moduli space topology, intersection theory, and the combinatorial models of ribbon graphs underpin calculations of random geometric and spectral quantities, drawing deep connections to both algebraic geometry and mathematical physics.

1. Weil–Petersson Measure and the Random Surface Model

The moduli space $\mathcal{M}_{g,n}$ of hyperbolic surfaces of genus $g$ with $n$ boundary components or cusps is equipped with a symplectic structure—Wolpert’s Weil–Petersson (WP) form—given in Fenchel–Nielsen coordinates by

$$\omega_{\mathrm{WP}} = \sum_{k=1}^{3g-3+n} d\ell_k \wedge d\tau_k$$

where $\ell_k$ are lengths of a pants decomposition and $\tau_k$ are the corresponding twist parameters (Do, 2011). The associated volume form,

$$\mathrm{vol}_{\mathrm{WP}} = \frac{1}{(3g-3+n)!}\omega_{\mathrm{WP}}^{3g-3+n}$$

after normalization, provides a probability measure on moduli space. Sampling a random hyperbolic surface with respect to the WP measure enables one to formulate and answer probabilistic questions about lengths of geodesics, spectral properties, or decomposition complexity (Mirzakhani et al., 2017, Monk, 2020).

The WP measure exhibits strong invariance properties under the mapping class group and is foundational for quantitative integration over $\mathcal{M}_{g,n}$ (Do, 2011). Mirzakhani's integration formula links geometric statistics of random surfaces to intersection numbers and volumes of lower-dimensional moduli spaces (Do, 2011), enabling recursive and asymptotic analysis.

2. Asymptotic Geometric Properties and Volume Growth

Weil–Petersson volumes exhibit precise super-exponential growth with genus: $V_{g,n} \sim (472)^{2g+n-3}(2g-3+n)!$ with explicit factorial and exponential prefactors, and satisfy stable recursion relations: $$V_{g,n+1} = (1 + O(1/g))\, 2g\, V_{g,n}, \quad V_{g,n}/V_{g-1, n+2} = 1 + O(1/g)$$ (Mirzakhani, 2010). The WP volume’s dependence on the boundary lengths is polynomial (Do, 2011), with coefficients explicitly encoding intersection numbers of tautological classes on the Deligne–Mumford compactification.

Sampling random surfaces by the WP measure and exploiting the volume asymptotics, one deduces robust large-genus geometric phenomena:

• The length of a shortest nonseparating simple closed geodesic stays bounded away from zero with fixed positive probability, while the shortest separating geodesic length grows as $2\log g$ with lower order corrections (Mirzakhani, 2010, Nie et al., 2020).
• The Cheeger constant $h(X)$ of a random surface is uniformly bounded below, $h(X) > \log 2$ with high probability, ensuring robust isoperimetric properties (Mirzakhani, 2010).

3. Models and Large-Scale Topology: From Graphs to Surfaces

A pivotal link between the geometry of random hyperbolic surfaces and combinatorial models is provided by the correspondence with decorated ribbon graphs. As the boundary lengths become large, the WP form converges to the piecewise linear form first defined combinatorially by Kontsevich (Do, 2010). The mapping from hyperbolic surfaces to metric ribbon graphs (Bowditch–Epstein spine construction) preserves quantitative properties asymptotically—for instance, the WP measure pushforward to the graph moduli space approximates the Kontsevich measure with explicit uniform error estimates (Talbott, 14 Jan 2025).

Intersection numbers of psi and kappa classes are computable via volumes of ribbon graph cells and associated Laplace transforms of polytope volumes. For combinatorial cycles defined by prescribed vertex degree distributions (Witten cycles), their dual cohomology classes are canonical and the associated intersection numbers are equivariant under these graph–surface correspondences (Do, 2010).

With the addition of a $\beta$-irreducibility constraint (all contractible cycles of a metric map have length at least $\beta$), volumes of irreducible maps for $\beta = 2\pi$ match the WP volumes of hyperbolic surfaces with boundary in the genus $0,1$ cases, and otherwise provide closely related generating functions; both families of volumes satisfy identical string and dilaton equations (Budd, 2020).

4. Spectral Theory and Statistical Mechanics of Random Surfaces

WP random surfaces are "locally" modeled on the hyperbolic plane: as genus grows, the proportion of points with large injectivity radius approaches $1$ (Benjamini–Schramm convergence), and the spectral measure of the Laplace–Beltrami operator converges weakly to that of $\mathbb{H}$ (Monk, 2020). Key statistical features include:

• For disjoint intervals $[a_i,b_i]$ in the length spectrum, the number of closed geodesics of length in $[a_i,b_i]$ jointly converge in law to independent Poisson random variables, with explicit asymptotics for the means

$$\lambda_{[a,b]} = \int_a^b \frac{e^t + e^{-t} - 2}{2t}dt$$

(Mirzakhani et al., 2017). The expected systole converges to a universal constant $\approx 1.61498$ as $g \to \infty$.

• The WP random surface exhibits a spectral gap: almost surely, the first nonzero Laplacian eigenvalue $\lambda_1$ satisfies

$$\lambda_1 \geq \frac{1}{4} - O\left(\frac{1}{g^c}\right)$$

for some $c>0$ (Hide et al., 20 Aug 2025). In particular, for all $\alpha > 0$, $$\mathbb{P}_{\mathrm{WP}}(\lambda_1 < 1/4 - \alpha^2) \to 0$$ as $g \to \infty$, achieving near-maximal "Ramanujan" (optimal) spectral properties (Anantharaman et al., 19 Mar 2024).

For surfaces with cusps, provided the number of cusps $n = O(g^\alpha)$ with $\alpha < 1/2$, there is a uniform positive lower bound on $\lambda_1$ (Hide, 2021).

Techniques

The main analytical tools are the Selberg trace formula with carefully chosen test functions, asymptotic expansions for WP volumes and intersection numbers, and probabilistic estimates via Markov's inequality and concentration of measure. The polynomial method from random matrix theory has been adapted to derive quantitative polynomial rates for the spectral gap (Hide et al., 20 Aug 2025). Conditioning on geometric regularity (e.g., $L$-tangle-free property) using Möbius inversion further refines control and removes rare, pathological configurations (Monk et al., 2020, Anantharaman et al., 3 Jan 2024).

5. Large-Scale Structure, Complexity, and Conditioning

Random WP surfaces are generically $L$-tangle-free for any $L = a\log g$ ($0 < a < 1$), meaning that every pair of pants or one-holed torus has total boundary length $> 2L$ (Monk et al., 2020). This property implies that all closed geodesics below length $L$ are simple, disjoint, and embedded in wide, disjoint collars of width at least $L/4$, and enables polynomial, rather than exponential, proliferation of local topological types for geodesics of length less than $L$. Conditioning on this property via an explicit Möbius inversion formula allows systematic integration against the WP measure (Anantharaman et al., 3 Jan 2024).

Such "tangle-free" conditioning, originally motivated by analogous concepts in random regular graphs, is crucial for controlling the combinatorics of geodesic distributions and allows efficient calculation of statistical properties of random surfaces even when integrating discontinuous or singular observables (e.g., counts of simple closed geodesics or spectrally "bad" regions).

6. Systolic, Intersection, and Extremal Invariants

Key invariants of WP random surfaces have been analyzed to high precision:

• The non-simple systole (length of the shortest non-simple closed geodesic) behaves asymptotically as $\log g$ (He et al., 2023).
• The minimal separating systole concentrates near $2\log g - 4\log\log g$ up to $o(1)$ corrections (Nie et al., 2020).
• The expected number of small Laplacian eigenvalues on WP random punctured surfaces grows linearly with the number of cusps (Hide et al., 2023).
• The distribution of the critical exponent of the graph zeta function and that of the corresponding hyperbolic surface converge in the long-boundary limit, under explicit uniform control via measures and local coordinates (Talbott, 14 Jan 2025).

Intersection numbers on $\overline{\mathcal{M}}_{g,n}$, including psi and kappa classes, are obtained as coefficients of the polynomial expansion of the WP volumes in squared boundary lengths (Do, 2011). In the asymptotic regime where boundary lengths become large, the WP symplectic form and intersection theory converge to a piecewise linear, combinatorial model reinterpreted in terms of metric ribbon graphs (Do, 2010).

7. Connections to Random Matrix Theory and Quantum Gravity

A precise duality links the geometry of WP random surfaces to Jackiw–Teitelboim (JT) quantum gravity and double-scaled random matrix ensembles (Weber et al., 2022). Universal correlation constraints in random matrix theory (RMT) place strong, explicit linear relations on the coefficients of the WP volume polynomials, and provide external validation and simplification for recursive computations. Such connections strengthen the understanding that the moduli space of hyperbolic surfaces encodes, and is subjected to, universal spectral dynamics observed in quantum chaos.

This duality is realized concretely in the expansion of genus-$g$, $n$-boundary correlation functions, with spectral universality mandating stringent cancellation constraints among volume coefficients. The long-term program is to use RMT universality as a guide to resolve structural and computational complexities in moduli space geometry, linking hyperbolic surfaces, intersection theory, and quantum gravitational models (Weber et al., 2022).

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The theory of Weil–Petersson random hyperbolic surfaces unifies deep aspects of hyperbolic geometry, combinatorics, probability, spectral theory, and mathematical physics. The asymptotic structure and fine statistical properties of these surfaces are now understood in terms of WP volume growth, intersection theory, and combinatorial models, with profound implications for universal spectral behavior and moduli space geometry in the high-genus limit.