Universality of the bottom of the length spectrum: finding an a priori mechanism relating Weil-Petersson and combinatorial models of random surfaces

For a random closed hyperbolic surface of large genus, sampled from the moduli space with the Weil-Petersson probability measure, Mirzakhani and Petri (2019) proved that the numbers of primitive closed geodesics with lengths in fixed disjoint intervals converge jointly, as the genus tends to infinity, to independent Poisson random variables with explicit mean given by the integral of $(\cosh t - 1)/t$, and deduced that the expected systole converges to $1.61498\ldots$. For the Brooks-Makover combinatorial model (uniform gluing of ideal triangles), the corresponding short-curve counts are also known to be asymptotically Poisson, with a slightly different mean, and Mirzakhani and Petri noted that the two very different models produce remarkably similar statistics even though 'no a priori reason for this is known'. Later work sharpened the coincidence into an exact matching of Poisson intensities for high-genus unicellular maps (Janson-Louf, 2023) and for multi-faced metric ribbon graphs (Barazer-Giacchetto-Liu, 2025), posed it as an explicit universality conjecture (a Gromov-Hausdorff coupling between the two models), and began developing a candidate mechanism via the Bowditch-Epstein spine construction and the asymptotic equivalence of the Weil-Petersson and Kontsevich measures (Talbott, 2026). The problem of establishing a rigorous a priori mechanism explaining the agreement - or proving that no such universality can hold - remains open.

Background

A closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric (a complete Riemannian metric of constant curvature $-1$), unique up to isometry once the conformal structure is fixed. The set of isometry classes of such surfaces is the moduli space $\mathcal{M}_g$, and it carries a natural symplectic volume form, the Weil-Petersson volume form, which can be normalized to a probability measure; sampling from it is the standard meaning of a 'random hyperbolic surface of genus $g$'. This model was introduced and developed in work of Mirzakhani, beginning with her computation of Weil-Petersson volumes and her study of random hyperbolic surfaces of large genus (Journal of Differential Geometry, 2013). The length spectrum of a hyperbolic surface $X$ is the multiset of lengths of its primitive closed geodesics (geodesics that traverse their image exactly once); its bottom part - the short geodesics - governs geometric quantities such as the systole $\mathrm{sys}(X)$, the length of the shortest closed geodesic.

Mirzakhani and Petri (Commentarii Mathematici Helvetici, 2019) proved that for disjoint intervals $[a_1,b_1],\ldots,[a_k,b_k] \subset \mathbb{R}_+$, the counts of primitive closed geodesics of a Weil-Petersson random surface $X \in \mathcal{M}_g$ with lengths in these intervals converge jointly, as $g \to \infty$, to independent Poisson random variables (a counting statistic is Poisson when the counted events are rare and approximately independent) with means

$$\lambda_{[a,b]} = \int_a^b \frac{\cosh t - 1}{t}\,dt,$$

and deduced that $\mathbb{E}_g[\mathrm{sys}] \to \int_0^\infty e^{-\lambda_{[0,R]}}\,dR = 1.61498\ldots$. Their proof combines the method of moments (Poisson convergence follows from convergence of factorial moments) with Mirzakhani's integration formula on moduli space and large-genus asymptotics of Weil-Petersson volumes.

In the same paper the authors compare this with the Brooks-Makover model, a purely combinatorial construction of random surfaces introduced in 2004: glue an even number $2N$ of ideal hyperbolic triangles together uniformly at random along their sides without shear and conformally compactify the result. The surfaces obtained are precisely the Belyi surfaces (compact Riemann surfaces admitting a meromorphic function with at most three critical values), which are dense in every $\mathcal{M}_g$, and the expected genus of the result is asymptotic to $N/2$. Despite the completely different notion of randomness, the known statistics of these two models are strikingly parallel: the number of short curves on a random Brooks-Makover surface is also asymptotically Poisson distributed (Petri, Journal of Topology, 2017; Petri and Thäle, Indiana University Mathematics Journal, 2018, with Chen-Stein error bounds - the Chen-Stein method measures total-variation distance to a Poisson law in terms of dependency structure), and the expected systole also converges to a constant as $N \to \infty$. The parallel is not literal: both the Poisson means and the systole limits differ slightly between the models, and Brooks-Makover surfaces asymptotically concentrate in the $b$-thick part of moduli space (the region where the systole is at least $b$, with $b = 2\cosh^{-1}(3/2)$, i.e. $\mathbb{P}_N[\mathrm{sys} \geq b] \to 1$), whereas for Weil-Petersson surfaces $\mathbb{P}_g[\mathrm{sys} \geq b] \to e^{-\lambda_{[0,b]}} \approx 0.339$. Mirzakhani and Petri state that 'many of the known results are very similar, even though no a priori reason for this is known'.

Subsequent work made the coincidence much sharper. A combinatorial map is a graph embedded in a surface, cutting it into faces (a unicellular map has one face; its 2-core is the maximal subgraph in which every vertex has degree at least 2). Janson and Louf (Annals of Probability, 2023) proved that in a uniform unicellular map of genus $g$ and size $n$ with $g = o(n)$, the lengths of short simple cycles, rescaled by $\sqrt{n/12g}$, converge to a Poisson point process with exactly the intensity $(\cosh t - 1)/t$ of the Weil-Petersson limit law, and they conjectured that the two models are 'somehow the same' in the limit, in the sense of a coupling realizing Gromov-Hausdorff convergence (a distance between abstract compact metric spaces) between the rescaled 2-core of the map and the random hyperbolic surface. Barazer, Giacchetto and Liu (Forum of Mathematics, Sigma, 2025) extended this to multi-faced metric ribbon graphs (a ribbon graph is a graph with a cyclic order of the edges at each vertex, equivalent to a graph thickened to an oriented surface; a metric ribbon graph carries positive edge lengths), using a combinatorial Teichmuller theory developed by Andersen, Borot, Charbonnier, Giacchetto, Lewanski and Wheeler that transports Mirzakhani's integration machinery to the combinatorial setting; they recover the same Poisson point process with the same intensity, call the exact matching of intensity functions 'somehow miraculous', and note that the naive combinatorial analogue fails when all closed curves (rather than just cycles) are counted, because metric ribbon graphs have zero curvature and can be rescaled.

The leading candidate for the missing a priori mechanism is the spine construction: to a hyperbolic surface with boundary one associates its spine, a metric ribbon graph (the natural triangulation of Bowditch and Epstein), and this identifies a space of hyperbolic surfaces with a space of metric ribbon graphs carrying the Kontsevich symplectic form (the piecewise-linear two-form governing the combinatorial moduli space of ribbon graphs). Do (2010) and Mondello (2009) proved that the Weil-Petersson form converges to the Kontsevich form in the long-boundary regime, and Talbott (Advances in Mathematics, 2026) proved quantitative uniform estimates showing that the normalized Kontsevich measure approximately pulls back to the Weil-Petersson measure in that regime, together with a convergence result for critical exponents (the exponential decay rate of the orbital counting function). However, none of these results covers the closed-surface, large-genus regime in which the length-spectrum coincidence lives, and none derives the Poisson limit laws. The question of whether a rigorous deeper connection - a universality class encompassing both model families - exists is therefore open, and it is now backed by several concrete open targets in the literature.

Problem Statement

Is there a rigorous a priori mechanism that explains why the bottom of the length spectrum of random hyperbolic surfaces has the same asymptotic statistics in the Weil-Petersson model and in combinatorial models of random surfaces, or is the agreement coincidental?

The facts to be explained are: (i) for Weil-Petersson random closed surfaces of genus $g \to \infty$, the counts of primitive closed geodesics with lengths in fixed disjoint intervals converge jointly to independent Poisson random variables with mean

$$\lambda_{[a,b]} = \int_a^b \frac{\cosh t - 1}{t}\,dt,$$

and the expected systole converges to $\int_0^\infty e^{-\lambda_{[0,R]}}\,dR = 1.61498\ldots$ (Mirzakhani-Petri, 2019); (ii) for random surfaces in the Brooks-Makover model, the short-curve counts are also asymptotically Poisson, with a slightly different mean, and the expected systole also converges to a constant (Petri, 2017; Petri-Thäle, 2018); and (iii) for high-genus unicellular maps in the regime $g = o(n)$ with lengths rescaled by $\sqrt{n/12g}$ (Janson-Louf, 2023) and for multi-faced metric ribbon graphs with boundary lengths of order $g$ (Barazer-Giacchetto-Liu, 2025), the rescaled length spectrum converges to a Poisson point process with exactly the intensity $(\cosh t - 1)/t$.

An a priori mechanism explaining the agreement would be a precise mathematical connection between the model families from which the parallel limit laws follow as corollaries. The connection must be established in a regime covering the large-genus limit, must reproduce the known Poisson intensities (including the rescaling factor), and must account for the observed discrepancies: the slightly different Poisson mean and systole constant in the Brooks-Makover model, its concentration in the $b$-thick part ($\mathbb{P}_N[\mathrm{sys} \geq 2\cosh^{-1}(3/2)] \to 1$ versus $\mathbb{P}_g[\mathrm{sys} \geq 2\cosh^{-1}(3/2)] \to e^{-\lambda_{[0,b]}} \approx 0.339$ for Weil-Petersson surfaces), and the failure of the naive closed-curve spectrum on metric ribbon graphs.

Alternatively, a rigorous refutation of the universality would prove that the parallel limit laws are coincidental or non-transferable, exhibiting a genuine discrepancy between the model families in a statistic in which they are asserted to agree, rather than one of the differences on which they are already known to disagree (the Brooks-Makover Poisson mean and systole constant, or its concentration in the $b$-thick part).

References

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