The universal constant and the critical many-markings regime in the Mirzakhani-Zograf large-genus asymptotics of Weil-Petersson volumes

Mirzakhani and Zograf proved that the Weil-Petersson volume $V_{g,n}$ of the moduli space of genus-$g$ curves with $n$ marked points admits a complete large-genus expansion $V_{g,n}=C\,\frac{(2g-3+n)!\,(4\pi^{2})^{2g-3+n}}{\sqrt{g}}\left(1+\frac{c_n^{(1)}}{g}+\frac{c_n^{(2)}}{g^{2}}+\cdots\right)$ with a universal constant $C\in(0,\infty)$ whose exact value their method cannot determine, although numerics and the physics of Jackiw-Teitelboim gravity (a two-dimensional quantum-gravity model whose genus expansion is governed by these volumes) both point to $C=1/\sqrt{\pi}$ (Zograf's conjecture). They conjectured that the coefficients lie in $\mathbb{Q}[\pi^{-2}]$ - resolved affirmatively by Huang in 2025 - and proved uniformity in the number of markings only in the subcritical regime $n(g)^{2}/g\to 0$. Two source questions remain open: (1) determine $C$ rigorously (in particular, prove or refute $C=1/\sqrt{\pi}$); (2) describe the large-genus asymptotics of $V_{g,n(g)}$ in the critical regime $n(g)\asymp\sqrt{g}$, where the fixed-$n$ expansion degenerates because the first correction $c_n^{(1)}$ is quadratic in $n$, and at supercritical growth of $n(g)$. The two deliverables share one normalization - the same $C$ multiplies the leading term in every regime - and one common objective: completing the leading large-genus theory of Weil-Petersson volumes, a load-bearing input for counting simple closed geodesics on random hyperbolic surfaces, for the spectral geometry of surfaces with many cusps, and for the genus expansion of Jackiw-Teitelboim gravity.

Background

The moduli space $\mathcal{M}_{g,n}$ parametrizes smooth compact Riemann surfaces of genus $g$ with $n$ marked points, up to isomorphism; it is an orbifold of complex dimension $3g-3+n$. It carries the Weil-Petersson metric, the natural $L^{2}$ Riemannian metric on infinitesimal deformations of the complex structure, and the total volume of $\mathcal{M}_{g,n}$ for the associated symplectic form is the Weil-Petersson volume $V_{g,n}$. Mirzakhani's integration recursion computes $V_{g,n}$ exactly (it is a rational multiple of a power of $\pi^{2}$), and with geodesic boundary components of prescribed lengths $x_1,\dots,x_n$ the volume becomes an explicit polynomial $V_{g,n}(x_1,\dots,x_n)$. Intersection-theoretically, on the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$ of stable nodal curves (curves allowed pairs of identified points, subject to a stability condition) the Weil-Petersson symplectic form represents $2\pi^{2}\kappa_1$ in cohomology, where $\kappa_1$ is the first Mumford kappa class - a tautological characteristic class defined as the pushforward of the square of a psi class along the universal curve - so $V_{g,n}$ is a top self-intersection number of $\kappa_1$. The psi classes $\psi_i=c_1(\mathcal{L}_i)$ are the first Chern classes of the cotangent line bundles $\mathcal{L}_i$ (the fiber of $\mathcal{L}_i$ over a curve is the cotangent line at its $i$-th marking), and their intersection numbers $\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g=\int_{\overline{\mathcal{M}}_{g,n}}\prod_i\psi_i^{d_i}$, the Witten-Kontsevich intersection numbers, are governed by the Virasoro constraints, an explicit recursive system of linear relations equivalent to Witten's conjecture as proved by Kontsevich.

Mirzakhani initiated the large-genus study of $V_{g,n}$, deriving asymptotic estimates used to analyze random hyperbolic surfaces (surfaces sampled by the Weil-Petersson measure), such as their Cheeger constants (an isoperimetric quantity controlling the spectral gap of the Laplacian) and their shortest geodesics of each combinatorial type. Mirzakhani and Zograf then proved the structural theorem of the field: for every fixed $n\ge 0$ and $k\ge 1$,

$$V_{g,n}=C\,\frac{(2g-3+n)!\,(4\pi^{2})^{2g-3+n}}{\sqrt{g}}\left(1+\frac{c_n^{(1)}}{g}+\cdots+\frac{c_n^{(k)}}{g^{k}}+O(g^{-k-1})\right)$$

as $g\to\infty$, where $C\in(0,\infty)$ is universal (independent of $n$ and $g$) and each $c_n^{(i)}$ is an effectively computable polynomial in $n$ of degree $2i$ with coefficients in $\mathbb{Q}[\pi^{-2},\pi^{2}]$. Their method, a Laplace-type analysis of the stable-graph recursion (stable graphs are the dual graphs of nodal curves, along which intersection numbers decompose into products of vertex factors), proves that $C$ exists but, as they state explicitly, provides no information about its exact value. Zograf conjectured from high-accuracy numerics the precise value $C=1/\sqrt{\pi}$; the same numerical data suggested that the coefficients $c_n^{(i)}$ in fact lie in the smaller ring $\mathbb{Q}[\pi^{-2}]$, involving no positive powers of $\pi^{2}$. For varying $n$, Mirzakhani-Zograf proved uniformity only in the subcritical regime: if $n(g)^{2}/g\to 0$ then $V_{g,n(g)}\sim\frac{C}{\sqrt{g}}(2g-3+n(g))!\,(4\pi^{2})^{2g-3+n(g)}$ with relative error $O(n(g)^{2}/g)$, remarking that the case of arbitrary $(g,n)$ "seems to be more complicated". Since $c_n^{(1)}$ is quadratic in $n$, the expansion degenerates exactly at the critical scale $n\asymp\sqrt{g}$.

A parallel program treats large-genus asymptotics of the psi-class numbers themselves. Delecroix-Goujard-Zograf-Zorich conjectured, and Aggarwal proved for $n=o(\sqrt{g})$, uniform leading asymptotics of suitably normalized correlators; Aggarwal also showed that this normalization genuinely fails at $n=O(\sqrt{g})$, with the most degenerate normalized correlator growing like $\exp(n^{2}/12g)$. Guo and Yang re-proved the uniform leading asymptotics via an explicit formula for the $n$-point generating function and formulated polynomiality conjectures for the subleading coefficients, which Eynard, Garcia-Failde, Giacchetto, Gregori and Lewanski established by resurgent analysis (a technique that extracts large-order asymptotics of the coefficients from the divergence structure of their generating series) of the $n$-point functions; Guo, Yang and Zagier recently obtained completely uniform leading asymptotics with the explicit constant $1/\pi$ for primitive psi-intersection numbers, including a critical-regime profile $\pi^{-1}e^{-k^{2}/30g}$ for a family with $k$ zero-degree insertions at $k=O(\sqrt{g})$. These results concern psi-correlators, not volumes: $V_{g,n}$ is a $\kappa_1$-intersection number, an exponentially large stable-graph sum of products of psi-numbers, and passing between the two requires a one-vertex-dominance analysis of that sum (a proof that stable graphs with a single vertex carry the entire leading contribution). For Masur-Veech volumes (volumes of strata of quadratic differentials, that is, of moduli spaces of pairs consisting of a curve and a meromorphic quadratic differential, which Delecroix-Goujard-Zograf-Zorich expressed as psi-intersection sums), Aggarwal carried out this analysis and determined the precise leading asymptotics with explicit constants; the analogous computation for Weil-Petersson volumes, which would determine $C$, has not been done.

The constant $C$ is also a benchmark in physics. In Jackiw-Teitelboim (JT) gravity - a two-dimensional theory of gravity with an auxiliary scalar (dilaton) field, whose path integral over geometries reduces to integrals of $V_{g,n}$ against boundary data - Saad, Shenker and Stanford showed that the theory is a double-scaled random-matrix integral (a matrix model in a scaling limit in which surfaces of each genus contribute one order of the expansion) and, by Borel-resumming its genus expansion (reconstructing a function from its divergent asymptotic series), recovered Zograf's conjectured asymptotics non-rigorously; Kimura conjectured the full large-genus asymptotics of $V_{g,n}(L_1,\dots,L_n)$ with explicit constants, whose zero-boundary-length specialization is Zograf's conjecture. For super-Weil-Petersson volumes (the supergeometric analogues of $V_{g,n}$ arising in JT supergravity), Griguolo, Papalini, Russo and Seminara proposed analogous large-genus formulae with the explicit constant $\pi^{-9/2}$.

Recent work has resolved part of the source problem while leaving its core untouched. Huang proved the coefficient-ring conjecture: all coefficients of the Mirzakhani-Zograf expansions of $V_{g,n}$ and of the mixed psi/kappa intersection numbers lie in $\mathbb{Q}[\pi^{-2}]$ (or $\mathbb{Q}[n,\pi^{-2}]$); the same work confirms Kimura's conjecture for $V_{g,n}(L)$ at fixed $n$ and boundary lengths $L_i=o(\sqrt{g})$ only up to the same undetermined constant $C$, and a companion paper proves the super-volume asymptotics up to a universal constant conjecturally equal to $\pi^{-9/2}$. At fixed $n$, Anantharaman and Monk derived the full high-genus expansion of the volume polynomials $V_{g,n}(x)$, with $C$ carried through as an unknown prefactor, and Guillermit computed the exact asymptotic ratio $V_{g,n}(x)/V_{g,n}$ for boundary lengths $x_j\to\infty$ with $x_j=O(\sqrt{g})$ - a critical regime in the boundary lengths at fixed marking number, in which $C$ cancels. Where $n$ grows with $g$, rigorous volume results stop short of the critical scale: Shen and Wu proved a one-sided volume bound valid for arbitrary $(g,n)$ and two-sided bounds with unknown multiplicative constants for $n(g)\prec\sqrt{g}$, leaving open a ratio question relevant to random surfaces with many cusps; Hide and Thomas treat the complementary large-$n$ limit with a fixed number of boundary components.

Problem Statement

Two linked questions from Mirzakhani-Zograf's large-genus theory of Weil-Petersson volumes remain open (conventions: $g\to\infty$; $V_{g,n}$ is the Weil-Petersson volume of the moduli space $\mathcal{M}_{g,n}$, as defined in the Background; the normalization is that of Theorem 1.2 of Mirzakhani-Zograf, Geom. Funct. Anal. 25 (2015), reproduced explicitly in the display below).

(1) The universal constant. Determine the exact value of the universal constant $C$ in

$$V_{g,n}=C\,\frac{(2g-3+n)!\,(4\pi^{2})^{2g-3+n}}{\sqrt{g}}\left(1+\frac{c_n^{(1)}}{g}+\frac{c_n^{(2)}}{g^{2}}+\cdots\right),$$

which Mirzakhani-Zograf proved to exist, to be positive, and to be independent of $n$, but which their method cannot evaluate; equivalently, evaluate the limit $\lim_{g\to\infty}\frac{\sqrt{g}\,V_{g,n}}{(2g-3+n)!\,(4\pi^{2})^{2g-3+n}}$ for any fixed $n\ge 0$ (the limit exists and is $n$-independent by the source theorem). In particular, prove or disprove Zograf's conjecture $C=1/\sqrt{\pi}$.

(2) The critical and supercritical growing-markings regimes. Describe the large-genus asymptotics of $V_{g,n(g)}$ along sequences $n(g)$ with $n(g)\asymp\sqrt{g}$, i.e. $n(g)/\sqrt{g}$ bounded above and below by positive constants (critical, where the fixed-$n$ expansion degenerates because the first correction $c_n^{(1)}$ is quadratic in $n$), and along supercritical sequences $\sqrt{g}\prec n(g)$, i.e. $n(g)/\sqrt{g}\to\infty$: determine the leading-order behavior, with the constant $C$ of (1) identified wherever it appears. The source theorem proves $C$ independent of $n$ but, as noted above, establishes uniformity only in the subcritical regime $n(g)^{2}/g\to 0$. That the same $C$ governs the critical and supercritical regimes is therefore part of what (2) must establish rather than a source fact; only once it is established does evaluating $C$ in these regimes also answer (1) for fixed $n$, which otherwise remains a separate requirement.

The common objective is the completion of the leading large-genus theory of Weil-Petersson volumes: both deliverables are normalizations of the same expansion, the same constant $C$ multiplies the leading term in every regime, and a critical-regime theorem must be consistent with the subcritical limit already proved (Theorem 1.8 of the source). The third open issue of the source - whether the coefficients $c_n^{(i)}$ involve positive powers of $\pi^{2}$ - has been resolved (Huang proved the coefficients lie in $\mathbb{Q}[\pi^{-2}]$) and is not part of the open core.

References

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