Weil-Petersson volume asymptotics in the mixed regime of large genus and growing number of boundary components, and large-degree coefficients of the volume polynomials

This problem is the surviving open core of the 'Questions' section of Mirzakhani's 'Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus' (J. Differential Geom. 94 (2013), 267-300). Let $V_{g,n}$ be the Weil-Petersson volume of the moduli space $\mathcal M_{g,n}$ and $V_{g,1}(L)=\sum_{k=0}^{3g-2}\frac{a_{g,k}}{(2k+1)!}L^{2k}$ the one-boundary volume polynomial. The two one-parameter regimes are settled: fixed $n$ with $g\to\infty$ (Mirzakhani; Mirzakhani-Zograf, uniform for $n=o(\sqrt g)$) and fixed $g$ with $n\to\infty$ (Manin-Zograf). The open problem is the mixed regime where both parameters grow. (a) Volumes: the asymptotics of $B_{g,n}=V_{g,n}/V_{g-1,n+2}$ along $n=n(g)\to\infty$, and of the whole sequence $V_g,V_{g-1,2},\ldots,V_{0,2g}$; the related ratio $D_{g,n}:=V_{g,n}^2/(V_{g,n-1}V_{g,n+1})$ is known to tend to 1 only for $n=o(g)$, Shen-Wu's general bounds are $1/2-\pi^2/20\le\liminf D_{g,n}\le\limsup D_{g,n}\le1$, and the linear regime $n\sim\lambda g$ is untouched. (b) Coefficients: the asymptotics of $a_{g,k}/a_{g,k+1}$ for degree $k$ growing with $g$; the two ends are known (fixed $k$: ratio $\to1$; top degree: ratio $\to0$) and the middle $k\asymp g$ is open, even though the analogous pure psi-class intersection-number uniformity problem was recently solved completely. Both parts govern the spectral theory of Weil-Petersson-random surfaces with many cusps.

Background

The moduli spaces and their volumes. For $2g-2+n>0$, let $\mathcal M_{g,n}(L_1,\ldots,L_n)$ be the moduli space of complete hyperbolic surfaces of genus $g$ with $n$ geodesic boundary components of lengths $L_1,\ldots,L_n$ (cusps when $L_i=0$). It carries the Weil-Petersson symplectic form $\omega$, of finite total volume

$$V_{g,n}(L)=\frac{1}{(3g-3+n)!}\int_{\mathcal M_{g,n}(L)}\omega^{3g-3+n},$$

which is a polynomial in $L_1^2,\ldots,L_n^2$ of total degree $3g-3+n$ with rational coefficients in powers of $\pi$ (Mirzakhani). Write $V_{g,n}=V_{g,n}(0,\ldots,0)$ for the cusped volume. Mirzakhani's integration formula expresses averages of geometric statistics of hyperbolic surfaces in terms of these polynomials, so volume asymptotics are the standard input to the study of Weil-Petersson random surfaces — surfaces $X\in\mathcal M_{g,n}(L)$ sampled from the probability measure proportional to $\omega^{3g-3+n}$, the random model analyzed by the source paper.

The coefficients are intersection numbers. Following the source (Section 3), for $|\mathbf d|\le 3g-3+n$ with $d_0=3g-3+n-|\mathbf d|$ define the brackets

$$[\tau_{d_1},\ldots,\tau_{d_n}]_{g,n}=\Big(\prod_{i=1}^n(2d_i+1)!!\Big)2^{2|\mathbf d|}\frac{(2\pi^2)^{d_0}}{d_0!}\int_{\overline{\mathcal M}_{g,n}}\psi_1^{d_1}\cdots\psi_n^{d_n}\kappa_1^{d_0},$$

where $\psi_i$ is the psi class (the first Chern class of the cotangent-line bundle at the $i$-th marking) and $\kappa_1=\omega/(2\pi^2)$ is the first Mumford kappa class; the integral is over the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$. These mixed psi-kappa pairings are exactly the volume coefficients (source, Eq. (3.1)):

$$V_{g,n}(2L)=\sum_{|\mathbf d|\le 3g-3+n}[\tau_{d_1},\ldots,\tau_{d_n}]_{g,n}\prod_{i=1}^n\frac{L_i^{2d_i}}{(2d_i+1)!}.$$

For $n=1$ the source writes $V_{g,1}(L)=\sum_{k=0}^{3g-2}\frac{a_{g,k}}{(2k+1)!}L^{2k}$; each $a_{g,k}$ is a rational multiple of a power of $\pi$, $a_{g,0}=V_{g,1}$, and $a_{g,k}=[\tau_k]_{g,1}/4^k$.

The two settled one-parameter regimes. For fixed $n$ and $g\to\infty$, the source's Theorem 1.2 gives $C_{g,n}:=\frac{V_{g,n+1}}{2gV_{g,n}}=4\pi^2+O(1/g)$ and $B_{g,n}:=\frac{V_{g,n}}{V_{g-1,n+2}}=1+O(1/g)$. Mirzakhani-Zograf upgraded this to a full asymptotic expansion, uniform for $n=o(\sqrt g)$ (as stated in He-Wu-Xue, Theorem 4.4):

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

with $C>0$ absolute, $C=1/\sqrt\pi$ conjectured (Zograf); Liu-Xu re-derived this expansion by other methods. For fixed $g$ and $n\to\infty$, Manin-Zograf (the source's Theorem 1.3) proved $V_{g,n}=n!\,C_{\mathrm{MZ}}^{\,n}\,n^{(5g-7)/2}(a_g+O(1/n))$ with $C_{\mathrm{MZ}}>0$ an absolute constant distinct from the large-genus constant $C$ above and $a_g>0$.

The barrier and the psi side. The Mirzakhani-Zograf expansion loses uniformity at the scale $n\asymp\sqrt g$, and this is known not to be an artifact: on the neighboring pure psi side — Witten intersection numbers $\int_{\overline{\mathcal M}_{g,n}}\psi_1^{d_1}\cdots\psi_n^{d_n}$ without kappa classes, on the Deligne-Mumford compactification — Aggarwal proved the Delecroix-Goujard-Zograf-Zorich conjecture in the stronger range $n=o(\sqrt g)$ for the normalized numbers $G_{\mathbf d}(g)$, and showed that $o(\sqrt g)$ cannot be replaced by $O(\sqrt g)$; Guo-Yang gave another proof of the $n\le C\log g$ statement for arbitrary $C$; and Guo-Yang-Zagier (2026) proved completely uniform asymptotics for all $g,n$, exhibiting an explicit Gaussian transition at the $\sqrt g$ scale ($C(0^k,2^{3g-3+k})\sim\pi^{-1}e^{-k^2/(30g)}$ for $k=O(\sqrt g)$, decaying to 0 for $k/\sqrt g\to\infty$). But these works treat pure psi-intersection numbers (their stated applications are to the Painleve I equation and to polynomiality); the volume coefficients are mixed psi-kappa brackets, related to pure psi-numbers by an alternating-sign stable-graph (Kaufmann-Manin-Zagier) transform, and on the volume side no uniform mixed-regime control is known.

Why the mixed regime matters. The mixed regime is exactly the regime of random hyperbolic surfaces with many cusps. Shen-Wu proved that for $\sqrt g\prec n(g)\prec g$ the spectral gap (the smallest nonzero eigenvalue $\lambda_1$ of the Laplace operator) of a WP-random surface in $\mathcal M_{g,n(g)}$ dies: for any $\epsilon>0$, the WP-probability that $\lambda_1(X)<\epsilon$ tends to 1. For $n=O(g^\alpha)$ with $\alpha<1/2$, He-Wu-Xue proved a uniform gap $\frac14-\frac{1}{(6(1-\alpha))^2}-\epsilon$ with WP-probability tending to 1 (approaching $5/36$ as $\alpha\to1/2$); the case $n\asymp g$ is open and is blocked on precisely the mixed-regime volume asymptotics: Shen-Wu state that a positive answer to their volume-ratio Question 3 is expected to remove the restriction $n(g)/g\to0$ from their gap theorem. The source paper initiated this spectral program, proving the first lower bounds for the spectral gap and for the Cheeger constant (an isoperimetric constant that controls $\lambda_1$ through Cheeger's inequality) of WP-random surfaces.

Problem Statement

The problem is the surviving open core of the 'Questions' section of Mirzakhani (J. Differential Geom. 94 (2013), 267-300) — its items 1 and 2, kept together because they are two faces of one missing theory: uniform two-parameter control of Weil-Petersson volumes of moduli spaces and of their polynomial coefficients. Part (a) below concerns the asymptotics of $B_{g,n}$ along growing-$n$ paths and the behavior of the sequence $V_g,V_{g-1,2},\ldots,V_{0,2g}$; the quantities $C_{g,n}$ and $D_{g,n}$ are closely related, with growing-$n$ behavior that is likewise unknown.

(a) Mixed-regime volume asymptotics. Determine the asymptotics of the ratio

$$B_{g,n}=\frac{V_{g,n}}{V_{g-1,n+2}}$$

along paths $n=n(g)\to\infty$ with $g\to\infty$ — the source: 'It would be useful to know the asymptotics of $V_{g,n(g)}/V_{g-1,n(g)+2}$, where $n(g)\to\infty$ as $g\to\infty$' — and determine the behavior of the whole sequence

$$V_g,\ V_{g-1,2},\ V_{g-2,4},\ \ldots,\ V_{0,2g}$$

as $g\to\infty$ (the source: 'we don't know much about the behavior of the sequence $V_g,V_{g-1,2},\ldots,V_{0,2g}$ as $g\to\infty$'; only its two ends are understood). Closely related quantities whose growing-$n$ behavior is likewise unknown are $C_{g,n}=\frac{V_{g,n+1}}{2gV_{g,n}}$ and $D_{g,n}=\frac{V_{g,n}^2}{V_{g,n-1}V_{g,n+1}}$: $D_{g,n}\to1$ is known only for $n=o(g)$, and in general Shen-Wu's Proposition 36 gives $1/2-\pi^2/20\le\liminf D_{g,n}\le\limsup D_{g,n}\le1$.

(b) Coefficients of growing degree. Understand the asymptotics of $\frac{a_{g,k}}{a_{g,k+1}}$ — the source: 'It would be helpful to understand the asymptotics of $a_{g,k}/a_{g,k+1}$ for an arbitrary $k$ (which can grow with $g$)' — for $k=k(g)\to\infty$ with $k\le 3g-2$; equivalently, of the one-point brackets $[\tau_k]_{g,1}=4^ka_{g,k}$. The fixed-degree end ($a_{g,k+1}/a_{g,k}\to1$ for each fixed $k$, the source's Theorem 3.5(a)) and the top end ($a_{g,3g-2}/a_{g,0}\to0$, from $\int\psi^{3g-2}=1/(24^g g!)$) are known; the middle regime $k\asymp g$ is open.

References

1. Maryam Mirzakhani, 'Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus', J. Differential Geom. 94 (2013), no. 2, 267-300. arXiv:1012.2167 (the arXiv version's title reads '...surfaces of large genus'). doi:10.4310/jdg/1367438650. https://doi.org/10.4310/jdg/1367438650. [The source paper.]
2. Yuri I. Manin, Peter Zograf, 'Invertible cohomological field theories and Weil-Petersson volumes', Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 519-535. arXiv:math/9902051. doi:10.5802/aif.1764. https://doi.org/10.5802/aif.1764. [Fixed genus, large n; kappa classes and the volume/intersection-number dictionary.]
3. Peter Zograf, 'On the large genus asymptotics of Weil-Petersson volumes', arXiv:0812.0544 (2008). https://arxiv.org/abs/0812.0544. [Conjectural value of the constant in the Mirzakhani-Zograf expansion.]
4. Maryam Mirzakhani, Peter Zograf, 'Towards large genus asymptotics of intersection numbers on moduli spaces of curves', Geom. Funct. Anal. 25 (2015), no. 4, 1258-1289. arXiv:1112.1151 (the arXiv listing carries the title typo 'asymtotics'). doi:10.1007/s00039-015-0336-5. https://doi.org/10.1007/s00039-015-0336-5. [Uniform large-genus expansion for n = o(sqrt g).]
5. Kefeng Liu, Hao Xu, 'A remark on Mirzakhani's asymptotic formulae', Asian J. Math. 18 (2014), no. 1, 29-52. arXiv:1103.5136. https://arxiv.org/abs/1103.5136. [Independent proof of the fixed-n expansion.]
6. Nalini Anantharaman, Laura Monk, 'A high-genus asymptotic expansion of Weil-Petersson volume polynomials', J. Math. Phys. 63 (2022), no. 4, 043502. arXiv:2011.14889. doi:10.1063/5.0039385. https://doi.org/10.1063/5.0039385.
7. Amol Aggarwal, 'Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials', Invent. Math. 226 (2021), no. 3, 897-1010. arXiv:2004.05042. doi:10.1007/s00222-021-01059-9. https://doi.org/10.1007/s00222-021-01059-9. [Uniformity of normalized psi-intersection numbers for n = o(sqrt g), and sharpness.]
8. Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich, 'Masur-Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves', Duke Math. J. 170 (2021), no. 12, 2633-2718. arXiv:2011.05306. doi:10.1215/00127094-2021-0054. https://doi.org/10.1215/00127094-2021-0054. [Origin of the DGZZ intersection-number conjecture.]
9. Jindong Guo, Di Yang, 'On the large genus asymptotics of psi-class intersection numbers', Math. Ann. 388 (2024), no. 1, 61-97. arXiv:2110.06774. doi:10.1007/s00208-022-02505-6. https://doi.org/10.1007/s00208-022-02505-6. [Another proof of uniformity for n <= C log g.]
10. Jindong Guo, Di Yang, Don Zagier, 'On uniform large genus asymptotics of Witten's intersection numbers', arXiv:2603.15233 (2026). https://arxiv.org/abs/2603.15233. [Completely uniform psi-side asymptotics with the Gaussian transition at the sqrt(g) scale.]
11. Yang Shen, Yunhui Wu, 'Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps', Comm. Math. Phys. 406 (2025), no. 8, Art. 190. arXiv:2203.15681. doi:10.1007/s00220-025-05366-7. https://doi.org/10.1007/s00220-025-05366-7. [Gap death for sqrt(g) << n << g; Question 3 and Proposition 36 on the volume ratio.]
12. Yuxin He, Yunhui Wu, Yuhao Xue, 'Uniform spectral gaps for random hyperbolic surfaces with not many cusps', arXiv:2602.08352 (2026). https://arxiv.org/abs/2602.08352. [Gap 1/4 - 1/(6(1-alpha))^2 - eps for n = O(g^alpha), alpha < 1/2; also the source of the uniform Mirzakhani-Zograf statement.]
13. Will Hide, 'Spectral gap for Weil-Petersson random surfaces with cusps', Int. Math. Res. Not. 2023 (2023), no. 20, 17411-17460. arXiv:2107.14555. doi:10.1093/imrn/rnac293. https://doi.org/10.1093/imrn/rnac293.
14. Will Hide, Joe Thomas, 'Large-n asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces', Comm. Math. Phys. 406 (2025), no. 9. arXiv:2312.11412. doi:10.1007/s00220-025-05369-4. https://doi.org/10.1007/s00220-025-05369-4. [Fixed-genus large-n regime.]
15. Yuxin He, Yunhui Wu, Yuhao Xue, 'Weil-Petersson random hyperbolic surfaces', Sci. Sin. Math. (2026). doi:10.1360/ssm-2025-0117. https://doi.org/10.1360/ssm-2025-0117. [Survey of Weil-Petersson random hyperbolic surfaces.]
16. Xuanyu Huang, 'Asymptotic coefficients of Weil-Petersson volumes in the large genus', arXiv:2501.06421 (2025). https://arxiv.org/abs/2501.06421. [Coefficients of the fixed-n expansion are polynomials in pi^{-2}.]
17. Marthe Guillermit, 'A refinement of the asymptotic expansion of Weil-Petersson volumes', arXiv:2607.20341 (2026). https://arxiv.org/abs/2607.20341. [Fixed-n expansion for boundary lengths O(sqrt g), n-dependent constants.]
18. Dawei Chen, Scott Mullane, 'A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech', arXiv:2603.07318 (2026). https://arxiv.org/abs/2603.07318.
19. Maryam Mirzakhani, Bram Petri, 'Lengths of closed geodesics on random surfaces of large genus', Comment. Math. Helv. 94 (2019), no. 4, 869-889. arXiv:1710.09727. doi:10.4171/CMH/477. https://doi.org/10.4171/CMH/477. [Poisson statistics of simple closed geodesics.]
20. Yunhui Wu, Yuhao Xue, 'Random hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - epsilon', Geom. Funct. Anal. 32 (2022), no. 2, 340-410. arXiv:2102.05581. doi:10.1007/s00039-022-00595-7. https://doi.org/10.1007/s00039-022-00595-7.
21. Michael Lipnowski, Alex Wright, 'Towards optimal spectral gaps in large genus', Ann. Probab. 52 (2024), no. 2, 545-575. arXiv:2103.07496. doi:10.1214/23-aop1657. https://doi.org/10.1214/23-aop1657.
22. Nalini Anantharaman, Laura Monk, 'Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I', arXiv:2304.02678 (2023-2026). https://arxiv.org/abs/2304.02678.
23. Nalini Anantharaman, Laura Monk, 'Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II', arXiv:2502.12268 (2025). https://arxiv.org/abs/2502.12268. [Asymptotically optimal spectral gap 1/4 - epsilon for typical hyperbolic surfaces.]
24. Nalini Anantharaman, Laura Monk, 'Typical hyperbolic surfaces have a spectral gap greater than 2/9 - epsilon', arXiv:2604.09792 (2026). https://arxiv.org/abs/2604.09792. [Self-described intermediate step towards the optimal gap.]
25. Hugo Parlier, Yunhui Wu, Yuhao Xue, 'The simple separating systole for hyperbolic surfaces of large genus', J. Inst. Math. Jussieu 21 (2022), no. 6, 2205-2214. doi:10.1017/S1474748021000190. https://doi.org/10.1017/S1474748021000190.
26. Xin Nie, Yunhui Wu, Yuhao Xue, 'Large genus asymptotics for lengths of separating closed geodesics on random surfaces', J. Topology 16 (2023), no. 1, 106-175. doi:10.1112/topo.12276. https://doi.org/10.1112/topo.12276.
27. Yuxin He, Yang Shen, Yunhui Wu, Yuhao Xue, 'Non-simple systoles on random hyperbolic surfaces for large genus', arXiv:2308.16447 (2023-2025). https://arxiv.org/abs/2308.16447.
28. Thomas Budzinski, Nicolas Curien, Bram Petri, 'On Cheeger constants of hyperbolic surfaces', Invent. Math. 242 (2025), no. 2, 511-530. arXiv:2207.00469. doi:10.1007/s00222-025-01361-w. https://doi.org/10.1007/s00222-025-01361-w.
29. Yang Shen, Yunhui Wu, 'Nearly optimal spectral gaps for random Belyi surfaces', arXiv:2511.02517 (2025-2026). https://arxiv.org/abs/2511.02517.
30. Irene Ren, 'Mirzakhani's frequencies of simple closed geodesics on hyperbolic surfaces in large genus and with many cusps', Geom. Dedicata 219 (2025), no. 1. arXiv:2304.03734. doi:10.1007/s10711-024-00972-6. https://doi.org/10.1007/s10711-024-00972-6.
31. Svante Janson, Baptiste Louf, 'Unicellular maps vs hyperbolic surfaces in large genus: simple closed curves', Ann. Probab. 51 (2023), no. 3. arXiv:2111.11903. doi:10.1214/22-aop1601. https://doi.org/10.1214/22-aop1601. [Partial progress on relating the Brooks-Makover and WP random models.]
32. Robert Brooks, Eran Makover, 'Random construction of Riemann surfaces', J. Differential Geom. 68 (2004), no. 1, 121-157. doi:10.4310/jdg/1102536712. https://doi.org/10.4310/jdg/1102536712.
33. R. Kaufmann, Yuri I. Manin, Don Zagier, 'Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves', Comm. Math. Phys. 181 (1996), no. 3, 763-787. doi:10.1007/BF02101297. https://doi.org/10.1007/BF02101297. [The KMZ stable-graph expression of WP volumes through psi-intersections; cited as [KMZ] by the source paper (whose bibliography misprints the pages as 736-787).]
34. C. Itzykson, J. Zuber, 'Combinatorics of the modular group. II. The Kontsevich integrals', Internat. J. Modern Phys. A 7 (1992), 5661-5705. doi:10.1142/S0217751X92002581. https://doi.org/10.1142/S0217751X92002581. [The identity int psi^{3g-2} = 1/(24^g g!) used for the top-degree coefficient; cited as [IZ] by the source paper.]