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Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

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We study the combinatorial geometry of a random closed multicurve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components \(\gamma _1, \dots ,\gamma _k\) of a random multicurve \(m_1\gamma _1+\dots +m_k\gamma _k\) represent linearly independent homology cycles with asymptotic probability 1 and that all its weights \(m_i\) are equal to 1 with asymptotic probability \(\sqrt{2}/2\). We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus g are both very well approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of \(3g-3\) elements. In particular, we prove that the expected value of these quantities has asymptotics \((\log (6g-6) + \gamma )/2 + \log 2\) as \(g\rightarrow \infty \), where \(\gamma \) is the Euler–Mascheroni constant. These results are based on our formula for the Masur–Veech volume \({\text {Vol}}{\mathcal {Q}}_g\) of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of \(\psi \)-classes on \({\overline{{\mathcal {M}}}}_{g,n}\) for large g proved by A. Aggarwal in 2020.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn., pp. 258–259. Dover, New York (1972)

    Google Scholar 

  2. Aggarwal, A.: Large genus asymptotics for volumes of strata of abelian differentials. J. Am. Math. Soc. 33(4), 941–989 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aggarwal, A.: Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials. Invent. Math. 226(3), 897–1010 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anantharaman, N., Monk, L.: A high-genus asymptotic expansion of Weil–Petersson volume polynomials (2020). arXiv:2011.14889 [math.GT]

  5. Angel, O., Schramm, O.: Uniform infinite planar triangulations. Commun. Math. Phys. 241(2–3), 191–213 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aggarwal, A., Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: Conjectural large genus asymptotics of Masur–Veech volumes and of area Siegel–Veech constants of strata of quadratic differentials. Arnold Math. J. 6(2), 149–161 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alexeev, A., Zograf, P.: Random matrix approach to the distribution of genomic distance. J. Comput. Biol. 21(8), 622–631 (2014)

    Article  MathSciNet  Google Scholar 

  8. Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS Monographs in Mathematics. European Mathematical Society, Zürich (2003)

    Book  MATH  Google Scholar 

  9. Arana-Herrera, F.: Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces. Geom. Dedicata 210, 65–102 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arana-Herrera, F.: Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani’s asymptotics for simple closed hyperbolic geodesics. J. Mod. Dyn. 16, 81–107 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Arana-Herrera, F.: Counting hyperbolic multi-geodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes, geometry and topology (to appear)

  12. Athreya, J., Eskin, A., Zorich, A.: Right-angled billiards and volumes of moduli spaces of quadratic differentials on \(\mathbb{C}{{\rm P}}^{1}\). Ann. Scient. ENS, 4ème série, 49, 1307–1381 (2016)

  13. Bettinelli, J.: Scaling limits for random quadrangulations of positive genus. Electron. J. Probab. 15, 1594–1644 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bettinelli, J.: The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40, 1897–1944 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brooks, R., Makover, E.: Random construction of Riemann surfaces. J. Differ. Geom. 68(1), 121–157 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Budzinski, T., Curien, N., Petri, B.: Universality for random surfaces in unconstrained genus. Electron. J. Combin. 26(4), Paper 4.2 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Budzinski, T., Curien, N., Petri, B.: The diameter of random Belyǐ surfaces Algebr. Geom. Topol. 21(6), 2929–2957 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Budzinski, T., Louf, B.: Local limits of uniform triangulations in high genus. Invent. Math. 223(1), 1–47 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Calegari, D.: Foliations and the Geometry Of 3-Manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  20. Chassaing, P., Durhuus, B.: Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34(3), 879–917 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, D., Möller, M., Sauvaget, A.: With an appendix by Borot, G., Giacchetto, A., Lewanski, D., Masur–Veech volumes and intersection theory: the principal strata of quadratic differentials. arXiv:1912.02267 [math.AG]

  22. Choi, J., Cvijović, D.: Values of the polygamma functions at rational arguments. J. Phys. A 40(50), 15019–15028 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, D., Möller, M., Sauvaget, A., Zagier, D.: Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials. Invent. Math. 222(1), 283–373 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chmutov, S., Pittel, B.: The genus of a random chord diagram is asymptotically normal. J. Comb. Theory Ser. A 120(1), 102–110 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Curien, N.: Planar stochastic hyperbolic triangulations. Probab. Theory Relat. Fields 165(3–4), 509–540 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves. arXiv:1908.08611 [math.GT] (2019)

  27. Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: Enumeration of meanders and Masur-Veech volumes. In: Forum of Mathematics \(\Pi \), vol. 8:e4 (2020)

  28. Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: With appendix of Ph. Engel, Contribution of one-cylinder square-tiled surfaces to Masur–Veech volumes. In: Crovisier, S., et al. (ed.) Some Aspects of the Theory of Dynamical Systems: A Tribute to Jean-Christophe Yoccoz, vol I, , vol. 415, pp. 223–274. Astérisque, SMF, Paris (2020)

  29. Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves. Duke Math. J. 170(12), 2633–2718 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Delecroix, V., Liu, M.: Length partition of random multicurves on large genus hyperbolic surfaces. arXiv:2202.10255 [math.GT] (2022)

  31. Erdős, P., Kac, M.: On the Gaussian law of errors in the theory of additive functions. Proc. Natl. Acad. Sci. USA 25, 206–207 (1939)

    Article  MATH  Google Scholar 

  32. Erlandsson, V., Parlier, H., Souto, J.: Counting curves, and the stable length of currents. J. EMS 22(6), 1675–1702 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Erlandsson, V., Souto, J.: Geodesic currents and Mirzakhani’s curve counting (to appear)

  34. Féray, V., Méliot, P.-L., Nikeghbali, A.: Mod-\(\phi \) convergence. Normality zones and precise deviations. Springer Briefs in Probability and Mathematical Statistics (2016)

  35. Flajolet, Ph., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  36. Gamburd, A.: Poisson–Dirichlet distribution for random Belyǐ surfaces. Ann. Probab. 34(5), 1827–1848 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Goncharov, V.L.: Some facts from combinatorics. Izvestia Akad. Nauk USSR Ser. Mat. 8, 3–48 (1944)

    Google Scholar 

  38. Goujard, E.: Volumes of strata of moduli spaces of quadratic differentials: getting explicit values. Ann. Inst. Fourier 66(6), 2203–2251 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Guo, J., Yang, D.: On the large genus asymptotics of psi-class intersection numbers (2021). arXiv:2110.06774 [math.AG]

  40. Guth, L., Parlier, H., Young, R.: Pants decomposition of random surfaces. Geom. Funct. Anal. 21(5), 1069–1090 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85(3), 457–485 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hide, W., Magee, M.: Near optimal spectral gaps for hyperbolic surfaces (2021). arXiv:2107.05292 [math.SP]

  43. Hwang, H.K.: Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques. Ph.D. thesis, École Polytechnique (1994)

  44. Hwang, H.K.: Asymptotics of Poisson approximation to random discrete distrubtions: an analytic approach. Adv. Appl. Probab. 31(2), 448–491 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  45. Hwang, H.K.: Asymptotic expansions for the Stirling numbers of the first kind. J. Comb. Theory Ser. A 71, 343–351 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  46. Janson, S., Louf, B.: Short cycles in high genus unicellular maps. arXiv:2103.02549 [math.PR]

  47. Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157(4), 295–300 (1985)

    Article  MathSciNet  Google Scholar 

  48. Kazarian, M.: Recursion for Masur-Veech volumes of moduli spaces of quadratic differentials. J. Inst. Math. Jussieu 1–6 (2021) https://doi.org/10.1017/S1474748020000638

  49. Krikun, M.: Local structure of random quadrangulations. arXiv:math/0512304 [math.PR]

  50. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kowalski, E., Nikeghbali, A.: Mod-Poisson convergence in probability and number theory. Int. Math. Res. Not. 18, 3549–3587 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. Le Gall, J.-F.: Brownian geometry (Takagi lectures). Jpn. J. Math. 14(2), 135–174 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lipnowski, M., Wright, A.: Towards optimal spectral gaps in large genus. arXiv:2103.07496 [math.GT]

  54. Liu, M.: Length statistics of random multicurves on closed hyperbolic surfaces. arXiv:1912.11155 [math.GT]; to appear in Groups, Geometry, Dynamics

  55. Louf, B.: Planarity and non-separating cycles in uniform high genus quadrangulations. arXiv:2012.06512 [math.PR]

  56. Magee, M., Naud, F., Puder, D.: A random cover of a compact hyperbolic surface has relative spectral gap \(\frac{3}{16}-\varepsilon \) Geom. Funct. Anal. 32(3), 595–661 (2022)

  57. Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. 115, 169–200 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  58. Masur, H., Rafi, K., Randecker, A.: Expected covering radius of a translation surface. Int. Math. Res. Not. arXiv:1809.10769

  59. Ménard, L.: The two uniform infinite quadrangulations of the plane have the same law. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 190–208 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Miermont, G.: Aspects of random maps. Lecture notes of Saint-Flour school 2014. http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf

  61. Mirzakhani, M.: Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves. Ph.D. Thesis, Harvard University (2004)

  62. Mirzakhani, M.: Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. (2) 168(1), 97–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Mirzakhani, M.: Ergodic theory of the earthquake flow. IMRN 3, 1–39 (2008)

    MATH  Google Scholar 

  64. Mirzakhani, M.: Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus. J. Differ. Geom. 94(2), 267–300 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  65. Mirzakhani, M.: Counting mapping class group orbits on hyperbolic surfaces. arXiv:1601.03342 [math.GT]

  66. Mirzakhani, M., Petri, B.: Lengths of closed geodesics on random surfaces of large genus. Comment. Math. Helv. 94(4), 869–889 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  67. Mirzakhani, M., Zograf, P.: Towards large genus asymtotics of intersection numbers on moduli spaces of curves. GAFA 25, 1258–1289 (2015)

    MATH  Google Scholar 

  68. Monk, L.: Benjamini–Schramm convergence and spectrum of random hyperbolic surfaces of high genus. Analysis & PDE (2021)

  69. Monk, L., Thomas, J.: The tangle-free hypothesis on random hyperbolic surfaces. Int. Math. Res. Not. rnab160 (2021)

  70. Monin, L., Telpukhovskiy, V.: On normalizations of Thurston measure on the space of measured laminations. Topol. Appl. 267, 1–12 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  71. Nikeghbali, A., Zeindler, D.: The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles. Ann. Inst. Henri Poincaré Probab. Stat. 49(4), 961–981 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  72. Pitman, J.: Combinatorial stochastic processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII—2002. Lecture Notes in Mathematics. Springer (2006)

  73. Rafi, K., Souto, J.: Geodesic currents and counting problems. GAFA 29(3), 871–889 (2019)

    MathSciNet  MATH  Google Scholar 

  74. Rényi, A., Turán, P.: On a theorem of Erdős-Kac. Acta Arith. 4, 71–84 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  75. Riordan, J.: Moment recurrence relations for binomial, Poisson and hypergeometric frequency distributions. Ann. Math. Stat. 8(2), 103–111 (1937)

    Article  MATH  Google Scholar 

  76. Selberg, A.: Note on a paper by L. G. Sathe. J. Indian Math. Soc. B 18, 83–87 (1954)

    MathSciNet  MATH  Google Scholar 

  77. Shrestha, S.: The topology and geometry of random square-tiled surfaces. arXiv:2005.00099 [math.GT]

  78. Tao, T.: The Poisson–Dirichlet process, and large prime factors of a random number. https://terrytao.wordpress.com/2013/09/21/the-poisson-dirichlet-process-and-large-prime-factors-of-a-random-number

  79. Thurston, W.P.: Geometry and Topology of Three-Manifolds. Lecture Notes. Princeton University, Princeton (1979)

    Google Scholar 

  80. Tutte, W.T.: A census of of planar maps. Can. J. Math. 15, 249–271 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  81. Veech, W.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115, 201–242 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  82. Vershik, A.M.: Statistical mechanics of combinatorial partitions, and their limit shapes. Funct. Anal. Appl. 30, 90–105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  83. Walsh, T.R.S., Lehman, A.B.: Counting rooted maps by genus. I. J. Comb. Theory B 13, 192–218 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  84. Witten, E., Two-dimensional gravity and intersection theory on moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1991), pp. 243–310. Lehigh Univ., Bethlehem (1990)

  85. Wu, Y., Xue, Y.: Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\varepsilon \). Geom. Funct. Anal. 32(2), 340–410 (2022)

  86. Yang, D., Zagier, D., Zhang, Y.: Masur–Veech volumes of quadratic differentials and their asymptotics. J. Geom. Phys. 158, 1–12 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are deeply indebted to A. Aggarwal for transforming our dreams into reality by proving all our conjectures from [26]. We also very much appreciate his suggestions, including the indication on how to compute multi-variate harmonic sums, which was crucial for making correct predictions in [26]. His numerous precious comments on the preliminary versions of this paper allowed us to correct a technical mistake and many typos, and improve the presentation. Results of this paper were directly or indirectly influenced by beautiful and deep ideas of Maryam Mirzakhani. We are grateful to the anonymous referee for careful reading of the manuscript and for helpful comments which allowed to clean some typos and to improve the presentation. We thank S. Schleimer, who was the first person to notice that our experimental data on statistics of cylinder decompositions of random Abelian square-tiled surfaces seems to have resemblance with statistics of cycle decomposition of random permutations. We thank F. Petrov for the reference to the paper [37] in the context of cycle decomposition of random permutations. We thank M. Bertola, A. Borodin, G. Borot, D. Chen, A. Eskin, V. Feray, M. Kazarian, S. Lando, M. Liu, H. Masur, M. Möller, B. Petri, K. Rafi, A. Sauvaget, J. Souto, D. Zagier and D. Zvonkine for useful discussions. We thank B. Green for the talk at the conference “CMI at 20” and T. Tao for his blog both of which were very inspiring for us. We thank D. Calegari for kind permission to use a picture from his book [19] in Fig. 1. Figure 2 originally appeared in our paper [29] (copyright Duke University Press). We are grateful to MPIM (Bonn) and EIMI (St. Petersburg), where part of the work was done, and to MSRI (Berkeley) and MFO (Oberwolfach) for providing us with friendly and stimulating environments.

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Authors and Affiliations

  1. LaBRI, Domaine Universitaire, 351 cours de la Libération, 33405, Talence, France

    Vincent Delecroix

  2. Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405, Talence, France

    Élise Goujard

  3. Euler International Mathematical Institute, Pesochnaya nab. 10, Saint Petersburg, Russia, 197022

    Peter Zograf

  4. Center for Advanced Studies, Skoltech, Moscow, Russia

    Anton Zorich

  5. Institut de Mathématiques de Jussieu – Paris Rive Gauche, Case 7012, 8 Place Aurélie Nemours, 75205, Paris Cedex 13, France

    Anton Zorich

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  1. Vincent Delecroix
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  4. Anton Zorich
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This work was supported by the Agence Nationale de la Recherche grants ANR-19-CE40-0021 and ANR-19-CE40-0003, by the NSF grant DMS-1440140, and by the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2019-162. In addition to that, Goujard’s work was partially supported by a Projets Exploratoires Premier Soutien (PEPS) JCJC grant.

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Delecroix, V., Goujard, É., Zograf, P. et al. Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves. Invent. math. 230, 123–224 (2022). https://doi.org/10.1007/s00222-022-01123-y

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