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

Abstract

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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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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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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