Abstract
A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.
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The authors declare that the data supporting the findings of this study are available within the article.
References
Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York (1999)
Brooks, R., Makover, E.: Random construction of Riemann surfaces. J. Differ. Geom. 68(1), 121–157 (2004)
Chmutov, S., Pittel, B.: The genus of a random chord diagram is asymptotically normal. J. Comb. Theory Ser. A 120(1), 102–110 (2013)
Chmutov, S., Pittel, B.: On a surface formed by randomly gluing together polygonal discs. Adv. Appl. Math. 73, 23–42 (2016)
Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. I. Wiley, New York (1968)
Fleming, K., Pippenger, N.: Large deviations and moments for the Euler characteristic of a random surface. Random Struct. Algorithms 37(4), 465–476 (2010)
Fomin, S., Lulov, N.: On the number of rim hook tableaux. J. Math. Sci. 87(6), 4118–4123 (1997)
Gamburd, A.: Poisson–Dirichlet distribution for random Belyi surfaces. Ann. Probab. 34(5), 1827–1848 (2006)
Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85(3), 457–485 (1986)
Larsen, M., Shalev, A.: Characters of symmetric groups: sharp bounds and applications. Invent. Math. 174(3), 645–687 (2008)
Linial, N., Nowik, T.: The expected genus of a random chord diagram. Discret. Comput. Geom. 45(1), 161–180 (2011)
Pippenger, N., Schleich, K.: Topological characteristics of random triangulated surfaces. Random Struct. Algorithms 28(3), 247–288 (2006)
Acknowledgements
Michael Farber was partially supported by a Grant from the Leverhulme Foundation. Chaim Even-Zohar was supported by the Lloyd’s Register Foundation/Alan Turing Institute programme on Data-Centric Engineering. We thank the anonymous reviewer for helpful suggestions which greatly improved the final version of the paper.
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Even-Zohar, C., Farber, M. Random Surfaces with Boundary. Discrete Comput Geom 66, 1463–1469 (2021). https://doi.org/10.1007/s00454-021-00301-8
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DOI: https://doi.org/10.1007/s00454-021-00301-8