×

Unicellular maps vs. hyperbolic surfaces in large genus: simple closed curves. (English) Zbl 1516.05045

Summary: We study uniformly random maps with a single face, genus \(g\), and size \(n\), as \(n,g \to \infty\) with \(g=o(n)\), in continuation of several previous works on the geometric properties of “high genus maps.” We calculate the number of short simple cycles, and we show convergence of their lengths (after a well-chosen rescaling of the graph distance) to a Poisson process, which happens to be exactly the same as the limit law obtained by M. Mirzakhani and B. Petri [Comment. Math. Helv. 94, No. 4, 869–889 (2019; Zbl 1472.57028)] when they studied simple closed geodesics on random hyperbolic surfaces under the Weil-Petersson measure as \(g \to \infty\).
This leads us to conjecture that these two models are somehow “the same” in the limit, which would allow to translate problems on hyperbolic surfaces in terms of random trees, thanks to a powerful bijection of G. Chapuy et al. [J. Comb. Theory, Ser. A 120, No. 8, 2064–2092 (2013; Zbl 1278.05081)].

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C80 Random graphs (graph-theoretic aspects)
60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
60B05 Probability measures on topological spaces
60B10 Convergence of probability measures
57M50 General geometric structures on low-dimensional manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] ANGEL, O., CHAPUY, G., CURIEN, N. and RAY, G. (2013). The local limit of unicellular maps in high genus. Electron. Commun. Probab. 18 no. 86. · Zbl 1310.60002 · doi:10.1214/ECP.v18-3037
[2] Angel, O. and Schramm, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191-213. · Zbl 1098.60010 · doi:10.1007/978-1-4419-9675-6_16
[3] BENDER, E. A. and CANFIELD, E. R. (1986). The asymptotic number of rooted maps on a surface. J. Combin. Theory Ser. A 43 244-257. · Zbl 0606.05031 · doi:10.1016/0097-3165(86)90065-8
[4] BETTINELLI, J. (2016). Geodesics in Brownian surfaces (Brownian maps). Ann. Inst. Henri Poincaré Probab. Stat. 52 612-646. · Zbl 1342.60043 · doi:10.1214/14-AIHP666
[5] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[6] BUDZINSKI, T. and LOUF, B. (2021). Local limits of uniform triangulations in high genus. Invent. Math. 223 1-47. · Zbl 1461.05182 · doi:10.1007/s00222-020-00986-3
[7] BUDZINSKI, T. and LOUF, B. (2022). Local limits of bipartite maps with prescribed face degrees in high genus. Ann. Probab. 50 1059-1126. · Zbl 1487.05240 · doi:10.1214/21-aop1554
[8] CHAPUY, G., FÉRAY, V. and FUSY, É. (2013). A simple model of trees for unicellular maps. J. Combin. Theory Ser. A 120 2064-2092. · Zbl 1278.05081 · doi:10.1016/j.jcta.2013.08.003
[9] CHAPUY, G., MARCUS, M. and SCHAEFFER, G. (2009). A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 1587-1611. · Zbl 1207.05087 · doi:10.1137/080720097
[10] DRMOTA, M. (2009). Random Trees. An Interplay Between Combinatorics and Probability. Springer, Vienna. · Zbl 1170.05022 · doi:10.1007/978-3-211-75357-6
[11] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge. · Zbl 1165.05001 · doi:10.1017/CBO9780511801655
[12] GILMORE, C., LE MASSON, E., SAHLSTEN, T. and THOMAS, J. (2021). Short geodesic loops and \[{L^p}\] norms of eigenfunctions on large genus random surfaces. Geom. Funct. Anal. 31 62-110. · Zbl 1478.58010 · doi:10.1007/s00039-021-00556-6
[13] GUTH, L., PARLIER, H. and YOUNG, R. (2011). Pants decompositions of random surfaces. Geom. Funct. Anal. 21 1069-1090. · Zbl 1242.32007 · doi:10.1007/s00039-011-0131-x
[14] JANSON, S. (2003). Cycles and unicyclic components in random graphs. Combin. Probab. Comput. 12 27-52. · Zbl 1010.05072 · doi:10.1017/S0963548302005412
[15] JANSON, S. and LOUF, B. (2022). Short cycles in high genus unicellular maps. Ann. Inst. Henri Poincaré Probab. Stat. 58 1547-1564. · Zbl 1493.05085 · doi:10.1214/21-aihp1218
[16] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York. · Zbl 0996.60001 · doi:10.1007/978-1-4757-4015-8
[17] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling 77. Springer, Cham. · Zbl 1376.60003 · doi:10.1007/978-3-319-41598-7
[18] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880-2960. · Zbl 1282.60014 · doi:10.1214/12-AOP792
[19] LE MASSON, M. and SAHLSTEN, T. (2020). Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus. Preprint. Available at arXiv:2006.14935.
[20] LIPNOWSKI, M. and WRIGHT, A. (2021). Towards optimal spectral gaps in large genus. Preprint. Available at arXiv:2103.07496.
[21] LOUF, B. (2022). Planarity and non-separating cycles in uniform high genus quadrangulations. Probab. Theory Related Fields 182 1183-1206. · Zbl 1489.60017 · doi:10.1007/s00440-021-01050-8
[22] LOUF, B. (2022). Large expanders in high genus unicellular maps. Comb. Theory 2 Paper No. 7. · Zbl 1498.05077
[23] MAGEE, M. (2020). Letter to Bram Petri. Available at https://www.mmagee.net/diameter.pdf.
[24] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319-401. · Zbl 1278.60124 · doi:10.1007/s11511-013-0096-8
[25] MIRZAKHANI, M. (2013). Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus. J. Differential Geom. 94 267-300. · Zbl 1270.30014
[26] MIRZAKHANI, M. and PETRI, B. (2019). Lengths of closed geodesics on random surfaces of large genus. Comment. Math. Helv. 94 869-889. · Zbl 1472.57028 · doi:10.4171/cmh/477
[27] MONK, L. (2022). Benjamini-Schramm convergence and spectra of random hyperbolic surfaces of high genus. Anal. PDE 15 727-752. · Zbl 1493.58015 · doi:10.2140/apde.2022.15.727
[28] MONK, L. and THOMAS, J. (2021). The tangle-free hypothesis on random hyperbolic surfaces. Int. Math. Res. Not.. Online.
[29] NIE, X., WU, Y. and XUE, Y. (2020). Large genus asymptotics for lengths of separating closed geodesics on random surfaces. Preprint. Available at arXiv:2009.07538.
[30] PARLIER, H., WU, Y. and XUE, Y. (2021). The simple separating systole for hyperbolic surfaces of large genus. J. Inst. Math. Jussieu. Online.
[31] RAY, G. (2015). Large unicellular maps in high genus. Ann. Inst. Henri Poincaré Probab. Stat. 51 1432-1456. · Zbl 1376.60011 · doi:10.1214/14-AIHP618
[32] SCHAEFFER, G. (1998). Conjugaison d’arbres et cartes combinatoires aléatoires. Thèse de doctorat, Université Bordeaux I.
[33] THOMAS, J. (2022). Delocalisation of eigenfunctions on large genus random surfaces. Israel J. Math. 250 53-83. · Zbl 1525.11056 · doi:10.1007/s11856-022-2327-1
[34] Tutte, W. T. (1963). A census of planar maps. Canad. J. Math. 15 249-271. · Zbl 0115.17305 · doi:10.4153/CJM-1963-029-x
[35] WRIGHT, A. (2020). A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces. Bull. Amer. Math. Soc. (N.S.) 57 359-408. · Zbl 1452.32003 · doi:10.1090/bull/1687
[36] WU, Y. and XUE, Y. (2022). Random hyperbolic surfaces of large genus have first eigenvalues greater than \[\frac{3}{16}-\epsilon \]. Geom. Funct. Anal. 32 340-410 · Zbl 1487.32072 · doi:10.1007/s00039-022-00595-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.