Local limits of uniform triangulations in high genus. (English) Zbl 1461.05182
Summary: We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in [N. Curien, Probab. Theory Relat. Fields 165, No. 3–4, 509–540 (2016; Zbl 1342.05137)]. The proof relies on a combinatorial argument and the Goulden-Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane \(T\) is weakly Markovian in the sense that the probability to observe a finite triangulation \(t\) around the root only depends on the perimeter and volume of \(t\), then \(T\) is a mixture of PSHT.
MSC:
| 05C80 | Random graphs (graph-theoretic aspects) |
| 05C81 | Random walks on graphs |
| 60C05 | Combinatorial probability |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
Keywords:
random infinite triangulations of the full-plane; Angel and Schramm’s uniform infinite planar triangulationCitations:
Zbl 1342.05137References:
| [1] | Addario-Berry, L.; Albenque, M., The scaling limit of random simple triangulations and random simple quadrangulations, Ann. Probab., 45, 5, 2767-2825 (2017) · Zbl 1417.60022 |
| [2] | Addario-Berry, L., Albenque, M.: Convergence of odd-angulations via symmetrization of labeled trees. arXiv:1904.04786, (2019) |
| [3] | Aldous, D.; Lyons, R., Processes on unimodular random networks, Electron. J. Probab., 54, 12, 1454-1508 (2007) · Zbl 1131.60003 |
| [4] | Angel, O.; Chapuy, G.; Curien, N.; Ray, G., The local limit of unicellular maps in high genus, Electron. Commun. Probab., 18, 86, 1-8 (2013) · Zbl 1310.60002 |
| [5] | Angel, O.; Nachmias, A.; Ray, G., Random walks on stochastic hyperbolic half planar triangulations, Random Struct. Algorithms, 49, 2, 213-234 (2016) · Zbl 1344.05127 |
| [6] | Angel, O.; Ray, G., Classification of half planar maps, Ann. Probab., 43, 3, 1315-1349 (2015) · Zbl 1354.60010 |
| [7] | Angel, O.; Schramm, O., Uniform infinite planar triangulations, Commun. Math. Phys., 241, 2-3, 191-213 (2003) · Zbl 1098.60010 |
| [8] | Bender, EA; Canfield, E., The asymptotic number of rooted maps on a surface, J. Comb. Theory Ser. A, 43, 2, 244-257 (1986) · Zbl 0606.05031 |
| [9] | Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O., Uniform spanning forests, Ann. Probab., 29, 1, 1-65 (2001) · Zbl 1016.60009 |
| [10] | Bettinelli, J., Geodesics in Brownian surfaces (Brownian maps), Ann. Inst. Henri Poincaré Probab. Stat., 52, 2, 612-646 (2016) · Zbl 1342.60043 |
| [11] | Budd, T., The peeling process of infinite Boltzmann planar maps, Electron. J. Comb., 23, 06 (2015) |
| [12] | Budzinski, T.: Cartes aléatoires hyperboliques. PhD thesis, Université Paris-Sud (2018) |
| [13] | Budzinski, T., The hyperbolic Brownian plane, Probab. Theory Relat. Fields, 171, 1, 503-541 (2018) · Zbl 1406.60050 |
| [14] | Budzinski, T., Infinite geodesics in hyperbolic random triangulations, Ann. Inst. H. Poincaré Probab. Stat., 56, 2, 1129-1161 (2020) · Zbl 1434.60036 |
| [15] | Budzinski, T., Curien, N., Petri, B.: Universality for random surfaces in unconstrained genus. Electr. J. Comb. 26(4), P4.2 (2019). doi:10.37236/8623 · Zbl 1422.05093 |
| [16] | Carrell, SR; Chapuy, G., Simple recurrence formulas to count maps on orientable surfaces, J. Comb. Theory Ser. A, 133, 58-75 (2015) · Zbl 1315.05010 |
| [17] | Chapuy, G., A new combinatorial identity for unicellular maps, via a direct bijective approach, Adv. Appl. Math., 47, 4, 874-893 (2011) · Zbl 1234.05037 |
| [18] | Chassaing, P.; Durhuus, B., Local limit of labeled trees and expected volume growth in a random quadrangulation, Ann. Probab., 34, 3, 879-917 (2006) · Zbl 1102.60007 |
| [19] | Chmutov, S.; Pittel, B., On a surface formed by randomly gluing together polygonal discs, Adv. Appl. Math., 73, 23-42 (2016) · Zbl 1328.05003 |
| [20] | Curien, N., Planar stochastic hyperbolic triangulations, Probab. Theory Relat. Fields, 165, 3, 509-540 (2016) · Zbl 1342.05137 |
| [21] | Curien, N.: Peeling random planar maps. Saint-Flour lecture notes (2019) |
| [22] | Curien, N.; Le Gall, J-F, The Brownian plane, J. Theor. Prob., 27, 4, 1249-1291 (2014) · Zbl 1305.05208 |
| [23] | David, F.; Kupiainen, A.; Rhodes, R.; Vargas, V., Liouville quantum gravity on the Riemann sphere, Commun. Math. Phys., 342, 3, 869-907 (2016) · Zbl 1336.83042 |
| [24] | David, F.; Rhodes, R.; Vargas, V., Liouville quantum gravity on complex tori, J. Math. Phys., 57, 2, 022302 (2016) · Zbl 1362.81082 |
| [25] | Eynard, B.: Counting Surfaces, Volume 70 of Progress in Mathematical Physics. Birkhäuser/Springer, Cham (2016). CRM Aisenstadt chair lectures · Zbl 1338.81005 |
| [26] | Gamburd, A., Poisson-Dirichlet distribution for random Belyi surfaces, Ann. Probab., 34, 5, 1827-1848 (2006) · Zbl 1113.60095 |
| [27] | Goulden, IP; Jackson, DM, The KP hierarchy, branched covers, and triangulations, Adv. Math., 219, 3, 932-951 (2008) · Zbl 1158.37026 |
| [28] | Hausdorff, F., Summationsmethoden und Momentfolgen I, Math. Z., 9, 74-109 (1921) · JFM 48.2005.01 |
| [29] | Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1, 1-23 (1992) · Zbl 0756.35081 |
| [30] | Krikun, M.: Local structure of random quadrangulations. arXiv:math/0512304 · Zbl 0964.05056 |
| [31] | Krikun, M.: Explicit enumeration of triangulations with multiple boundaries. Electron. J. Comb. 14(1):Research Paper 61, 14 pp. (electronic) (2007) · Zbl 1157.05031 |
| [32] | Le Gall, J-F, Uniqueness and universality of the Brownian map, Ann. Probab., 41, 2880-2960 (2013) · Zbl 1282.60014 |
| [33] | Marzouk, C.: Scaling limits of random bipartite planar maps with a prescribed degree sequence. Random Struct. Algorithms (2018) · Zbl 1397.05164 |
| [34] | Miermont, G., The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., 210, 2, 319-401 (2013) · Zbl 1278.60124 |
| [35] | Miwa, T.; Jimbo, M.; Date, E., Solitons. Cambridge Tracts in Mathematics (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0986.37068 |
| [36] | Mullin, R., On the enumeration of tree-rooted maps, Can. J. Math., 19, 174-183 (1967) · Zbl 0148.17705 |
| [37] | Okounkov, A., Toda equations for Hurwitz numbers, Math. Res. Lett., 7, 4, 447-453 (2000) · Zbl 0969.37033 |
| [38] | Ray, G., Geometry and percolation on half planar triangulations, Electron. J. Probab., 19, 47, 1-28 (2014) · Zbl 1360.60034 |
| [39] | Ray, G., Large unicellular maps in high genus, Ann. Inst. H. Poincaré Probab. Stat., 51, 4, 1432-1456 (2015) · Zbl 1376.60011 |
| [40] | Stephenson, R., Local convergence of large critical multi-type Galton-Watson trees and applications to random maps, J. Theor. Probab., 31, 1, 159-205 (2018) · Zbl 1393.05244 |
| [41] | Tutte, WT, A census of planar maps, J. Can. Math., 15, 249-271 (1963) · Zbl 0115.17305 |
| [42] | Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC ’96, New York, NY, USA, pp. 296-303. ACM (1996) · Zbl 0946.60070 |
| [43] | Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge. MA, 1990), pp. 243-310. Lehigh Univ, Bethlehem, PA (1991) · Zbl 0757.53049 |
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.
