Addario-Berry, Louigi; Wen, Yuting Joint convergence of random quadrangulations and their cores. (English. French summary) Zbl 1382.60017 Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 4, 1890-1920 (2017). Summary: We show that a uniform quadrangulation, its largest \(2\)-connected block, and its largest simple block jointly converge to the same Brownian map in distribution for the Gromov-Hausdorff-Prokhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result in [C. Banderier et al., Random Struct. Algorithms 19, No. 3–4, 194–246 (2001; Zbl 1016.68179)]. The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov-Hausdorff convergence. To extend the convergence to the Gromov-Hausdorff-Prokhorov topology, we show that exchangeable “uniformly asymptotically negligible” attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure. Cited in 10 Documents MSC: 60C05 Combinatorial probability 68P10 Searching and sorting 68W40 Analysis of algorithms Keywords:Brownian map; Gromov-Hausdorff-Prokhorov convergence; singularity analysis; connectivity; random quadrangulations Citations:Zbl 1016.68179 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] L. Addario-Berry and M. Albenque. The scaling limit of random simple triangulations and random simple quadrangulations.Ann. Probab.45(5) (2017) 2767-2825. Available atarXiv:1306.5227. · Zbl 1417.60022 [2] D. J. Aldous.Exchangeability and Related Topics1-198. Springer, Berlin Heidelberg, 1985. [3] C. Banderier, P. Flajolet, G. Schaeffer and M. Soria. Random maps, coalescing saddles, singularity analysis, and Airy phenomena.Random Structures Algorithms19(3-4) (2001) 194-246. · Zbl 1016.68179 · doi:10.1002/rsa.10021 [4] E. A. Bender and E. R. Canfield. Face sizes of 3-polytopes.J. Combin. Theory Ser. B46(1) (1989) 58-65. · Zbl 0666.05005 · doi:10.1016/0095-8956(89)90007-5 [5] D. Burago, Y. Burago and S. Ivanov. Gromov-Hausdorff limits. InA Course in Metric Geometry371-374.Graduate Studies in Mathematics33. American Mathematical Society, Providence, 2001. · Zbl 0981.51016 [6] P. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion.Probab. Theory Related Fields128(2) (2004) 161-212. · Zbl 1041.60008 · doi:10.1007/s00440-003-0297-8 [7] S. Even.Graph Algorithms. Cambridge University Press, Cambridge, 2011. · Zbl 0441.68072 [8] P. Flajolet and R. Sedgewick.Analytic Combinatorics. Cambridge University Press, Cambridge, 2009. · Zbl 1165.05001 [9] Z. Gao and N. C. Wormald. The size of the largest components in random maps.SIAM J. Discrete Math.12(2) (1999) 217-228. · Zbl 0923.05030 · doi:10.1137/S0895480195292053 [10] I. P. Goulden and D. M. Jackson.Combinatorial Enumeration. John Wiley & Sons, New York, 1983. · Zbl 0519.05001 [11] J. F. Le Gall. Uniqueness and universality of the Brownian map.Ann. Probab.41(4) (2013) 2880-2960. · Zbl 1282.60014 · doi:10.1214/12-AOP792 [12] C. McDiarmid. Concentration. InProbabilistic Methods for Algorithmic Discrete Mathematics195-248. Springer, Berlin Heidelberg, 1998. · Zbl 0927.60027 [13] G. Miermont. Tessellations of random maps of arbitrary genus.Ann. Sci. Éc. Norm. Supér. (4)42(5) (2009) 725-781. · Zbl 1228.05118 [14] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations.Acta Math.210(2) (2013) 319-401. · Zbl 1278.60124 · doi:10.1007/s11511-013-0096-8 [15] V. Strassen. The existence of probability measures with given marginals.Ann. Math. Stat.36(1965) 423-439. · Zbl 0135.18701 · doi:10.1214/aoms/1177700153 [16] W. T. Tutte. A census of planar maps.Canad. J. Math.15(2) (1963) 249-271. · Zbl 0115.17305 · doi:10.4153/CJM-1963-029-x [17] T. R. Walsh and A. B. Lehman. Counting rooted maps by genus III: Nonseparable maps.J. Combin. Theory Ser. B18(3) (1975) 222-259. · Zbl 0299.05110 · doi:10.1016/0095-8956(75)90050-7 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.