Unified study of the phase transition for block-weighted random planar maps

EUROCOMB’23

Abstract
In [Fleurat, Salvy 2023], we introduced a model of block-weighted random maps that undergoes a phase transition as the density of separating elements changes. The purpose of this note is to demonstrate that the methodology we developed can be extended to many other families of maps. We prove that a phase transition exists and provide detailed information about the size of the largest blocks in each regime.

Pages:
790–798
References

Louigi Addario-Berry. A probabilistic approach to block sizes in random maps. ALEA - Latin American Journal of Probability and Mathematical Statistics, XVI:1-13, 2019.
https://doi.org/10.30757/ALEA.v16-01

Louigi Addario-Berry and Marie Albenque. The scaling limit of random simple triangulations and random simple quadrangulations. The Annals of Probability, 45(5):2767 - 2825, 2017.
https://doi.org/10.1214/16-AOP1124

Louigi Addario-Berry and Marie Albenque. Convergence of odd-angulations via symmetrization of labeled trees. Annales Henri Lebesgue, 4:653-683, 2021.
https://doi.org/10.5802/ahl.84

David Aldous. The continuum random tree. II. an overview. Stochastic analysis, 167:23-70, 1991.
https://doi.org/10.1017/CBO9780511662980.003

Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, and Michèle Soria. Random Maps, Coalescing Saddles, Singularity Analysis, and Airy Phenomena. Random Struct. Algorithms, 19(3-4):194-246, oct 2001.
https://doi.org/10.1002/rsa.10021

Valentin Bonzom. Large N limits in tensor models: Towards more universality classes of colored triangulations in dimension d ⩾ 2. Symmetry, Integrability and Geometry: Methods and Applications, 12(073):39, 2016.
https://doi.org/10.3842/SIGMA.2016.073

Ariane Carrance. Convergence of Eulerian triangulations. Electronic Journal of Probability, (26):1-48, 2021.
https://doi.org/10.1214/21-EJP579

William Fleurat and Zéphyr Salvy. A phase transition in block-weighted random maps. 2023.

Omer Giménez and Marc Noy. Asymptotic enumeration and limit laws of planar graphs. Journal of the American Mathematical Society, 22(2):309-329, 2009.
https://doi.org/10.1090/S0894-0347-08-00624-3

Xavier Gourdon. Largest component in random combinatorial structures. Discrete Mathematics, 180(1):185-209, 1998.
https://doi.org/10.1016/S0012-365X(97)00115-5

Svante Janson. Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Probability Surveys, 9(none):103 - 252, 2012.
https://doi.org/10.1214/11-PS188

Jean-François Le Gall. Uniqueness and universality of the Brownian map. The Annals of Probability, 41(4):2880 - 2960, 2013.
https://doi.org/10.1214/12-AOP792

Cyril Marzouk. Scaling limits of random bipartite planar maps with a prescribed degree sequence. Random Struct. Algorithms, 53(3):448-503, 2018.
https://doi.org/10.1002/rsa.20773

Grégory Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Mathematica, 210(2):319-401, 2013.
https://doi.org/10.1007/s11511-013-0096-8

Jacques Neveu. Arbres et processus de Galton-Watson. Annales de l'I.H.P. Probabilités et statistiques, 22(2):199-207, 1986.

Gilles Schaeffer. Random sampling of large planar maps and convex polyhedra. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, STOC '99, pages 760-769, New York, NY, USA, 1999. Association for Computing Machinery.
https://doi.org/10.1145/301250.301448

Benedikt Stufler. Random enriched trees with applications to random graphs. Electronic Journal of Combinatorics, 25(3), 2018.
https://doi.org/10.37236/7328

Benedikt Stufler. Limits of random tree-like discrete structures. Probability Surveys, 17(none):318 - 477, 2020.
https://doi.org/10.1214/19-PS338

Benedikt Stufler. On the maximal offspring in a subcritical branching process. Electronic Journal of Probability, 25(none):1 - 62, 2020.
https://doi.org/10.1214/20-EJP506

W. T. Tutte. A census of planar maps. Canadian Journal of Mathematics, 15:249-271, 1963.
https://doi.org/10.4153/CJM-1963-029-x

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