On scaling limits of random Halin-like maps
DOI:
https://doi.org/10.7146/math.scand.a-139930Abstract
We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the α-stable looptrees of Curien and Kortchemski.
References
Addario-Berry, L., Broutin, N., and Goldschmidt, C., The continuum limit of critical random graphs, Probab. Theory Related Fields 152 (2012), no. 3–4, 367–406. https://doi.org/10.1007/s00440-010-0325-4
Aldous, D., The continuum random tree. I, Ann. Probab. 19 (1991), no. 1, 1–28.
Aldous, D., The continuum random tree. II. An overview, Stochastic analysis (Durham, 1990), 23–70, London Math. Soc. Lecture Note Ser., 167, Cambridge Univ. Press, Cambridge, 1991. https://doi.org/10.1017/CBO9780511662980.003
Aldous, D., The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289.
Bertoin, J., Lévy processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996
Bingham, N. H., Goldie, C. M., and Teugels, J. L., Regular variation, Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989.
Bouttier, J., Di Francesco, P., and Guitter, E., Planar maps as labeled mobiles, Electron. J. Combin. 11 (2004), no. 1, Research Paper 69, 27pp. https://doi.org/10.37236/1822
Burago, D., Burago, Y., and Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/033
Caraceni, A., The scaling limit of random outerplanar maps, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 4, 1667–1686. https://doi.org/10.1214/15-AIHP694
Chapuy, G., The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees, Probab. Theory Related Fields 147 (2010), no. 3–4, 415–447. https://doi.org/10.1007/s00440-009-0211-0
Cori, R., and Vauquelin, B., Planar maps are well labeled trees, Canadian J. Math. 33 (1981), no. 5, 1023–1042. https://doi.org/10.4153/CJM-1981-078-2
Curien, N., Haas, B., and Kortchemski, I., The CRT is the scaling limit of random dissections, Random Structures Algorithms 47 (2015), no. 2, 304–327. https://doi.org/10.1002/rsa.20554
Curien, N., and Kortchemski, I., Random stable looptrees, Electron. J. Probab. 19 (2014), no. 108, 35pp. https://doi.org/10.1214/EJP.v19-2732
Duquesne, T., A limit theorem for the contour process of conditioned Galton-Watson trees, Ann. Probab. 31 (2003), no. 2, 996–1027. https://doi.org/10.1214/aop/1048516543
Duquesne, T., and Le Gall, J.-F., Random trees, Lévy processes and spatial branching processes, Astérisque No. 281 (2002), vi+147pp.
Foss, S., Korshunov, D., and Zachary, S., An introduction to heavy-tailed and subexponential distributions, Second edition. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-7101-1
Han, R., and Yanpei, L., Enumeration of rooted planar Halin maps, Appl. Math. Chin. Univ. 14 (1999), 117–121. https://doi.org/10.1007/s11766-999-0063-5
Janson, S., Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation, Probab. Surv. 9 (2012), 103–252. https://doi.org/10.1214/11-PS188
Jonsson, T., and Stefánsson, S. O., Condensation in nongeneric trees, J. Stat. Phys. 142 (2011), no. 2, 277–313. https://doi.org/10.1007/s10955-010-0104-8
Kortchemski, I., Invariance principles for Galton-Watson trees conditioned on the number of leaves, Stochastic Process. Appl. 122 (2012), no. 9, 3126–3172. https://doi.org/10.1016/j.spa.2012.05.013
Kortchemski, I., Limit theorems for conditioned non-generic Galton–Watson trees, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 2, 489–511. https://doi.org/10.1214/13-AIHP580
Kortchemski, I., and Marzouk, C., Triangulating stable laminations, Electron. J. Probab. 21 (2016), Paper No. 11, 31 pp. https://doi.org/10.1214/16-EJP4559
Kortchemski, I., and Richier, L., The boundary of random planar maps via looptrees, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), no. 2, 391–430 ttps://doi.org/10.5802/afst.1636
Le Gall, J.-F., Random geometry on the sphere, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1, 421–442, Kyung Moon Sa, Seoul, 2014.
Miermont, G., Aspects of random maps, Lecture notes of the 2014 Saint-Flour Probability Summer School.
Schaeffer, G., Conjugaison d’arbres et cartes combinatoires aléatoires, PhD thesis, Université Bordeaux I, 1998.
Sénizergues, D., Stefánsson, S. O., and Stufler, B., Decorated stable trees, arXiv:2205.02968.
Stefánsson, S. O., and Stufler, B., Geometry of large Boltzmann outerplanar maps, Random Structures Algorithms 55 (2019), no. 3, 742–771. https://doi.org/10.1002/rsa.20834
Stufler, B., Gibbs partitions: The convergent case, Random Structures Algorithms 53 (2018), no. 3, 537–558. https://doi.org/10.1002/rsa.20771
Stufler, B., Limits of random tree-like discrete structures, Probab. Surv. 17 (2020), 318–477. https://doi.org/10.1214/19-PS338
Stufler, B., Gibbs partitions: a comprehensive phase diagram, arXiv:2204.06982.
