An invariance principle for random planar maps

Grégory Miermont

doi:10.46298/dmtcs.3505

Grégory Miermont - An invariance principle for random planar maps

dmtcs:3505 - Discrete Mathematics & Theoretical Computer Science, January 1, 2006, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities - https://doi.org/10.46298/dmtcs.3505

An invariance principle for random planar maps

Authors: Grégory Miermont ¹

1 Laboratoire de Mathématiques d'Orsay

We show a new invariance principle for the radius and other functionals of a class of conditioned `Boltzmann-Gibbs'-distributed random planar maps. It improves over the more restrictive case of bipartite maps that was discussed in Marckert and Miermont (2006). As in the latter paper, we make use of a bijection between planar maps and a class of labelled multitype trees, due to Bouttier et al. (2004). We also rely on an invariance principle for multitype spatial Galton-Watson trees, which is proved in a companion paper.

https://doi.org/10.46298/dmtcs.3505

Source: HAL:hal-01184709v1

Volume: DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities

Section: Proceedings

Published on: January 1, 2006

Imported on: May 10, 2017

Keywords: Random planar map,invariance principle,multitype spatial Galton-Watson tree,Brownian snake,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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16 Documents citing this article

Source : OpenCitations

Addario-Berry, Louigi; Albenque, Marie, 2021, Convergence Of Non-Bipartite Maps Via Symmetrization Of Labeled Trees, Annales Henri Lebesgue, 4, pp. 653-683, 10.5802/ahl.84.

Albenque, Marie; Marckert, Jean-Francois, 2008, Some Families Of Increasing Planar Maps, Electronic Journal Of Probability, 13, none, 10.1214/ejp.v13-563.

Bernardi, Olivier; Curien, Nicolas; Miermont, GrĂŠgory, 2019, A Boltzmann Approach To Percolation On Random Triangulations, Canadian Journal Of Mathematics, 71, 1, pp. 1-43, 10.4153/cjm-2018-009-x.

Bettinelli, Jérémie; Jacob, Emmanuel; Miermont, Grégory, 2014, The Scaling Limit Of Uniform Random Plane Maps, Via The Ambjørn–Budd Bijection, Electronic Journal Of Probability, 19, none, 10.1214/ejp.v19-3213.

Bjรถrnberg, Jakob; Stefรกnsson, Sigurdur, 2014, Recurrence Of Bipartite Planar Maps, Electronic Journal Of Probability, 19, none, 10.1214/ejp.v19-3102.

Curien, Nicolas; Le Gall, Jean-François; Miermont, Grégory, 2013, The Brownian Cactus I. Scaling Limits Of Discrete Cactuses, Annales De l'Institut Henri Poincaré, Probabilités Et Statistiques, 49, 2, 10.1214/11-aihp460.

Gall, Jean-Franรงois, 2010, Geodesics In Large Planar Maps And In The Brownian Map, Acta Mathematica, 205, 2, pp. 287-360, 10.1007/s11511-010-0056-5.

Le Gall, Jean-FranĂ§ois; Miermont, GrĂŠgory, 2011, Scaling Limits Of Random Planar Maps With Large Faces, The Annals Of Probability, 39, 1, 10.1214/10-aop549.

Le Gall, Jean-Franรงois, 2013, Uniqueness And Universality Of The Brownian Map, The Annals Of Probability, 41, 4, 10.1214/12-aop792.

Miermont, Grégory, 2008, Invariance Principles For Spatial Multitype Galton–Watson Trees, Annales De l'Institut Henri Poincaré, Probabilités Et Statistiques, 44, 6, 10.1214/07-aihp157.

Miermont, GrĂŠgory, 2009, Random Maps And Their Scaling Limits, Fractal Geometry And Stochastics IV, pp. 197-224, 10.1007/978-3-0346-0030-9_7.

Miermont, GrĂŠgory, 2013, The Brownian Map Is The Scaling Limit Of Uniform Random Plane Quadrangulations, Acta Mathematica, 210, 2, pp. 319-401, 10.1007/s11511-013-0096-8.

Richier, LoĂŻc, 2017, Limits Of The Boundary Of Random Planar Maps, Probability Theory And Related Fields, 172, 3-4, pp. 789-827, 10.1007/s00440-017-0820-y.

Stephenson, Robin, 2016, Local Convergence Of Large Critical Multi-type GaltonâWatson Trees And Applications To Random Maps, Journal Of Theoretical Probability, 31, 1, pp. 159-205, 10.1007/s10959-016-0707-3.

Stufler, Benedikt, 2021, Quenched Local Convergence Of Boltzmann Planar Maps, Journal Of Theoretical Probability, 35, 2, pp. 1324-1342, 10.1007/s10959-021-01089-2.

Stufler, Benedikt, 2021, Rerooting Multi-type Branching Trees: The Infinite Spine Case, Journal Of Theoretical Probability, 35, 2, pp. 653-684, 10.1007/s10959-020-01069-y.

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