Tessellations of random maps of arbitrary genus

Grégory Miermont

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 5, page 725-781
  • ISSN: 0012-9593

Abstract

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We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points is linked by a unique geodesic.

How to cite

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Miermont, Grégory. "Tessellations of random maps of arbitrary genus." Annales scientifiques de l'École Normale Supérieure 42.5 (2009): 725-781. <http://eudml.org/doc/272240>.

@article{Miermont2009,
abstract = {We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points is linked by a unique geodesic.},
author = {Miermont, Grégory},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {random maps; scaling limits; random snakes; asymptotic enumeration; geodesics},
language = {eng},
number = {5},
pages = {725-781},
publisher = {Société mathématique de France},
title = {Tessellations of random maps of arbitrary genus},
url = {http://eudml.org/doc/272240},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Miermont, Grégory
TI - Tessellations of random maps of arbitrary genus
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 5
SP - 725
EP - 781
AB - We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points is linked by a unique geodesic.
LA - eng
KW - random maps; scaling limits; random snakes; asymptotic enumeration; geodesics
UR - http://eudml.org/doc/272240
ER -

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