We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter . We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes , and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime , with parameter
This is the counterpart in random planar maps of the duality property of Schramm–Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner [33]. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.
On étudie la percolation par arête critique sur une carte planaire de Boltzmann “stable” de paramètre . On montre en particulier que la composante connexe de l’origine est elle-même une carte de Boltzmann “stable” de paramètre
C’est le pendant dans la théorie des cartes planaires de la dualité des processus Schramm–Loewner (SLE) et des ensembles de boucles conformes (CLE) récemment établie par Miller, Sheffield et Werner [33]. En bonus, on identifie la limite d’échelle du bord des grands amas de percolation critiques et on prouve la décroissance exponentielle de la taille des amas dans le régime sous-critique.
Revised:
Accepted:
Published online:
Keywords: Random planar maps, bond percolation, peeling process
Mot clés : Cartes planaires aléatoires, percolation, processus d’épluchage
Curien, Nicolas 1; Richier, Loïc 2
@article{AIF_2020__70_6_2425_0, author = {Curien, Nicolas and Richier, Lo{\"\i}c}, title = {Duality of random planar maps via percolation}, journal = {Annales de l'Institut Fourier}, pages = {2425--2471}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {6}, year = {2020}, doi = {10.5802/aif.3369}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3369/} }
TY - JOUR AU - Curien, Nicolas AU - Richier, Loïc TI - Duality of random planar maps via percolation JO - Annales de l'Institut Fourier PY - 2020 SP - 2425 EP - 2471 VL - 70 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3369/ DO - 10.5802/aif.3369 LA - en ID - AIF_2020__70_6_2425_0 ER -
%0 Journal Article %A Curien, Nicolas %A Richier, Loïc %T Duality of random planar maps via percolation %J Annales de l'Institut Fourier %D 2020 %P 2425-2471 %V 70 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3369/ %R 10.5802/aif.3369 %G en %F AIF_2020__70_6_2425_0
Curien, Nicolas; Richier, Loïc. Duality of random planar maps via percolation. Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2425-2471. doi : 10.5802/aif.3369. https://aif.centre-mersenne.org/articles/10.5802/aif.3369/
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