no. 6
Duality of random planar maps via percolation
[Dualité des cartes planaires aléatoires via la percolation]
Curien, Nicolas1 ; Richier, Loïc2
1 Institut de mathématiques d’Orsay Université Paris-Saclay 91405 Orsay (France)
2 CMAP Ecole Poytechnique 91120 Palaiseau (France)
Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2425-2471.

On étudie la percolation par arête critique sur une carte planaire de Boltzmann “stable” de paramètre α∈(3/2,2]. On montre en particulier que la composante connexe de l’origine est elle-même une carte de Boltzmann “stable” de paramètre

α ′ :=2α+3 4α-2.

C’est le pendant dans la théorie des cartes planaires de la dualité κ↔16/κ 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.

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 α∈(1,2]. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes α∈(3/2,2], and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime α∈(1,3/2), with parameter

α ′ :=2α+3 4α-2.

This is the counterpart in random planar maps of the duality property κ↔16/κ 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.

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DOI : 10.5802/aif.3369
Classification : 60K35, 60D05, 05A16
Keywords: Random planar maps, bond percolation, peeling process
Mot clés : Cartes planaires aléatoires, percolation, processus d’épluchage

Curien, Nicolas 1 ; Richier, Loïc 2

1 Institut de mathématiques d’Orsay Université Paris-Saclay 91405 Orsay (France)
2 CMAP Ecole Poytechnique 91120 Palaiseau (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Curien, Nicolas; Richier, Loïc. Duality of random planar maps via percolation. Annales de l'Institut Fourier, Tome 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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