Duality of random planar maps via percolation
Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2425-2471.

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 κ16/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.

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 κ16/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.

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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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

[1] Angel, Omer Growth and percolation on the uniform infinite planar triangulation, Geom. Funct. Anal., Volume 13 (2003) no. 5, pp. 935-974 | DOI | MR | Zbl

[2] Angel, Omer Scaling of Percolation on Infinite Planar Maps, I (2004) (http://arxiv.org/abs/math/0501006)

[3] Angel, Omer; Curien, Nicolas Percolations on random maps I: Half-plane models, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 2, pp. 405-431 | DOI | Numdam | MR | Zbl

[4] Bernardi, Olivier; Curien, Nicolas; Miermont, Grégory A Boltzmann approach to percolation on random triangulations, Can. J. Math., Volume 71 (2019) no. 1, pp. 1-43 | DOI | MR | Zbl

[5] Bertoin, Jean; Budd, Timothy; Curien, Nicolas; Kortchemski, Igor Martingales in self-similar growth-fragmentations and their connections with random planar maps, Probab. Theory Relat. Fields, Volume 172 (2018) no. 3-4, pp. 663-724 | DOI | MR | Zbl

[6] Bingham, Nicholas H.; Goldie, Charles M.; Teugels, Jozef L. Regular Variation, Cambridge University Press, 1989 | Zbl

[7] Björnberg, Jakob E.; Stefánsson, Sigurdur O. On Site Percolation in Random Quadrangulations of the Half-Plane, J. Stat. Phys., Volume 160 (2015) no. 2, pp. 336-356 | DOI | MR | Zbl

[8] Borot, Gaëtan; Bouttier, Jérémie; Guitter, Emmanuel A recursive approach to the O(n) model on random maps via nested loops, J. Phys. A, Math. Theor., Volume 45 (2012) no. 4, p. 045002 | DOI | MR | Zbl

[9] Borovkov, Aleksandr A.; Borovkov, Konstantin A. Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2008 | Zbl

[10] Bouttier, Jérémie; Di Francesco, Philippe; Guitter, Emmanuel Planar maps as labeled mobiles, Electron. J. Comb., Volume 11 (2004) no. 1, R69, 27 pages | MR | Zbl

[11] Budd, Timothy The Peeling Process of Infinite Boltzmann Planar Maps, Electron. J. Comb., Volume 23 (2016) no. 1, pp. 1-28 | MR | Zbl

[12] Budd, Timothy; Curien, Nicolas Geometry of infinite planar maps with high degrees, Electron. J. Probab., Volume 22 (2017) no. 35, pp. 1-37 | DOI | MR | Zbl

[13] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A Course in Metric Geometry, Graduate Studies in Mathematics, American Mathematical Society, 2001

[14] Curien, Nicolas Peeling random planar maps, available at www.u-psud.fr/ curien/, 2019 (Saint-Flour summer course)

[15] Curien, Nicolas; Kortchemski, Igor Percolation on random triangulations and stable looptrees, Probab. Theory Relat. Fields, Volume 163 (2014) no. 1-2, pp. 303-337 | DOI | MR | Zbl

[16] Curien, Nicolas; Kortchemski, Igor Random stable looptrees, Electron. J. Probab., Volume 19 (2014) no. 108, pp. 1-35 | DOI | MR | Zbl

[17] Denisov, Denis; Dieker, A. B.; Shneer, Vsevolod Large deviations for random walks under subexponentiality: The big-jump domain, Ann. Probab., Volume 36 (2008) no. 5, pp. 1946-1991 | DOI | MR | Zbl

[18] Doney, Ronald A. On the exact asymptotic behaviour of the distribution of ladder epochs, Stochastic Processes Appl., Volume 12 (1982) no. 2, pp. 203-214 | DOI | MR | Zbl

[19] Duplantier, Bertrand; Sheffield, Scott Liouville quantum gravity and KPZ, Invent. Math., Volume 185 (2011), pp. 333-393 | DOI | MR | Zbl

[20] Feller, William An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1971

[21] Gorny, Matthias; Maurel-Segala, Édouard; Singh, Arvind The geometry of a critical percolation cluster on the UIPT, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 4, pp. 2203-2238 | MR | Zbl

[22] Gwynne, Ewain; Miller, Jason Convergence of percolation on uniform quadrangulations with boundary to SLE 6 on 8/3-Liouville quantum gravity (2017) (https://arxiv.org/abs/1701.05175)

[23] Ibragimov, Ilʼdar A.; Linnik, Yuriĭ V. Independent and stationary sequences of random variables, Wolters-Noordhof Publishing, 1971

[24] Janson, Svante; Stefánsson, Sigurdur O. Scaling limits of random planar maps with a unique large face, Ann. Probab., Volume 43 (2015) no. 3, pp. 1045-1081 | DOI | MR | Zbl

[25] Knizhnik, Vadim G.; Polyakov, Alexsandr M.; Zamolodchikov, Alexander B. Fractal structure of 2D-quantum gravity, Mod. Phys. Lett. A, Volume 03 (1988) no. 08, pp. 819-826 | DOI

[26] Kortchemski, Igor Invariance principles for Galton-Watson trees conditioned on the number of leaves, Stochastic Processes Appl., Volume 122 (2012) no. 9, pp. 3126-3172 | DOI | MR | Zbl

[27] Le Gall, Jean-François Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR | Zbl

[28] Le Gall, Jean-François; Miermont, Grégory Scaling limits of random trees and planar maps (2011) (http://arxiv.org/abs/1101.4856)

[29] Marckert, Jean-François; Miermont, Grégory Invariance principles for random bipartite planar maps, Ann. Probab., Volume 35 (2007) no. 5, pp. 1642-1705 | DOI | MR | Zbl

[30] Marzouk, Cyril Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Struct. Algorithms, Volume 53 (2018) no. 3, pp. 448-503 | DOI | MR

[31] Ménard, Laurent; Nolin, Pierre Percolation on uniform infinite planar maps, Electron. J. Probab., Volume 19 (2014) no. 0 | DOI | MR

[32] Miermont, Grégory The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., Volume 210 (2013) no. 2, pp. 319-401 | DOI | MR

[33] Miller, Jason; Sheffield, Scott; Werner, Wendelin CLE percolations, Forum Math. Pi, Volume 5 (2017) | DOI | MR | Zbl

[34] Richier, Loïc Universal aspects of critical percolation on random half-planar maps, Electron. J. Probab., Volume 20 (2015) no. 0 | DOI | MR | Zbl

[35] Richier, Loïc Limits of the boundary of random planar maps, Probab. Theory Relat. Fields (2017), pp. 1-39 | DOI | MR | Zbl

[36] Richier, Loïc The incipient infinite cluster of the uniform infinite half-planar triangulation, Electron. J. Probab., Volume 23 (2018), 89, 38 pages | MR | Zbl

[37] Watabiki, Yoshiyuki Construction of Non-critical String Field Theory by Transfer Matrix Formalism in Dynamical Triangulation, Nucl. Phys., B, Volume 441 (1995) no. 1-2, pp. 119-163 | DOI | MR | Zbl

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