Percolations on random maps I: Half-plane models
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 405-431

We study Bernoulli percolations on random maps in the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process (Geom. Funct. Anal. 13 (2003) 935–974) of these random maps we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical annealed exponents related to percolation clusters such as the probabilities of a cluster having a large volume or perimeter.

Nous étudions différentes percolations de Bernoulli sur les cartes aléatoires du demi-plan obtenues comme limites locales de triangulations ou quadrangulations planaires uniformes. En utilisant la propriété de Markov spatiale – ou épluchage (Geom. Funct. Anal. 13 (2003) 935–974) – de ces réseaux, nous prouvons une formule simple et universelle pour le paramètre critique de percolation par arêtes ou par sites sur ces cartes. Nos techniques nous permettent également de calculer certains exposants « annealed » presque-critiques et critiques comme la probabilité qu’un cluster ait un grand volume ou un grand périmètre.

DOI : https://doi.org/10.1214/13-AIHP583
Classification:  60K37,  60K35,  05C80
Keywords: random planar map, percolation, critical exponent
@article{AIHPB_2015__51_2_405_0,
     author = {Angel, Omer and Curien, Nicolas},
     title = {Percolations on random maps I: Half-plane models},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {405-431},
     doi = {10.1214/13-AIHP583},
     mrnumber = {3335009},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_405_0}
}
Angel, Omer; Curien, Nicolas. Percolations on random maps I: Half-plane models. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 405-431. doi : 10.1214/13-AIHP583. http://www.numdam.org/item/AIHPB_2015__51_2_405_0/

[1] L. Addario-Berry and B. A. Reed. Ballot theorems, old and new. In Horizons of Combinatorics. Bolyai Soc. Math. Stud. 17 9–35. Springer, Berlin, 2008. | MR 2432525 | Zbl 1151.91412

[2] D. Aldous and J. M. Steele. The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin, 2004. | MR 2023650 | Zbl 1037.60008

[3] J. Ambjørn, B. Durhuus and T. Jonsson. Quantum Geometry: A Statistical Field Theory Approach. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 1997. | MR 1465433 | Zbl 1096.82500

[4] O. Angel. Scaling of percolation on infinite planar maps, I. Available at arXiv:math/0501006.

[5] O. Angel. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003) 935–974. | MR 2024412 | Zbl 1039.60085

[6] O. Angel and N. Curien. Percolations on infinite random maps II, full-plane models. Unpublished manuscript.

[7] O. Angel and G. Ray. Classification of domain Markov half planar maps. Ann. Probab. To appear, 2015. Available at arXiv:1303.6582. | MR 3342664

[8] O. Angel and O. Schramm. Uniform infinite planar triangulation. Comm. Math. Phys. 241 (2003) 191–213. | MR 2013797 | Zbl 1098.60010

[9] V. Beffara. Hausdorff dimensions for SLE 6 . Ann. Probab. 32 (2004) 2606–2629. | MR 2078552 | Zbl 1055.60036

[10] I. Benjamini and N. Curien. Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points. Geom. Funct. Anal. 23 (2013) 501–531. | MR 3053754 | Zbl 1274.60143

[11] I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001) 23 (electronic). | MR 1873300 | Zbl 1010.82021

[12] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | Zbl 0938.60005

[13] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) Research Paper 69 (electronic). | MR 2097335 | Zbl 1060.05045

[14] J. Bouttier and E. Guitter. Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A 42 (2009) 465208. | MR 2552016 | Zbl 1179.82069

[15] P. Chassaing and B. Durhuus. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006) 879–917. | MR 2243873 | Zbl 1102.60007

[16] N. Curien and J.-F. Le Gall. The Brownian plane. Available at arXiv:1204.5921. | MR 3278940 | Zbl 1305.05208

[17] N. Curien, L. Ménard and G. Miermont. A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 45–88. | MR 3083919 | Zbl 1277.05151

[18] N. Curien and G. Miermont. Uniform infinite planar quadrangulations with a boundary. Random Structures Algorithms. To appear, 2015. Available at arXiv:1202.5452. | MR 3366810

[19] R. A. Doney. On the exact asymptotic behaviour of the distribution of ladder epochs. Stochastic Process. Appl. 12 (1982) 203–214. | MR 651904 | Zbl 0482.60066

[20] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2011) 333–393. | MR 2819163 | Zbl 1226.81241

[21] I. P. Goulden and D. M. Jackson. Combinatorial Enumeration. Wiley-Interscience Series in Discrete Mathematics. Wiley, New York, 1983. | MR 702512 | Zbl 0519.05001

[22] O. Gurel-Gurevich and A. Nachmias. Recurrence of planar graph limits. Ann. of Math. (2) 177 (2013) 761–781. | MR 3010812 | Zbl 1262.05031

[23] V. A. Kazakov. Percolation on a fractal with the statistics of planar Feynman graphs: Exact solution. Modern Phys. Lett. A 17 (1989) 1691–1704. | MR 1016993

[24] V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov. Fractal structure of 2D-quantum gravity. Modern Phys. Lett. A 3 (1988) 819–826. | MR 947880

[25] M. Krikun. Local structure of random quadrangulations. Available at arXiv:math/0512304.

[26] M. Krikun. On one property of distances in the infinite random quadrangulation. Available at arXiv:0805.1907.

[27] M. Krikun. Explicit enumeration of triangulations with multiple boundaries. Electron. J. Combin. 14 (2007) Research Paper 61 (electronic). | MR 2336338 | Zbl 1157.05031

[28] J.-F. Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013) 2880–2960. | MR 3112934 | Zbl 1282.60014

[29] J.-F. Le Gall and L. Ménard. Scaling limits for the uniform infinite quadrangulation. Illinois J. Math. 54 (2010) 1163–1203. | MR 2928350 | Zbl 1259.60035

[30] J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642–1705. | MR 2349571 | Zbl 1208.05135

[31] L. Ménard and P. Nolin. Percolation on uniform infinite planar maps. Available at arXiv:1302.2851. | Zbl 1300.60114

[32] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013) 319–401. | MR 3070569 | Zbl 1278.60124

[33] G. Schaeffer. Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. thesis, 1998.

[34] V. A. Vatutin and V. Wachtel. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009) 177–217. | MR 2449127 | Zbl 1158.60014

[35] Y. Watabiki. Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 (1995) 119–163. | MR 1329946 | Zbl 0990.81657