Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Percolations on random maps I: Half-plane models

Omer Angel and Nicolas Curien

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 405-431.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672675

Digital Object Identifier
doi:10.1214/13-AIHP583

Mathematical Reviews number (MathSciNet)
MR3335009

Zentralblatt MATH identifier
1315.60105

Subjects
Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Keywords
Random planar map Percolation Critical exponent

Citation

Angel, Omer; Curien, Nicolas. Percolations on random maps I: Half-plane models. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 405--431. doi:10.1214/13-AIHP583. https://projecteuclid.org/euclid.aihp/1428672675


Export citation

References

  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [8] O. Angel and O. Schramm. Uniform infinite planar triangulation. Comm. Math. Phys. 241 (2003) 191–213.
  • [9] V. Beffara. Hausdorff dimensions for $\mathrm{SLE}_{6}$. Ann. Probab. 32 (2004) 2606–2629.
  • [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.
  • [11] I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001) 23 (electronic).
  • [12] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
  • [13] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) Research Paper 69 (electronic).
  • [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.
  • [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.
  • [16] N. Curien and J.-F. Le Gall. The Brownian plane. Available at arXiv:1204.5921.
  • [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.
  • [18] N. Curien and G. Miermont. Uniform infinite planar quadrangulations with a boundary. Random Structures Algorithms. To appear, 2015. Available at arXiv:1202.5452.
  • [19] R. A. Doney. On the exact asymptotic behaviour of the distribution of ladder epochs. Stochastic Process. Appl. 12 (1982) 203–214.
  • [20] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2011) 333–393.
  • [21] I. P. Goulden and D. M. Jackson. Combinatorial Enumeration. Wiley-Interscience Series in Discrete Mathematics. Wiley, New York, 1983.
  • [22] O. Gurel-Gurevich and A. Nachmias. Recurrence of planar graph limits. Ann. of Math. (2) 177 (2013) 761–781.
  • [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.
  • [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.
  • [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).
  • [28] J.-F. Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013) 2880–2960.
  • [29] J.-F. Le Gall and L. Ménard. Scaling limits for the uniform infinite quadrangulation. Illinois J. Math. 54 (2010) 1163–1203.
  • [30] J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642–1705.
  • [31] L. Ménard and P. Nolin. Percolation on uniform infinite planar maps. Available at arXiv:1302.2851.
  • [32] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013) 319–401.
  • [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.
  • [35] Y. Watabiki. Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 (1995) 119–163.