Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits for the peeling process on random maps

Nicolas Curien and Jean-François Le Gall

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or quadrangulation (UIPQ). In particular, our results apply to the metric exploration or peeling by layers algorithm, where the discovered regions are (almost) completed balls, or hulls, centered at the root vertex. The scaling limits of the perimeter and volume of hulls can be expressed in terms of the hull process of the Brownian plane studied in our previous work. Other applications include the metric exploration of the dual graph of our infinite random lattices, and first-passage percolation with exponential edge weights on the dual graph, also known as the Eden model or uniform peeling.

Résumé

Nous étudions la limite d’échelle du processus des volumes et des périmètres des régions explorées par un algorithme « d’épluchage » sur les cartes infinies aléatoires telles que l’UIPT (la triangulation infinie uniforme du plan) ou son analogue quadrangulaire l’UIPQ. Nos résultats s’appliquent en particulier à l’exploration des boules (pour la distance de graphe) complétées et centrées à la racine de la carte. Dans ce cas, la limite d’échelle coïncide avec le processus du périmètre et du volume des boules complétées dans le plan brownien. Parmi les autres applications, mentionnons l’exploration des boules complétées sur la carte duale et la percolation de premier passage avec poids exponentiels sur la carte duale. Ce dernier modèle, équivalent au modèle d’Eden sur la carte initiale, correspond à l’algorithme d’épluchage uniforme.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 322-357.

Dates
Received: 19 December 2014
Revised: 19 July 2015
Accepted: 1 October 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544894

Digital Object Identifier
doi:10.1214/15-AIHP718

Mathematical Reviews number (MathSciNet)
MR3606744

Zentralblatt MATH identifier
1358.05255

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F17: Functional limit theorems; invariance principles
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Random planar maps Peeling process Scaling limits Lévy process

Citation

Curien, Nicolas; Le Gall, Jean-François. Scaling limits for the peeling process on random maps. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 322--357. doi:10.1214/15-AIHP718. https://projecteuclid.org/euclid.aihp/1486544894


Export citation

References

  • [1] J. Ambjørn and T. Budd. Multi-point functions of weighted cubic maps. Ann. Inst. Henri Poincaré D 3 (1) (2016) 1–44.
  • [2] O. Angel. Scaling of percolation on infinite planar maps, I. Available at arXiv:math/0501006.
  • [3] O. Angel. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003) 935–974.
  • [4] O. Angel and N. Curien. Percolations on infinite random maps, half-plane models. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 405–431.
  • [5] O. Angel and G. Ray. Classification of half-planar maps. Ann. Probab. 43 (2015) 1315–1349.
  • [6] O. Angel and O. Schramm. Uniform infinite planar triangulation. Comm. Math. Phys. 241 (2003) 191–213.
  • [7] 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.
  • [8] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996.
  • [9] J. Bertoin and R. A. Doney. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994) 2152–2167.
  • [10] J. Bouttier and E. Guitter. Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A 42 (2009) 465208.
  • [11] T. Budd. The peeling process of infinite Boltzmann planar maps. Electron. J. Combin. 23 (1) (2016) Paper 1.28.
  • [12] F. Caravenna and L. Chaumont. Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 170–190.
  • [13] P. Chassaing and B. Durhuus. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006) 879–917.
  • [14] N. Curien. Planar stochastic hyperbolic triangulations. Probab. Theory Related Fields 165 (3–4) (2016) 509–540.
  • [15] N. Curien. A glimpse of the conformal structure of random planar maps. Comm. Math. Phys. 333 (2015) 1417–1463.
  • [16] N. Curien and J.-F. Le Gall. First-passage percolation and local modifications of distances in random planar maps. Available at arxiv:1511.04264.
  • [17] N. Curien and J.-F. Le Gall. The hull process of the Brownian plane. Probab. Theory Related Fields 166 (1–2) (2016) 187–231.
  • [18] N. Curien and J.-F. Le Gall. The Brownian plane. J. Theoret. Probab. 27 (2014) 1249–1291.
  • [19] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen, 1971.
  • [20] T. Jeulin. Sur la convergence absolue de certaines intégrales. In Séminaire de Probabilités XVI 248–256. Lecture Notes in Mathematics 920. Springer, Berlin, 1982.
  • [21] M. Krikun. Local structure of random quadrangulations. Available at arXiv:0512304.
  • [22] M. Krikun. A uniformly distributed infinite planar triangulation and a related branching process. J. Math. Sci. (N. Y.) 131 (2005) 5520–5537.
  • [23] M. Krikun. Explicit enumeration of triangulations with multiple boundaries. Electron. J. Combin. 14 (2007) P61.
  • [24] R. Lyons and Y. Peres. Probability on trees and networks. Available at http://mypage.iu.edu/~rdlyons/.
  • [25] L. Ménard. The two uniform infinite quadrangulations of the plane have the same law. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 190–208.
  • [26] L. Ménard and P. Nolin. Percolation on uniform infinite planar maps. Electron. J. Probab. 19 (2014) 79.
  • [27] J. Miller and S. Sheffield. Quantum Loewner evolution. Duke Math. J. 165 (17) (2016) 3241–3378.
  • [28] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer, Berlin, 2006.
  • [29] G. Ray. Geometry and percolation on half planar triangulations. Electron. J. Probab. 19 (2014) 47.
  • [30] R. Stephenson. Local convergence of large critical multi-type Galton–Watson trees and applications to random maps. Available at arXiv:1412.6911 and http://link.springer.com/article/10.1007/s10959-016-0707-3.
  • [31] Y. Watabiki. Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 (1995) 119–163.