Infinite random planar maps related to Cauchy processes

Timothy Budd; Nicolas Curien; Cyril Marzouk

doi:10.5802/jep.82

Infinite random planar maps related to Cauchy processes
[Cartes planaires aléatoires infinies reliées aux processus de Cauchy]

Timothy Budd ¹ ; Nicolas Curien ² ; Cyril Marzouk ³

¹ Institut de Physique Théorique, CEA, Université Paris-Saclay Orme des Merisiers, F-91191 Gif-sur-Yvette Cedex, France & Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Faculty of Science, HEF 79, 6500 GL Nijmegen, The Netherlands
² Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay Bâtiment 307, 91405 Orsay, France & Institut Universitaire de France
³ Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay Bâtiment 307, 91405 Orsay, France & Fédération mathématique Jacques Hadamard

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 749-791.

Résumé
Abstract

Nous étudions la géométrie de cartes planaires aléatoires infinies de Boltzmann associées à des poids qui décroissent polynomialement de l’ordre de $k^{- 2}$ pour chaque sommet de degré $k$ . Elles correspondent au dual des « cartes stables » discrètes de Le Gall et Miermont [26] étudiées dans [12] et reliées aux processus de Cauchy symétriques, ou encore aux cartes obtenues à partir de la « décomposition en gasket » d’un modèle de boucles $O (2)$ critique sur une carte planaire aléatoire. Nous montrons que ces cartes ont une géométrie surprenante et peu commune. En particulier, nous prouvons que le volume des boules (complétées) de rayon $r$ pour la distance de graphe a une croissance intermédiaire, de l’ordre de $e^{c \sqrt{r}}$ . Nous étudions également la percolation de premier passage avec des poids exponentiels sur les arêtes et montrons que la croissance du volume des boules pour cette distance est désormais de l’ordre de $e^{c r}$ . Finalement, nous étudions la percolation par site sur ces réseaux : bien que le phénomène de percolation ne se produise qu’à $p = 1$ , nous identifions une transition de phase à $p = 1 / 2$ pour la longueur des interfaces ; pour cela, nous prouvons de nouvelles estimées sur les marches aléatoires dans le bassin d’attraction d’un processus de Cauchy asymétrique.

We study the geometry of infinite random Boltzmann planar maps having weight of polynomial decay of order $k^{- 2}$ for each vertex of degree $k$ . These correspond to the dual of the discrete “stable maps” of Le Gall and Miermont [26] studied in [12] related to a symmetric Cauchy process, or alternatively to the maps obtained after taking the gasket of a critical $O (2)$ -loop model on a random planar map. We show that these maps have a striking and uncommon geometry. In particular we prove that the volume of the (hull of the) ball of radius $r$ for the graph distance has an intermediate rate of growth and scales roughly as $e^{c \sqrt{r}}$ . We also perform first passage percolation with exponential edge-weights and show that the volume growth for the fpp-distance scales as $e^{c r}$ . Finally we consider site percolation on these lattices: although percolation occurs only at $p = 1$ , we identify a phase transition at $p = 1 / 2$ for the length of interfaces. On the way we also prove new estimates on random walks attracted to an asymmetric Cauchy process.

Reçu le : 2017-05-04
Accepté le : 2018-09-20
Publié le : 2018-10-09

MR Zbl

DOI : 10.5802/jep.82

Classification : 05C80, 60G52, 60K35
Keywords: Random planar maps, Cauchy processes, Lévy process, peeling exploration, percolation, volume growth, scaling limits
Mot clés : Cartes planaires aléatoires, processus de Cauchy, processus de Lévy, épluchage, percolation, limite d’échelle

Affiliations des auteurs :

Timothy Budd ¹ ; Nicolas Curien ² ; Cyril Marzouk ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2018__5__749_0,
     author = {Timothy Budd and Nicolas Curien and Cyril Marzouk},
     title = {Infinite random planar maps related to {Cauchy} processes},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {749--791},
     publisher = {\'Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.82},
     zbl = {1401.05268},
     mrnumber = {3877166},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.82/}
}

TY  - JOUR
AU  - Timothy Budd
AU  - Nicolas Curien
AU  - Cyril Marzouk
TI  - Infinite random planar maps related to Cauchy processes
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2018
SP  - 749
EP  - 791
VL  - 5
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.82/
DO  - 10.5802/jep.82
LA  - en
ID  - JEP_2018__5__749_0
ER  -

%0 Journal Article
%A Timothy Budd
%A Nicolas Curien
%A Cyril Marzouk
%T Infinite random planar maps related to Cauchy processes
%J Journal de l’École polytechnique — Mathématiques
%D 2018
%P 749-791
%V 5
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.82/
%R 10.5802/jep.82
%G en
%F JEP_2018__5__749_0

Timothy Budd; Nicolas Curien; Cyril Marzouk. Infinite random planar maps related to Cauchy processes. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 749-791. doi : 10.5802/jep.82. https://jep.centre-mersenne.org/articles/10.5802/jep.82/

Bibliographie
Cité par

[1] J. Ambjørn & T. Budd - “Multi-point functions of weighted cubic maps”, Ann. Inst. H. Poincaré D 3 (2016) no. 1, p. 1-44 | MR | Zbl

[2] O. Angel & N. Curien - “Percolations on random maps I: Half-plane models”, Ann. Inst. H. Poincaré Probab. Statist. 51 (2015) no. 2, p. 405-431 | DOI | Numdam | MR | Zbl

[3] Q. Berger - “Notes on random walks in the Cauchy domain of attraction” (2017), arXiv:1706.07924

[4] O. Bernardi, N. Curien & G. Miermont - “A Boltzmann approach to percolation on random triangulations”, Canad. J. Math. (2018), online, arXiv:1705.04064

[5] J. Bertoin - Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996 | Zbl

[6] J. Bertoin & M. Yor - “The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes”, Potential Anal. 17 (2002) no. 4, p. 389-400 | DOI | Numdam | Zbl

[7] N. H. Bingham, C. M. Goldie & J. L. Teugels - Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[8] J. Björnberg & S. Ö. Stefánsson - “Recurrence of bipartite planar maps”, Electron. J. Probab. 19 (2014), article no. 31 | MR | Zbl

[9] G. Borot, J. Bouttier & E. Guitter - “A recursive approach to the $O (n)$ model on random maps via nested loops”, J. Phys. A 45 (2012) no. 4, article no. 045002 | MR | Zbl

[10] T. Budd - “The peeling process of infinite Boltzmann planar maps”, Electron. J. Combin. 23 (2016) no. 1, article no. 1.28 | MR | Zbl

[11] T. Budd - “The peeling process on random planar maps coupled to an $O (n)$ loop model (with an appendix by L. Chen)” (2018), arXiv:1809.02012

[12] T. Budd & N. Curien - “Geometry of infinite planar maps with high degrees”, Electron. J. Probab. 22 (2017), article no. 35 | MR | Zbl

[13] F. Caravenna & L. Chaumont - “Invariance principles for random walks conditioned to stay positive”, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) no. 1, p. 170-190 | DOI | Numdam | MR | Zbl

[14] L. Chaumont - “Excursion normalisée, méandre et pont pour les processus de Lévy stables”, Bull. Sci. Math. 121 (1997) no. 5, p. 377-403 | MR | Zbl

[15] L. Chen, N. Curien & P. Maillard - “The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O (n)$ loop model” (2017), arXiv:1702.06916

[16] N. Curien - “Peeling random planar maps (lecture notes)” (2017), available at https://www.math.u-psud.fr/~curien/cours/peccot.pdf

[17] N. Curien & J.-F. Le Gall - “Scaling limits for the peeling process on random maps”, Ann. Inst. H. Poincaré Probab. Statist. 53 (2017) no. 1, p. 322-357 | DOI | MR | Zbl

[18] N. Curien & J.-F. Le Gall - “First-passage percolation and local modifications of distances in random triangulations”, Ann. Sci. École Norm. Sup. (4) (to appear), arXiv:1511.04264

[19] D. A. Darling - “The maximum of sums of stable random variables”, Trans. Amer. Math. Soc. 83 (1956), p. 164-169 | DOI | MR | Zbl

[20] R. A. Doney - “Local behaviour of first passage probabilities”, Probab. Theory Related Fields 152 (2012) no. 3-4, p. 559-588 | DOI | MR | Zbl

[21] R. A. Doney & M. S. Savov - “The asymptotic behavior of densities related to the supremum of a stable process”, Ann. Probab. 38 (2010) no. 1, p. 316-326 | DOI | MR | Zbl

[22] W. Feller - An introduction to probability theory and its applications. Vol. II., John Wiley & Sons, Inc., New York-London-Sydney, 1971 | Zbl

[23] P. Flajolet & A. Odlyzko - “Singularity analysis of generating functions”, SIAM J. Discrete Math. 3 (1990) no. 2, p. 216-240 | DOI | MR | Zbl

[24] J. Jacod & A. N. Shiryaev - Limit theorems for stochastic processes, Grundlehren Math. Wiss., vol. 288, Springer-Verlag, Berlin, 2003 | MR | Zbl

[25] A. E. Kyprianou, J. C. Pardo & V. Rivero - “Exact and asymptotic $n$ -tuple laws at first and last passage”, Ann. Appl. Probab. 20 (2010) no. 2, p. 522-564 | MR | Zbl

[26] J.-F. Le Gall & G. Miermont - “Scaling limits of random planar maps with large faces”, Ann. Probab. 39 (2011) no. 1, p. 1-69 | MR | Zbl

[27] J.-F. Marckert & G. Miermont - “Invariance principles for random bipartite planar maps”, Ann. Probab. 35 (2007) no. 5, p. 1642-1705 | DOI | MR | Zbl

[28] L. Richier - “Universal aspects of critical percolation on random half-planar maps”, Electron. J. Probab. 20 (2015), article no. 129 | MR | Zbl

[29] S. Sheffield & W. Werner - “Conformal loop ensembles: the Markovian characterization and the loop-soup construction”, Ann. of Math. (2) 176 (2012) no. 3, p. 1827-1917 | DOI | MR | Zbl

[30] R. Stephenson - “Local convergence of large critical multi-type Galton-Watson trees and applications to random maps”, J. Theoret. Probab. 31 (2018) no. 1, p. 159-205 | MR | Zbl

[31] V. A. Vatutin & V. Wachtel - “Local probabilities for random walks conditioned to stay positive”, Probab. Theory Related Fields 143 (2009) no. 1-2, p. 177-217 | DOI | MR | Zbl

Cité par Sources :