[Cartes planaires aléatoires infinies reliées aux processus de Cauchy]
Nous étudions la géométrie de cartes planaires aléatoires infinies de Boltzmann associées à des poids qui décroissent polynomialement de l’ordre de pour chaque sommet de degré . 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 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 pour la distance de graphe a une croissance intermédiaire, de l’ordre de . 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 . Finalement, nous étudions la percolation par site sur ces réseaux : bien que le phénomène de percolation ne se produise qu’à , nous identifions une transition de phase à 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 for each vertex of degree . 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 -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 for the graph distance has an intermediate rate of growth and scales roughly as . We also perform first passage percolation with exponential edge-weights and show that the volume growth for the fpp-distance scales as . Finally we consider site percolation on these lattices: although percolation occurs only at , we identify a phase transition at for the length of interfaces. On the way we also prove new estimates on random walks attracted to an asymmetric Cauchy process.
Accepté le :
Publié le :
DOI : 10.5802/jep.82
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
Timothy Budd 1 ; Nicolas Curien 2 ; Cyril Marzouk 3
CC-BY-ND 4.0
@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/
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