The mesoscopic geometry of sparse random maps
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1305-1345.

We investigate the structure of large uniform random maps with a given number of vertices, edges, faces and on a surface of a given genus. We focus on two regimes: the planar case and the unicellular case, letting the three other parameters tend to infinity in a sparse regime, in which the ratio between the number of vertices and edges tends to 1. Albeit different at first sight, these two models can be treated in a unified way, using a probabilistic version of the classical core–kernel decomposition. In both cases, we identify a mesoscopic scale at which the scaling limits of these random maps can be obtained by first taking the local limit of their kernels (or scheme) – which turns out to be the dual of the Uniform Infinite Planar Triangulation in the planar case and the infinite three-regular tree in the unicellular case – and then replacing each edge by an independent (mass-biased) Brownian tree with two marked points.

Nous étudions la structure de grandes cartes aléatoires choisies uniformément au hasard avec un nombre donné de sommets, d’arêtes et de faces et sur une surface de genre donné. Nous nous concentrons sur deux cas : le cas planaire et le cas unicellulaire, en faisant tendre les trois autres paramètres vers l’infini dans un régime clairsemé, dans lequel le rapport entre le nombre de sommets et d’arêtes tend vers 1. Si les deux cas semblent différents, ils peuvent être traités dans un cadre unifié en utilisant une version probabiliste de la décomposition classique en cœur-noyau. Dans les deux cas, nous identifions une échelle mésoscopique à laquelle les limites d’échelles de ces cartes s’obtiennent en prenant la limite locale de leur noyau (ou schéma) – qui est le dual de la Triangulation Planaire Infinie Uniforme dans le cas planaire et l’arbre infini 3-régulier dans le cas unicellulaire – et en remplaçant chaque arête par des arbres browniens indépendants (biaisés par la taille) avec deux points marqués.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.207
Classification: 05C80, 60D05, 05C07, 05C10, 60C05
Keywords: Random maps, intermediate scaling limits, kernel, drifted Brownian motion
Mot clés : Cartes aléatoires, limites d’échelle intermédiaire, noyau, mouvement brownien avec drift
Nicolas Curien 1; Igor Kortchemski 2; Cyril Marzouk 2

1 Département de Mathématique, Université Paris-Saclay, Faculté des Sciences d’Orsay Orsay, France
2 CNRS et Centre de Mathématiques Appliquées, École Polytechnique Palaiseau, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Nicolas Curien; Igor Kortchemski; Cyril Marzouk. The mesoscopic geometry of sparse random maps. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1305-1345. doi : 10.5802/jep.207. https://jep.centre-mersenne.org/articles/10.5802/jep.207/

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