The boundary of random planar maps via looptrees

Abstract

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://afst. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Annales de la faculté des sciences de Toulouse Volume XXIX, n o 2, 2020 pp. 391-430 The boundary of random planar maps via looptrees (*)

Local convergence of large random triangulations coupled with an Ising model

“…We will see in this work that in this regime, the root spin cluster corresponds to a non generic critical Boltzmann map. Kortchemski and Richier [49] proved that, in the bipartite case, the boundary of such random maps converges to the circle C 1 in the scaling limit, agreeing with our result. This result at criticality can shed some light on the link between random maps coupled with an Ising model at the critical temperature and the O √ 2 model in the dilute regime, where interfaces are supposed to be simple in the scaling limit, see for example the papers by Borot, Bouttier, Duplantier and Guitter [12,13,14,15].…”

“…In the same article, the authors also prove that the scaling limit boundary of a supercritical site percolation cluster in the UIPT is the unit length cycle C 1 for the Gromov-Hausdorff topology. Similar results were obtained for the boundary of random bipartite Boltzmann maps by Kortchemski and Richier [49].…”

“…This implies that the probability to see a given root spin cluster in a random triangulation with spins is driven by the degrees of its faces. One of the most important models of random maps having this kind of property are the so-called Boltzmann maps which have been intensively studied in the recent years [23,34,49,55,57,58,60].…”

“…To study the root spin cluster, we will interpret it in the context of Boltzmann planar maps associated to a given weight sequence, which has been studied intensively in the literature [23,49,55,57,58,60]. In this section, we first review the necessary material about this model of maps.…”

The phase transition for parking on Galton--Watson trees

“…To define a spine in T n , conditionally on T n we sample a uniform vertex V n ∈ T n . It is standard that the height |V n | of V n converges once renormalized by √ n towards a Rayleigh distribution, more precisely, we have the following local limit law established in [10,Eq (12)]…”