The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection (original) (raw)

The scaling limit of uniform random plane maps, via the

2013

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambjørn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

Scaling limits of random planar maps with a unique large face

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n −1/2 is described by a Brownian excursion. The planar maps, with the graph metric rescaled by n −1/2 , are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

The scaling limit of random outerplanar maps

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2016

A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with n vertices suitably rescaled by a factor 1/ √ n converge in the Gromov-Hausdorff sense to 7 √ 2/9 times Aldous' Brownian tree. The proof uses the bijection of Bonichon, Gavoille and Hanusse (J. Graph Algorithms Appl. 9 (2005) 185-204). Résumé. Une carte planaire est dite outerplanaire si tous ses sommets appartiennent à la même face. Nous montrons que les cartes outerplanaires aléatoires uniformes à n sommets, multipliées par le facteur d'échelle 1/ √ n, convergent au sens de Gromov-Hausdorff vers 7 √ 2/9 fois l'arbre Brownien d'Aldous. La preuve utilise la bijection de Bonichon, Gavoille et Hanusse (J. Graph Algorithms Appl. 9 (2005) 185-204).

Scaling limit of random planar quadrangulations with a boundary

Arxiv preprint arXiv:1111.7227, 2011

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence (σn) of integers such that σn/ √ 2n tends to some σ ∈ [0, ∞]. For every n ≥ 1, we call qn a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having n faces and 2σn half-edges on the boundary. For σ ∈ (0, ∞), we view qn as a metric space by endowing its set of vertices with the graph metric, rescaled by n −1/4 . We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov-Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the two-dimensional disc. For σ = 0, the same convergence holds without extraction and the limit is the so-called Brownian map. For σ = ∞, the proper scaling becomes σ −1/2 n and we obtain a convergence toward Aldous's CRT.

On scaling limits of random Halin-like maps

arXiv (Cornell University), 2021

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the α-stable looptrees of Curien and Kortchemski.

On scaling limits of random Halin-like maps with large faces

2021

We consider maps which are constructed from plane trees by connecting each pair of consecutive leaves on their contour by a single edge. When every non-leaf vertex of the underlying tree has exactly one child which is a leaf, these maps are bijectively related to plane trees with marked corners. We study the scaling limits of such maps where Boltzmann weights are assigned to each face. The main result is that when the degree of a typical face is in the domain of attraction of an α-stable law, with α ∈ (1, 2), the maps, properly rescaled, converge to the α-stable looptrees of Curien and Kortchemski.

The topology of scaling limits of positive genus random quadrangulations

Arxiv preprint arXiv:1012.3726, 2010

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every n ≥ 1, a random quadrangulation qn uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph distance. As n tends to infinity, this metric space, with distances rescaled by the factor n −1/4 , converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus g-torus. * This work is partially supported by ANR-08- in the sense of the Gromov-Hausdorff topology, where

Scaling limits for random quadrangulations of positive genus

Electronic Journal of Probability}, 2010

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every n ≥ 1, a random quadrangulation qn uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as n tends to infinity, this metric space, with distances rescaled by the factor n −1/4 , converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4.

A random mapping with preferential attachment

Random Structures and Algorithms, 2009

In this paper we investigate the asymptotic structure of a random mapping model with preferential attachment, T ρ n , which maps the set {1, 2, ..., n} into itself. The model T ρ n was introduced in a companion paper and the asymptotic structure of the associated directed graph G ρ n which represents the action of T ρ n on the set {1, 2, ..., n} was investigated in [11] and [12] in the case when the attraction parameter ρ > 0 is fixed as n → ∞. In this paper we consider the asymptotic structure of G ρ n when the attraction parameter ρ ≡ ρ(n) is a function of n as n → ∞. We show that there are three distinct regimes during the evolution of G ρ n : (i) ρn → ∞ as n → ∞, (ii) ρn → β > 0 as n → ∞, and (iii) ρn → 0 as n → ∞. It turns out that the asymptotic structure of G ρ n is, in some cases, quite different from the asymptotic structure of well-known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function.

A cutting process for random mappings

Random Structures and Algorithms, 2007

In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph G n on n labelled vertices, and we delete (if possible), uniformly and at random, m non-cyclic directed edges from G n . The maximal random digraph consisting of the uni-cyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by G m n . If the number of non-cyclic directed edges is less than m, then G m n consists of the cycles, including loops, of the initial mapping G n . We consider the component structure of the trimmed mapping G m n . In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of G m n as n, m → ∞. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: