The CRT is the scaling limit of random dissections

Curien, Nicolas; Haas, Bénédicte; Kortchemski, Igor

doi:10.1002/rsa.20554

Cited by 37 publications

(49 citation statements)

References 28 publications

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“…Since then, the study of scaling limits of random discrete structures such as trees, graphs and planar maps has developed into a very active field with contributions by a wide variety of researchers [CHK15,HM12,Bet15,JS15,PS15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Scaling limits of random outerplanar maps with independent link-weights

Stufler¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edgerooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us to treat subclasses such as bipartite outerplanar maps.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Scaling limits of random outerplanar maps with independent link-weights

Stufler¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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“…The CRT plays a central role in the study of the geometric shape of large discrete structures. It arises as scaling limit for a variety of models [5,[7][8][9]15,18,19,30,36]. Although scaling limits describe asymptotic global properties, they do not contain information on local properties, such as the limiting degree distribution of a randomly chosen vertex in a graph.…”

Section: Figurementioning

confidence: 99%

The continuum random tree is the scaling limit of unlabeled unrooted trees

Stufler

2018

Random Struct Algorithms

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We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.

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“…As mentioned in Remark 2.8, we do not know if this is satisfied for all subcritical sequences. However, we believe that [25,Theorem 14] holds under a finite variance assumption, by proving tightness of the sequence of laws and identifying the finite-dimensional marginals.…”

Section: Remark 42mentioning

confidence: 99%

Limits of the boundary of random planar maps

Richier

2017

Probab. Theory Relat. Fields

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We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α ∈ (1, 2). First, in the dense phase corresponding to α ∈ (1, 3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α − 1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α ∈ (3/2, 2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien & Kortchemski.

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The CRT is the scaling limit of random dissections

Cited by 37 publications

References 28 publications

Scaling limits of random outerplanar maps with independent link-weights

Scaling limits of random outerplanar maps with independent link-weights

The continuum random tree is the scaling limit of unlabeled unrooted trees

Limits of the boundary of random planar maps

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