Scaling limits of random outerplanar maps with independent link-weights

Stufler, Benedikt

doi:10.1214/16-aihp741

Cited by 18 publications

(20 citation statements)

References 26 publications

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“…Caraceni showed that for uniform outerplanar maps the inner faces typically have small degree, and the map has a tree‐like geometric shape. A similar behavior was observed later for further natural combinatorial classes of outerplanar maps . But which interesting phenomena may be observed in weighted outerplanar maps?…”

Section: Introductionsupporting

confidence: 80%

Geometry of large Boltzmann outerplanar maps

Stefánsson

Stufler

2018

Random Struct Algorithms

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We study the phase diagram of random outerplanar maps sampled according to nonnegative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The Boltzmann outerplanar maps are then shown to converge in the Gromov-Hausdorff sense towards the -stable looptree introduced by Curien and Kortchemski (2014), with the parameter depending on the specific weight-sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree. KEYWORDSenriched trees, looptrees, outerplanar maps, random trees INTRODUCTIONA planar map may be described as a proper embedding of a connected graph into the sphere, considered up to continuous orientation-preserving deformations. In order to eliminate possible internal symmetries one usually distinguishes and orients a root edge. The probabilistic study of these objects has lead to a rich and beautiful theory, see for example the survey paper [31] and references given therein. The connected components of the complement of a planar map are its faces, and the unique face that lies to the right of the oriented edge is termed the outer face. This face is usually drawn as the unbounded face in plane representations. The number of edges adjacent to a face is its degree. A planar map is termed outerplanar, if all its vertices are adjacent to the outer face. Classes of outerplanar maps have received some attention in recent literature: Bonichon, Gavoille, and Hanusse [5] gave a combinatorial encoding of outerplanar maps in terms of certain bi-colored plane trees. Combining this encoding with probabilistic techniques, Caraceni [8] described the asymptotic geometric shape of uniform outerplanar maps, establishing Aldous' Brownian tree as their Gromov-Hausdorff scaling limit. The scaling limit and the asymptotic enumerative formula for Random Struct Alg. 2019;55:742-771.wileyonlinelibrary.com/journal/rsa

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Section: Introductionsupporting

confidence: 80%

Geometry of large Boltzmann outerplanar maps

Stefánsson

Stufler

2018

Random Struct Algorithms

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“…We refer to [1,5,15,16,17,30,48,52,53] for a zoology of random discrete structures which are not trees, but whose scaling limits are T e , the Brownian CRT.…”

Section: Scaling Limits Of Looptrees (Crt Regime)mentioning

confidence: 99%

The boundary of random planar maps via looptrees

Kortchemski¹,

Richier²

2020

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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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 (*)

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“…We naturally distinguish the point o e = p e (0), where p e is the canonical projection, and will usually write T e instead of (T e , d e , o e ). This random metric space (or more precisely its isometry class) appears as the universal scaling limit of many tree-like random objects that naturally appear in combinatorics and probability, see for instance [20] for a survey, and [11,12,19,24,25,26,28,30] for some recent developments on the topic. Here we show that the CRT also appears naturally in this more geometric context.…”

Section: Resultsmentioning

confidence: 99%

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Chen¹,

Miermont

2017

Electron. J. Probab.

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We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of • A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion,• A property of invariance under re-rooting,• The hyperbolicity of the ambient space in the sense of Gromov.A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

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Scaling limits of random outerplanar maps with independent link-weights

Cited by 18 publications

References 26 publications

Geometry of large Boltzmann outerplanar maps

Geometry of large Boltzmann outerplanar maps

The boundary of random planar maps via looptrees

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

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