Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Chen, Xinxin; Miermont, Grégory

doi:10.1214/17-ejp68

Cited by 4 publications

(4 citation statements)

References 33 publications

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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 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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“…Both when taking the ambient metric on the connected set, as an embedded subset, or the internal metric. See [6] for a related result and definition of the scaling limit. We conjecture that Brownian tree is also the scaling limit of self avoiding loop in the ambient metric.…”

Section: Comments On Saw On Large Graphsmentioning

confidence: 99%

Self avoiding walk on the seven regular triangulation

Benjamini

2016

Preprint

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“…This limit theorem for R b (n) is related to earlier works on scaling limits of the range of tree-valued critical or near-critical biased random walks (RWs): in particular we refer to D. [12] who deals with near-critical biased RWs on b-ary trees, to Y. Peres and O. Zeitouni [30] who show that the distance to the root of a critical biased RW in a Galton-Watson environment is diffusive, to A. Dembo and N. Sun [8] who study the cases of critical biased RWs on N -type GW-trees, to E. Aïdékon and L. de Raphélis [2] who improve Y. Peres and O. Zeitouni's result and who show that the range of the same RW converges when suitably rescaled, to a variant of the Brownian CRT, and to X. Chen and G. Miermont [6] who show that rescaled Brownian bridges and loops in hyperbolic spaces converge to the Brownian CRT. Their work is based on a previous results due to P. Bougerol and T. Jeulin [5].…”

Section: Introductionmentioning

confidence: 99%

Scaling limits of tree-valued branching random walks

Duquesne

Khanfir

Lin

2022

Electron. J. Probab.

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We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (i.e. the set of finite words written in the alphabet {1, . . . , b}, with b ≥ 2). The BRW is indexed by a critical Galton-Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, γ ∈ (1, 2].The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 1/2 to move closer to the root of Wb and probability 1/(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)n∈N satisfying an → ∞ such that the metric space (Rb(n), a −1 n dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

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Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Cited by 4 publications

References 33 publications

The boundary of random planar maps via looptrees

The boundary of random planar maps via looptrees

Self avoiding walk on the seven regular triangulation

Scaling limits of tree-valued branching random walks

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