Scaling limits of random outerplanar maps with independent link-weights

Stufler, Benedikt

doi:10.48550/arxiv.1505.07600

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2017

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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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