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arXiv · 1609.01907

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Xinxin Chen·Grégory Miermont
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Abstract

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 $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ 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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Xinxin Chen, Grégory Miermont. 2016-09-07. Long Brownian bridges in hyperbolic spaces converge to Brownian trees. https://arxiv.org/abs/1609.01907

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