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.
"Long Brownian bridges in hyperbolic spaces converge to Brownian trees." Electron. J. Probab. 22 1 - 15, 2017. https://doi.org/10.1214/17-EJP68