Infinite geodesics in hyperbolic random triangulations

Thomas Budzinski

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arXiv · 1804.07711

Infinite geodesics in hyperbolic random triangulations

Thomas Budzinski

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Abstract

We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_{\lambda}$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_{\lambda}$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain new hyperbolicity properties of $\mathbb{T}_{\lambda}$: it satisfies a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.

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BibTeX RIS

Thomas Budzinski. 2018-04-20. Infinite geodesics in hyperbolic random triangulations. https://arxiv.org/abs/1804.07711

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