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2019 Supercritical causal maps: geodesics and simple random walk
Thomas Budzinski
Electron. J. Probab. 24: 1-43 (2019). DOI: 10.1214/19-EJP341


We study the random planar maps obtained from supercritical Galton–Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in [15], and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the underlying tree has no leaf, we prove that the random walk has positive speed. Some of the methods used here are robust, and allow us to obtain more general results about planar maps containing a supercritical Galton–Watson tree.


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Thomas Budzinski. "Supercritical causal maps: geodesics and simple random walk." Electron. J. Probab. 24 1 - 43, 2019.


Received: 14 November 2018; Accepted: 7 July 2019; Published: 2019
First available in Project Euclid: 10 September 2019

zbMATH: 07107393
MathSciNet: MR4003139
Digital Object Identifier: 10.1214/19-EJP341

Primary: 60D05


Vol.24 • 2019
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