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July 2018 Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces
Itai Benjamini, Elliot Paquette, Joshua Pfeffer
Ann. Probab. 46(4): 1917-1956 (July 2018). DOI: 10.1214/17-AOP1216


We show that a random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson–Voronoi tessellation and the hyperbolic Poisson–Delaunay triangulation, have $1$-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive hyperbolic speed. Finally, we include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson–Voronoi tessellation.


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Itai Benjamini. Elliot Paquette. Joshua Pfeffer. "Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces." Ann. Probab. 46 (4) 1917 - 1956, July 2018.


Received: 1 October 2014; Revised: 1 July 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919015
MathSciNet: MR3813982
Digital Object Identifier: 10.1214/17-AOP1216

Primary: 60D05
Secondary: 52C20 , 60G55

Keywords: anchored isoperimetric constant , Expansion , hyperbolic geometry , Hyperbolic space , isoperimetric constant , Poisson process , tessellation , unimodular random graph , Voronoi tiling

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 4 • July 2018
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