The Annals of Probability

Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces

Itai Benjamini, Elliot Paquette, and Joshua Pfeffer

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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.

Article information

Ann. Probab., Volume 46, Number 4 (2018), 1917-1956.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]

Hyperbolic space hyperbolic geometry Poisson process Voronoi tiling tessellation isoperimetric constant anchored isoperimetric constant expansion unimodular random graph


Benjamini, Itai; Paquette, Elliot; Pfeffer, Joshua. Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces. Ann. Probab. 46 (2018), no. 4, 1917--1956. doi:10.1214/17-AOP1216.

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