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Anchored expansion of Delaunay complexes in real hyperbolic space and stationary point processes

This paper is dedicated to Harry Kesten, whom the first author holds in his loving memory and whom we all admire

Abstract

We give sufficient conditions for a discrete set of points in any dimensional real hyperbolic space to have positive anchored expansion. The first condition is an anchored bounded density property, ensuring not too many points can accumulate in large regions. The second is an anchored bounded vacancy condition, effectively ensuring there is not too much space left vacant by the points over large regions. These properties give as an easy corollary that stationary Poisson–Delaunay graphs have positive anchored expansion, as well as Delaunay graphs built from stationary determinantal point processes.

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Fig. 1

Notes

  1. For unimodular random graphs, anchored nonamenability implies the weaker invariant nonamenability. When the degree of root of such an invariantly nonamenable graph has finite expectation, random walk will escape from the root with linear rate in any (stationary) pseudometric on the graph in which balls grow subexponentially. In particular, random walk will have positive speed almost surely in the hyperbolic metric when run on a stationary point process (c.f. the proof of [19, Corollary 1.10] or [2]).

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Correspondence to Elliot Paquette.

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Benjamini, I., Krauz, Y. & Paquette, E. Anchored expansion of Delaunay complexes in real hyperbolic space and stationary point processes. Probab. Theory Relat. Fields 181, 197–209 (2021). https://doi.org/10.1007/s00440-021-01076-y

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Mathematics Subject Classification

  • Primary 60D05
  • Secondary 52C20
  • 60G55