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Poisson-Voronoi percolation in higher rank

Jan Grebík, Konstantin Recke

TL;DR

This work analyzes Poisson-Voronoi percolation on symmetric spaces of connected higher rank semisimple Lie groups with property (T), establishing that the uniqueness threshold $\lim_{\lambda\to0} p_u(\lambda)=0$ in the low-intensity limit. The authors develop finitary touching results, a corona-space coupling framework, and a long-range-order criterion to relate local density to global connectivity, using the Delaunay graph and unimodular random graph theory to connect continuum percolation to discrete percolation concepts. As applications, they construct FIID sparse unique infinite cluster processes on non-amenable Cayley graphs and provide FIID sparse unique infinite clusters on lattices in higher rank groups, with arbitrarily small expected degree, addressing questions related to Hutchcroft-Pete and fixed price$^{ }$. The results substantially advance understanding of percolation in non-amenable, highly symmetric spaces, and open pathways to new FIID constructions and cost-one phenomena in property (T) groups.

Abstract

We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally different from situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (arXiv:2307.01194) and provides an alternative proof strategy for Gaboriau's fixed price problem. As a further application of our result, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and arbitrarily small expected degree, answering a question inspired by Hutchcroft-Pete (Invent. math. 221 (2020)).}

Poisson-Voronoi percolation in higher rank

TL;DR

This work analyzes Poisson-Voronoi percolation on symmetric spaces of connected higher rank semisimple Lie groups with property (T), establishing that the uniqueness threshold in the low-intensity limit. The authors develop finitary touching results, a corona-space coupling framework, and a long-range-order criterion to relate local density to global connectivity, using the Delaunay graph and unimodular random graph theory to connect continuum percolation to discrete percolation concepts. As applications, they construct FIID sparse unique infinite cluster processes on non-amenable Cayley graphs and provide FIID sparse unique infinite clusters on lattices in higher rank groups, with arbitrarily small expected degree, addressing questions related to Hutchcroft-Pete and fixed price. The results substantially advance understanding of percolation in non-amenable, highly symmetric spaces, and open pathways to new FIID constructions and cost-one phenomena in property (T) groups.

Abstract

We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally different from situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (arXiv:2307.01194) and provides an alternative proof strategy for Gaboriau's fixed price problem. As a further application of our result, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and arbitrarily small expected degree, answering a question inspired by Hutchcroft-Pete (Invent. math. 221 (2020)).}

Paper Structure

This paper contains 32 sections, 52 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Applications to the sparse FIID unique infinite cluster property
  3. Strategy of proof of the main result, Theorem \ref{['maintheorem']}
  4. Organization
  5. Poisson-Voronoi percolation on symmetric spaces
  6. Lie groups and symmetric spaces
  7. The Poisson-Voronoi percolation model
  8. Finitary conditions: statements
  9. Finitary conditions: proofs of Theorems \ref{['thm:A1']} and Theorem \ref{['thm:A2']}
  10. A lemma about convergence of Poisson processes
  11. The corona space
  12. Coupling of Poisson point processes on ${\bf D}$
  13. Voronoi diagrams, revisited
  14. A lemma about geodesics
  15. Proof of Theorem \ref{['thm:A2']}
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $G$ be a connected higher rank semisimple real Lie group with property (T) and let $(X,d_X)$ be its symmetric space. Then

Theorems & Definitions (122)

  • Theorem 1.1: Vanishing uniqueness thresholds
  • Remark 1.2: Continuity of $p_u$
  • Definition 1.3
  • Theorem 1.5: Cayley graphs with the FIID sparse unique infinite cluster property
  • Remark 1.7
  • Corollary 1.8: FIID sparse unique infinite clusters
  • Theorem 1.9: Long-range order implies uniqueness
  • Theorem 1.10: Intersection of cells at small intensities
  • Remark 1.11
  • Theorem 1.12: Long-range order threshold
  • ...and 112 more