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Optimal factor matchings for point processes on non-amenable unimodular graphs

Yinon Spinka, Oren Yakir

TL;DR

This work develops a framework for optimal invariant matchings between point processes on non-amenable unimodular graphs. By constructing a random bipartite graph with radii-based connection thresholds and leveraging strong expansion, the authors adapt the Lyons–Nazarov algorithm to produce a factor perfect matching between two processes (Poisson or perturbed vertex sets) with an exponential tail in the matching distance, quantified in terms of ball volume $b_r$. The main contribution is proving the existence of such factor matchings under a total-order assumption and showing the tail is optimal up to constants, with the result robust to dependence between the two processes as long as they are i.i.d. factors of a common source. Additionally, the paper provides verifiable sufficient conditions for the total-order property and develops key lemmas on tail bounds and absence of infinite clusters to ensure the construction is well-controlled. This yields deterministic, equivariant matchings on hyperbolic-like graphs with strong expansion, advancing understanding of point-process couplings in non-Euclidean settings and connecting to prior Lyons–Nazarov type results in non-amenable graphs.

Abstract

Consider a unit-intensity point process $Π$ on the vertex set $V$ of a transitive non-amenable unimodular graph. We study invariant matchings between $Π$ and $V$ having small typical matching distances. When $Π$ is either a Poisson process or i.i.d. perturbations of the vertex set, we determine the optimal matching distance and show that it can be attained by a factor matching scheme (that is, a deterministic and equivariant function of $Π$).

Optimal factor matchings for point processes on non-amenable unimodular graphs

TL;DR

This work develops a framework for optimal invariant matchings between point processes on non-amenable unimodular graphs. By constructing a random bipartite graph with radii-based connection thresholds and leveraging strong expansion, the authors adapt the Lyons–Nazarov algorithm to produce a factor perfect matching between two processes (Poisson or perturbed vertex sets) with an exponential tail in the matching distance, quantified in terms of ball volume . The main contribution is proving the existence of such factor matchings under a total-order assumption and showing the tail is optimal up to constants, with the result robust to dependence between the two processes as long as they are i.i.d. factors of a common source. Additionally, the paper provides verifiable sufficient conditions for the total-order property and develops key lemmas on tail bounds and absence of infinite clusters to ensure the construction is well-controlled. This yields deterministic, equivariant matchings on hyperbolic-like graphs with strong expansion, advancing understanding of point-process couplings in non-Euclidean settings and connecting to prior Lyons–Nazarov type results in non-amenable graphs.

Abstract

Consider a unit-intensity point process on the vertex set of a transitive non-amenable unimodular graph. We study invariant matchings between and having small typical matching distances. When is either a Poisson process or i.i.d. perturbations of the vertex set, we determine the optimal matching distance and show that it can be attained by a factor matching scheme (that is, a deterministic and equivariant function of ).
Paper Structure (13 sections, 7 theorems, 78 equations, 2 figures)

This paper contains 13 sections, 7 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Optimal tail for matching distance
  4. Related works
  5. A random bipartite graph with good expansion
  6. The bipartite graph
  7. The expansion
  8. The Lyons-Nazarov matching algorithm
  9. Proof of main result
  10. Proofs of supporting lemmas
  11. Tail bounds for the matching distance
  12. Absence of infinite clusters
  13. Sufficient conditions for \ref{['eq:assumption_total_order']}

Key Result

Theorem 1.1

Let $G$ be a connected transitive non-amenable unimodular graph and let $b_r$ denote the volume of a ball of radius $r$ in $G$. Let $\Pi$ be a perturbed vertex set or a Poisson process on $G$. Let $\Pi'$ be another such process (of either type), independent of $\Pi$. Assume that eq:assumption_total_ for all $v\in V$ and for all $r$ large enough.

Figures (2)

  • Figure 1: Left: A chain of length 9, beginning at the red vertex $x_1$ and ending at the blue vertex $y_5$. Dashed edges correspond to edges not contained in the partial matching. Right: The corresponding chain after flipping. Solid edges correspond to edges contained in the (new) partial matching.
  • Figure 2: The infinite ladder graph with added diagonal edges.

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lemma:lyons_nazarov_style_chebyshev']}
  • proof : Proof of Lemma \ref{['lemma:boosted_hall']}
  • Claim 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:factor_matching_for_transitive_non-amenable_graphs']}
  • ...and 10 more