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

TL;DR

This paper investigates optimal invariant matchings between a point process and graph vertices on non-amenable unimodular graphs, establishing the minimal matching distances and their realizability via deterministic schemes.

Contribution

It characterizes the optimal matching distances for Poisson and i.i.d. perturbation processes on non-amenable unimodular graphs and constructs explicit factor matching schemes.

Findings

Determines the minimal matching distances for specific point processes.

Shows these optimal matchings can be achieved by deterministic, equivariant functions.

Provides explicit constructions of factor matching schemes.

Abstract

Consider a unit-intensity point process $\Pi$ on the vertex set $V$ of a transitive non-amenable unimodular graph. We study invariant matchings between $\Pi$ and $V$ having small typical matching distances. When $\Pi$ 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 $\Pi$).