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

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 $\Pi$).