Optimal matchings of randomly perturbed lattices

TL;DR

This paper constructs a translation-invariant perfect matching between a randomly perturbed lattice and the original lattice, demonstrating that the matching distance's tail behavior matches the hole probability of the point process, under mild assumptions.

Contribution

It introduces a method to create a translation-invariant perfect matching with optimal tail behavior for perturbed lattices.

Findings

Matching distance has tail behavior matching hole probability

Constructs translation-invariant perfect matching

Applicable under mild perturbation assumptions

Abstract

Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.