This research demonstrates a perfect matching between a perturbed lattice and point process, suggesting implications for geometric probability.
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.
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Elboim et al. (2026) studied this question.
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