Void probabilities for perturbed lattices under mixing

Source abstract

We study void probabilities for stationary perturbed lattices with possibly dependent perturbations. Our main result gives a general upper bound in terms of two ingredients: the number of lattice sites having a uniformly positive probability of hitting the observation set, and the $α$-mixing rate of the perturbation field. For unions of lattice cubes, this yields explicit bounds without moment or regularity assumptions on the marginal distribution of the perturbations. As a consequence, we obtain quantitative estimates for growing bounded Borel sets with non-empty interior, including stretched-exponential decay under exponential mixing and polynomial decay under polynomial mixing.