A perturbed lattices is a point process \Pi = {x + Yx : x ∈ Z^d } where the lattice points in Z^d are perturbed by independent and identically distributed random variables {Y_x}_{x\in Z^d} . Deletion tolerance concerns the question of whether it is possible to distinguish with certainty if a point has been removed. Holroyd and Soo showed that in dimensions d = 1, 2 the Gaussian...