(Non)-hyperuniformity of perturbed lattices

We ask whether a stationary lattice in dimension d whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite d-moment, and that this condition is sharp. When d ≥ 3, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.