Daniela Flimmel - Hyperuniformity of perturbed lattices

We construct a perturbed lattice by performing possibly spatially correlated displacements to each of the components of a standard d-dimensional lattice. Such models are celebrated in physics and material science as an easy to handle model for crystalic types of matter, but nowadays also in probability theory. To study the asymptotic fluctuations in the number of points, a standard tool is assessing the small scale behaviour of the structure factor. However, in its common definition, it is not rigorously defined for lattices. We introduce a new, generalized version of the structure factor using theory of distributions and find conditions for hyperuniformity (suppressed density fluctuations) of perturbed lattices.