A point process (random point pattern) is said to be hyperuniform if the variance of the number of points in a large ball grows more slowly than its volume, i.e., if this number has a significant smaller fluctuation than that for a homogeneous Poisson process. A point process is said to be (number) rigid if the number of points inside a given compact set is (almost surely) determined by the number of points outside. The local behavior of such processes can very much resemble that of a weakly correlated point process as for instance a Poisson process or a Gibbs process with short-range interactions. A regular geometric pattern, homogeneous like a crystal, might become visible only on a global scale. Examples of hyperuniform processes are perturbed lattices, the Ginibre process (describing the eigenvalues of a Gaussian matrix), some other determinantal processes, stable submatchings of a Poisson process, and the Coulomb gas. Examples of rigid point processes are the aforementioned stable submatchings of a Poisson process, Gibbs processes with certain long-range interactions, zeros of Gaussian entire functions, and determinantal processes with a projection kernel. Physicists have observed these exotic states of matter in random jammed packings, disordered quantum ground states, quasicrystals, Coulomb systems, and photonic disordered solids. The first goal of this project is to study stationary transports of point processes (or more general random measures) and to establish assumptions guaranteeing the persistence of the asymptotic variance. This should be used for the construction of new interesting examples of hyperuniform point processes. The second goal of this project is the mathematical analysis of the relationships between rigidity and hyperuniformity within a certain class of point processes. This class will be defined by a suitable coupling of the stationary and the Palm distribution and will also be used to quantify the local degree of attraction and repulsion. The third goal is to extend a test on hyperuniformity developed in the first stage of this project to random closed sets and other hyperuniform processes.

A point process (random point pattern) is said to be (number) rigid if the number of points inside a given compact set is (almost surely) determined by the number of points outside. It is said to be hyperuniform if the variance of the number of points in a large ball grows more slowly than its volume, that is, if this number has a significant smaller fluctuation than the corresponding number for a Poisson process with the same intensity (mean) measure. The local behavior of such processes can very much resemble that of a weakly correlated point process as for instance a Poisson process or a Gibbs process with short-range interactions. Only on a global scale a regular crystalline geometric pattern might become visible. Examples of hyperuniform processes are perturbed lattices, the Ginibre process (describing the eigenvalues of a Gaussian matrix), some other determinantal processes, stable submatchings of a Poisson process and the three-dimensional hierarchical Coulomb gas. Examples of rigid point processes are the stable submatchings of a Poisson process, Gibbs processes with certain long-range interactions, zeros of Gaussian entire functions and some determinantal processes with a projection kernel. Physicists have observed these exotic states of matter in random jammed packings, disordered quantum ground states, quasicrystals, Coulomb systems, and photonic disordered solids. Hyperuniform structures have also been observed in biology.

It has been recently proved that in one or two dimensions hyperuniformity implies rigidity provided that the spatial air-correlation function decays sufficiently fast. In dimension three and higher this has been shown to be wrong. The first goal of this project is the mathematical analysis of the exact relationships between rigidity, decay of correlations and hyperuniformity.

A second goal of this project is to establish persistence properties of hyper- uniformity under matchings and more general transports of random measures. This should be used for the construction of new interesting examples of hyperuniform point processes. More generally it is planned to study monotonicity of asymptotic variances under transports of general random mea- sures.

A third goal is the construction of explicit couplings of the stationary and the Palm distribution for a preferably large class of point processes and to use this to characterize hyperuniformity and to quantify the local degree of attraction and repulsion. Such couplings are also of relevance in sextreme value theory (Poisson approximation) and statistical physics.