Hyperuniform point processes
A point process in Euclidean space is said to be hyperuniform,
if the variance of the number of points in a large ball grows significantly
more slowly than its volume. It turns out that this
property is closely related to number rigidity. The latter property
means that the number of points inside a given compact set is almost
surely determined by the configuration of points outside.
Rigidity and hyperuniformity are properties that unify crystals and
exceptional random systems.
The local behavior of processes with those properties
can very much resemble that of a weakly correlated point process.
Only on a global scale a regular geometric pattern might become visible.
In the first part of the talk we provide some examples
and discuss a few fundamental properties of hyperuniform
point processes. Then we shall present
a new class of hyperuniform point processes introduced and studied
in recent joint work with M. Klatt (Princeton) and D. Yogeshwaran (Bangalore).
It arises by a peculiar thinning of a stationary Poisson process
(or a more general determinantal point process) based on a stable
matching procedure. This thinning
is also number rigid.