Hyperuniformity and Number-Rigidity for Point Processes

David Dereudre (Univ. Lille)

Abstract:

Recently two notions of rigidity for spatial stationary point processes appear in the literature. 
The first one is the Hyperuniformity property, which claims that the variance of the number of points 
in any bounded domain D is strictly sub-linear with respect to the volume of D. This notion has been 
introduced by physicists to describe the nature of crystals or pseudo-crystals. The second notion is 
the Number-rigidity property, which claims that in any bounded domain D, the configuration of points 
outside D determines almost-surely the number of points inside D. This notion has been introduced 
by mathematicians to describe unusual rigidity properties for point processes. It is not difficult 
to see that the standard Poisson Point Process is not Hyperuniform nor Number-rigid. The Poisson Point 
process is not rigid enough! During the talk we will give the main results involving these rigidity 
properties. In particular, we will see that they are not equivalent but not so far. We will see that 
some determinantal point processes, Gibbs point processes or perturbed lattices are Hyperuniform or/and 
Number-rigid. Several conjectures will be presentedas well.