Number Rigidity in Superhomogeneous Random Point Fields (vol 166, pg 1016, 2017)

We give sufficient conditions for the number rigidity of a large class of point processes in dimension and 2, based on the decay of correlations. Number rigidity implies that the probability distribution of the number of particles in a bounded domain , conditional on the configuration on , is concentrated on a single integer . Our conditions are: (a) for all x, where and are the intensity and the truncated pair correlation function resp., and (b) is bounded by in and by in . Condition (a) covers a wide class of processes, including translation invariant or periodic point process on , , that are superhomogeneous or hyperuniform (that is the variance of the number of particles in a bounded domain grows slower than the volume of ). It also covers determinantal point processes having a projection kernel. Our conditions for number rigidity are satisfied by all known processes with number rigidity in . We also observe, in the light of the results of [26], that no such criteria exist in d > 2.