Number rigidity in superhomogeneous random point fields

We give sufficient conditions for the number rigidity of a translation invariant or periodic point process on $\mathbb{R}^d$, where $d=1,2$. That is, the probability distribution of the number of particles in a bounded domain $\Lambda \subset \mathbb{R}^d$, conditional on the configuration on $\Lambda^\complement$, is concentrated on a single integer $N_\Lambda$. These conditions are : (a) the variance of the number of particles in a bounded domain $\mathcal{O} \subset \mathbb{R}^d$ grows slower than the volume of $\mathcal{O}$ (a.k.a. superhomogeneous point processes), when $\mathcal{O} \uparrow \mathbb{R}^d$ (in a self-similar manner), and (b) the truncated pair correlation function is bounded by $C_1[|x-y|+1]^{-2}$ in $d=1$ and by $C_2[|x-y|+1]^{-(4+\epsilon)}$ in $d=2$. These conditions are satisfied by all known processes with number rigidity ([GP],[G],[PS],[AM],[Bu],[BuDQ], [BBNY], and many more) in $d=1,2$. We also observe, in the light of the results of [PS], that no such criteria exist in $d>2$.