  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 $d>2$.
