Number rigidity in superhomogeneous random point fields

Abstract

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 $d>2$.

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