Number rigidity in superhomogeneous random point fields

TL;DR

This paper establishes sufficient conditions for number rigidity in certain translation-invariant or periodic point processes in one and two dimensions, based on variance growth and correlation decay, encompassing all known examples.

Contribution

It provides the first known criteria for number rigidity in superhomogeneous point processes in 1 and 2 dimensions, extending understanding of rigidity phenomena.

Findings

Conditions include sub-volume variance growth and bounded pair correlation functions.

All known number-rigid processes in 1 and 2 dimensions satisfy these conditions.

No such criteria are known or expected to exist in dimensions higher than 2.

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 $d>2$.