Number Rigidity in Superhomogeneous Random Point Fields
Abstract
We give sufficient conditions for the number rigidity of a large class of point processes in dimension d=1 and 2, based on the decay of correlations. Number rigidity implies that the probability distribution of the number of particles in a bounded domain Λ subset R^d, conditional on the configuration on Λ ^\complement , is concentrated on a single integer N_Λ . Our conditions are: (a) ρ _1(x)= - int _{R^d} ρ _tr^{(2)}(x,y) dy for all x, where ρ _1 and ρ _tr^{(2)} are the intensity and the truncated pair correlation function resp., and (b)|ρ _tr^{(2)}(x,y)| is bounded by C_1[|x-y|+1]^{-2} in d=1 and by C_2[|x-y|+1]^{-(4+ɛ)} in d=2. Condition (a) covers a wide class of processes, including translation invariant or periodic point process on R^d, d=1,2, that are superhomogeneous or hyperuniform (that is the variance of the number of particles in a bounded domain Ω subset R^d grows slower than the volume of Ω ). 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 d=1,2. We also observe, in the light of the results of [26], that no such criteria exist in d>2.
- Publication:
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Journal of Statistical Physics
- Pub Date:
- February 2017
- DOI:
- 10.1007/s10955-016-1633-6
- arXiv:
- arXiv:1601.04216
- Bibcode:
- 2017JSP...166.1016G
- Keywords:
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- Mathematics - Probability;
- Condensed Matter - Statistical Mechanics;
- Mathematical Physics
- E-Print:
- doi:10.1007/s10955-016-1633-6