Generic point processes are intermittently hyperuniform and maximally rigid

Abstract

We study the prevalence of hyperuniformity and spatial rigidity among translation- invariant ergodic simple point processes of fixed intensity, in a topology controlling local second moments. Hyperuniform processes are dense but meagre. We prove that intermittent hyperuniformity is residual: the number variance exhibits hyperuniform behavior along some sequences of radii and hyperfluctuating behavior along others. Maximal rigidity is also residual. Thus generic processes are maximally rigid but not hyperuniform. A local-to-global principle reduces the proofs to cross-section constructions for an arbitrary fixed essentially free ergodic $\mathbb{R}^d$-action. We also prove that processes admitting revival sequences form a residual set and exhibit near-quadratic variance along suitable van Hove sequences.