Existence of disordered stealthy hyperuniform point processes in d ≥ 2

Establish whether there exist stationary disordered (e.g., isotropic and/or mixing) stealthy hyperuniform point processes on Euclidean spaces of dimension at least two by constructing such processes whose spectral density vanishes on a non-empty open set and which are not finite unions of shifted lattices, or prove that no such processes exist.

Background

Stealthy processes are wide-sense stationary random measures whose spectral density vanishes on a non-empty open set. They exhibit very strong forms of rigidity and are of major interest in physics and image analysis, where simulated samples appear both disordered and stealthy.

Mathematically, the only rigorously established examples in Euclidean space are finite unions of shifted lattices, which are not representative of disordered configurations. Demonstrating the existence (or impossibility) of truly disordered stealthy hyperuniform point processes would resolve a fundamental gap between empirical observations and current theory.

References

The existence of such models has not been established mathematically, except some toy models such as unions of shifted lattices, see Example \ref{ex:irrational-lattices}.

— Hyperuniform random measures, transport and rigidity  (2510.18392 - Lachièze-Rey, 21 Oct 2025) in Section 5.2, Further questions on stealthy point processes (Rigidity chapter)

While a growing family of systems spanning physics, chemistry, biology, ecology, and social systems have been shown to display emergent long-range correlations in disordered structures, many open questions remain on the realizability and properties of subclasses of correlated disorder, making numerical generation strategies invaluable, and even useful in practice to manufacture structures.

— Fast generation of spectrally-shaped disorder, on the sphere  (2608.24867 - Casiulis et al., 25 Aug 2026) in Section 1, Introduction