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.