Nearest Neighbor Functions for Stealthy Hyperuniform Many-particle Ground States

We present new analytical theories backed by simulation for the nearest neighbor functions of disordered stealthy hyperuniform many-particle ground states. Stealthy systems display a variety of ordered and disordered phases, and have been related to novel photonic band gaps in disordered systems. They can also be shown to have optimal transport properties, such as the fluid permeability, trapping constant, and elastic properties. The nearest neighbor functions, which include the probability to find a hole of a certain radius, are intrinsically related to the geometric properties of stealthy systems, such as their bounded hole size and quantizer error. The manner in which these functions approach the critical (bounded) hole size is a fundamental geometrical question (in particular, whether they behave like ordered lattices), and is also related to their thermodynamics. We use insights from simulation and previous analytical work to develop new analytical theories of the core and tail regions of these functions, and comment on their implications for geometric problems and material properties.