  Hyperuniform point patterns are characterized by vanishing infinite
wavelength density fluctuations and encompass all crystal structures, certain
quasi-periodic systems, and special disordered point patterns. This article
generalizes the notion of hyperuniformity to include also two-phase random
heterogeneous media. Hyperuniform random media do not possess
infinite-wavelength volume fraction fluctuations, implying that the variance in
the local volume fraction in an observation window decays faster than the
reciprocal window volume as the window size increases. For microstructures of
impenetrable and penetrable spheres, we derive an upper bound on the asymptotic
coefficient governing local volume fraction fluctuations in terms of the
corresponding quantity describing the variance in the local number density
(i.e., number variance). Extensive calculations of the asymptotic number
variance coefficients are performed for a number of disordered (e.g.,
quasiperiodic tilings, classical stealth disordered ground states, and certain
determinantal point processes), quasicrystal, and ordered (e.g., Bravais and
non-Bravais periodic systems) hyperuniform structures in various Euclidean
space dimensions, and our results are consistent with a quantitative order
metric characterizing the degree of hyperuniformity. We also present
corresponding estimates for the asymptotic local volume fraction coefficients
for several lattice families. Our results have interesting implications for a
certain problem in number theory.
