  We study the variance in the number of points contained within a window
$\Omega$ of arbitrary size, and to further illuminate our understanding of {\it
hyperuniform} systems, i.e., point patterns that do not possess long-wavelength
fluctuations. For large windows, hyperuniform systems are characterized by a
local variance that grows only as the surface area (rather than the volume) of
the window. We show that a homogeneous point pattern in a hyperuniform state is
at a ``critical-point'' of a type with appropriate scaling laws and critical
exponents, but one in which the {\it direct correlation function} (rather than
the pair correlation function) is long-ranged.variance.We prove that the simple
periodic linear array yields the global minimum value of the average variance
among all infinite one-dimensional hyperuniform patterns. We also evaluate the
variance for common infinite periodic lattices as well as certain nonperiodic
point patterns in one, two, and three dimensions for spherical windows.Our
results suggest that the local variance may serve as a useful order metric for
general point patterns. Contrary to the conjecture that the lattices associated
with the densest packing of congruent spheres have the smallest variance
regardless of the space dimension, we show that for $d=3$, the body-centered
cubic lattice has a smaller variance than the face-centered cubic lattice.
Finally, for certain hyperuniform disordered point patterns, we evaluate the
direct correlation function, structure factor, and associated critical
exponents exactly.
