  We study the Voronoi and void statistics of super-homogeneous (or
hyperuniform) point patterns in which the infinite-wavelength density
fluctuations vanish. Super-homogeneous or hyperuniform point patterns arise in
one-component plasmas, primordial density fluctuations in the Universe, and in
jammed hard-particle packings. We specifically analyze a certain
one-dimensional model by studying size fluctuations and correlations of the
associated Voronoi cells. We derive exact results for the complete joint
statistics of the size of two Voronoi cells. We also provide a sum rule that
the correlation matrix for the Voronoi cells must obey in any space dimension.
In contrast to the conventional picture of super-homogeneous systems, we show
that infinitely large Voronoi cells or voids can exist in super-homogeneous
point processes in any dimension.
  We also present two heuristic conditions to identify and classify any
super-homogeneous point process in terms of the asymptotic behavior of the void
size distribution.
