  It is well known that one can map certain properties of random matrices,
fermionic gases, and zeros of the Riemann zeta function to a unique point
process on the real line. Here we analytically provide exact generalizations of
such a point process in d-dimensional Euclidean space for any d, which are
special cases of determinantal processes. In particular, we obtain the
n-particle correlation functions for any n, which completely specify the point
processes. We also demonstrate that spin-polarized fermionic systems have these
same n-particle correlation functions in each dimension. The point processes
for any d are shown to be hyperuniform. The latter result implies that the pair
correlation function tends to unity for large pair distances with a decay rate
that is controlled by the power law r^[-(d+1)]. We graphically display one- and
two-dimensional realizations of the point processes in order to vividly reveal
their "repulsive" nature. Indeed, we show that the point processes can be
characterized by an effective "hard-core" diameter that grows like the square
root of d. The nearest-neighbor distribution functions for these point
processes are also evaluated and rigorously bounded. Among other results, this
analysis reveals that the probability of finding a large spherical cavity of
radius r in dimension d behaves like a Poisson point process but in dimension
d+1 for large r and finite d. We also show that as d increases, the point
process behaves effectively like a sphere packing with a coverage fraction of
space that is no denser than 1/2^d.
