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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 pointprocess on the real line. Here we analytically provide exact generalizations ofsuch a point process in d-dimensional Euclidean space for any d, which arespecial cases of determinantal processes. In particular, we obtain then-particle correlation functions for any n, which completely specify the pointprocesses. We also demonstrate that spin-polarized fermionic systems have thesesame n-particle correlation functions in each dimension. The point processesfor any d are shown to be hyperuniform. The latter result implies that the paircorrelation function tends to unity for large pair distances with a decay ratethat is controlled by the power law r^[-(d+1)]. We graphically display one- andtwo-dimensional realizations of the point processes in order to vividly revealtheir "repulsive" nature. Indeed, we show that the point processes can becharacterized by an effective "hard-core" diameter that grows like the squareroot of d. The nearest-neighbor distribution functions for these pointprocesses are also evaluated and rigorously bounded. Among other results, thisanalysis reveals that the probability of finding a large spherical cavity ofradius r in dimension d behaves like a Poisson point process but in dimensiond+1 for large r and finite d. We also show that as d increases, the pointprocess behaves effectively like a sphere packing with a coverage fraction ofspace that is no denser than 1/2^d.

Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory