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

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 in . We also demonstrate that spin-polarized fermionic systems in have these same n-particle correlation functions in each dimension. The point processes for any d are shown to be hyperuniform, i.e., infinite wavelength density fluctuations vanish, and the structure factor (or power spectrum) S(k) has a non-analytic behavior at the origin given by S(k)∼|k| (). The latter result implies that the pair correlation function g₂(r) tends to unity for large pair distances with a decay rate that is controlled by the power law 1/r^d+1, which is a well-known property of bosonic ground states and more recently has been shown to characterize maximally random jammed sphere packings. 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, i.e., this probability is given by exp[−κ(d)r^d+1] for large r and finite d, where κ(d) is a positive d-dependent constant. 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. This coverage fraction has a special significance in the study of sphere packings in high-dimensional Euclidean spaces.