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

Abstract

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 ind-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain then-particle correlationfunctions 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 anyd are shown to be hyperuniform, i.e., infinite wavelength densityfluctuations vanish, and the structure factor (or power spectrum)S(k) has a non-analytic behavior at the origin given byS(k)∼|k| (). The latter result implies that the pair correlation functiong2(r) tends to unity for large pair distances with a decay rate that is controlled by the power law1/rd+1, which is a well-known property of bosonic ground states and more recently has beenshown to characterize maximally random jammed sphere packings. We graphically displayone-and two-dimensional realizations of the point processes in order to vividlyreveal their ‘repulsive’ nature. Indeed, we show that the point processes can becharacterized by an effective ‘hard core’ diameter that grows like the square root ofd. The nearest-neighbor distribution functions for these point processes arealso evaluated and rigorously bounded. Among other results, this analysisreveals that the probability of finding a large spherical cavity of radiusr indimension d behaves like a Poisson point process but in dimensiond+1, i.e., this probabilityis given by exp[−κ(d)rd+1] for large r andfinite d, whereκ(d) is a positived-dependent constant.We also show that as d increases, the point process behaves effectively like a spherepacking with a coverage fraction of space that is no denser than1/2d. This coverage fraction has a special significance in the study of sphere packings inhigh-dimensional Euclidean spaces.