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

Abstract

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 R. Here we analytically provide exact generalizations of such a point process in d-dimensional Euclidean space R d 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 R d . We also demonstrate that spinpolarized fermionic systems in R d 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 nonanalytic behavior at the origin given by S(k) ∼ |k| (k → 0). The latter result implies that the pair correlation function g 2 (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 expPoint processes in arbitrary dimension 2 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.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres

“…This property is markedly different from typical liquids, in which pair correlations decay exponentially fast [2,5,6]. Similar QLR behavior has also been observed in noninteracting spin-polarized fermionic ground states [7,8], the ground state of liquid helium [9], and the Harrison-Zeldovich power spectrum of the density fluctuations of the early Universe [10].…”

“…Although the expansion (9) will hold for all periodic and quasiperiodic point patterns with Bragg peaks, this behavior is not generally true for disordered hyperuniform systems. For example, it is known that if the total correlation function [7]. Such behavior occurs in maximally random jammed monodisperse sphere packings in three dimensions [4] and noninteracting spin-polarized fermion ground states [7,8].…”

“…For example, it is known that if the total correlation function [7]. Such behavior occurs in maximally random jammed monodisperse sphere packings in three dimensions [4] and noninteracting spin-polarized fermion ground states [7,8]. Other examples of "anomalous" local density fluctuations have been characterized by Zachary and Torquato [6].…”

Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres

“…Disordered hyperuniform materials, for example, behave more like crystals in the manner they suppress density fluctuations over large length scales, and yet they also resemble traditional liquids and glasses with statistically isotropic structures with no Bragg peaks. During the last decade, hyperuniform disordered states have been identified in maximally random jammed packings of hard particles [3][4][5], jammed athermal granular systems [6], jammed thermal colloidal packings [7], cold atoms [8], certain Coulombic systems [9], and "stealthy" disordered classical ground states [10]. Furthermore, the hyperuniformity property has been suggested to endow materials with novel physical properties potentially important for applications in photonics [11][12][13] and electronics [14][15][16] .…”

Local Central Limit Theorem for Determinantal Point Processes

“…Spin polarized free fermions in dimension d provide examples of determinantal point processes in higher dimensions [33]. With k F = 2 √ π(Γ(1 + d/2)) 1/d , the corresponding bulk scaled (unit density) kernel is computed to equal c…”