Reformulation of the covering and quantizer problems as ground states of interacting particles

Abstract

It is known that the sphere packing problem and the number variance problem (closely related to an optimization problem in number theory) can be posed as energy minimizations associated with an infinite number of point particles in d-dimensional Euclidean space R d interacting via certain repulsive pair potentials. We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in R d that generally involve singlebody, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the "void" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems. The covering and quantizer problems have relevance in numerous applications, including wireless communication network layouts, the search of high-dimensional data parameter spaces, stereotactic radiation therapy, data compression, digital communications, meshing of space for numerical analysis, and coding and cryptography, among other examples. In the first three space dimensions, the best known solutions of the sphere packing and number variance problems (or their "dual" solutions) are directly related to those of the covering and quantizer problems, but such relationships may or may not exist for d ≥ 4, depending on the peculiarities of the dimensions involved. Our reformulation sheds light on the reasons for these similarities and differences. We also show that disordered saturated sphere packings provide relatively thin (economical) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions.In the case of the quantizer problem, we derive improved upper bounds on the quantizer error using sphere-packing solutions, which are generally substantially sharper than an existing upper bound in low to moderately large dimensions. We also demonstrate that disordered saturated sphere packings yield relatively good quantizers. Finally, we remark on possible applications of our results for the detection of gravitational waves.

Can exotic disordered “stealthy” particle configurations tolerate arbitrarily large holes?

“…Thus, to study the tail behavior of E V (r), we choose sufficiently large χ values (0.45-0.46) in Fig. 3 14 and saturated RSA sphere packings; and contrast our results to Poisson point processes, in which hole sizes are unbounded. As Fig.…”

“…[7][8][9][10][11][12] However, rather than considering the particles themselves, it has been suggested that the space between the particles may be even more fundamental and contain greater statisticalgeometrical information. 13,14 A major focus of this paper is the study of a particular property of the void space between point particles in disordered "stealthy" systems, [15][16][17][18][19] which are disordered many-particle configurations that anomalously suppress large-scale density fluctuations, endowing them with unique physical properties. [20][21][22][23][24][25] The specific question that we investigate is whether disordered stealthy systems can contain arbitrarily large holes.…”

“…where v 1 (r) = π d/2 r d /Γ(1 + d/2) is the volume of a ddimensional sphere of radius r, 14 and Γ(x) is the gamma function. Although E V decays exponentially as v 1 (r) increases, it is always positive for any finite r. Thus, no matter how large a hole is desired, the rare event of forming such a hole can always be observed in the infinite system.…”

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

“…These include the pore-size functions (the distribution of the distance from a randomly chosen location in the void phase to the closest phase boundary), 46 the quantizer error (a moment of the poresize function, which is related to the principal relaxation time), 32,46 the order metric τ (a measure of the translational order of point configurations), 18 and the percolation threshold or the critical radius (the radius of the spheres at which a specific phase becomes connected) of each phase. [47][48][49][50] We compare the aforementioned physical and geometrical properties of our two-phase system derived from decorated stealthy ground states, as a function of the tuning parameter χ, with those of two other two-phase media: (1) equilibrium disordered (fluid) hard-sphere systems and (2) decorated Poisson point processes (idealgas configurations).…”

New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

“…Some aperiodic tilings of Euclidean space underlie quasicrystals (12,13), which possess forbidden crystallographic symmetries, and glassy states of matter (14). Tilings of high-dimensional Euclidean space also have important applications in communications, cryptography, information theory, and in the search for gravitational waves (2,15).…”