Estimates of the optimal density and kissing number of sphere packings in high dimensions

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound that is controlled asymptotically by $1/2^d$, where $d$ is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski's bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a "test" pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found with an asymptotic behavior controlled by $1/2^{(0.77865...)d}$. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions correponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same exponential improvement and therefore the asymptotic form $1/2^{(0.77865...)d}$ is much more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.

Spherical codes, maximal local packing density, and the golden ratio

The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of the fixed sphere to the centers of any of the N surrounding spheres is minimized. Solutions to the DLP problem are relevant to the realizability of pair correlation functions for packings of nonoverlapping spheres and might prove useful in improving upon the best known upper bounds on the maximum packing fraction of sphere packings in dimensions greater than three. The optimal spherical code problem in Rd involves the placement of the centers of N nonoverlapping spheres of unit diameter onto the surface of a sphere of radius R such that R is minimized. It is proved that in any dimension, all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem. It follows that for any packing of nonoverlapping spheres of unit diameter, a spherical region of radius less than or equal to the golden ratio centered on an arbitrary sphere center cannot enclose a number of sphere centers greater than one more than the number that can be placed on the region's surface.Comment: 12 pages, 1 figure. Accepted for publication in the Journal of Mathematical Physic

Application of Edwards' statistical mechanics to high dimensional jammed sphere packings

The isostatic jamming limit of frictionless spherical particles from Edwards' statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629 (2008)] is generalized to arbitrary dimension $d$ using a liquid-state description. The asymptotic high-dimensional behavior of the self-consistent relation is obtained by saddle-point evaluation and checked numerically. The resulting random close packing density scaling $\phi\sim d\,2^{-d}$ is consistent with that of other approaches, such as replica theory and density functional theory. The validity of various structural approximations is assessed by comparing with three- to six-dimensional isostatic packings obtained from simulations. These numerical results support a growing accuracy of the theoretical approach with dimension. The approach could thus serve as a starting point to obtain a geometrical understanding of the higher-order correlations present in jammed packings.Comment: 13 pages, 7 figure

Mean field theory of hard sphere glasses and jamming

Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important problems in information theory, such as digitalization of signals, error correcting codes, and optimization problems. In three dimensions the densest packing of identical hard spheres has been proven to be the FCC lattice, and it is conjectured that the closest packing is ordered (a regular lattice, e.g, a crystal) in low enough dimension. Still, amorphous packings have attracted a lot of interest, because for polydisperse colloids and granular materials the crystalline state is not obtained in experiments for kinetic reasons. We review here a theory of amorphous packings, and more generally glassy states, of hard spheres that is based on the replica method: this theory gives predictions on the structure and thermodynamics of these states. In dimensions between two and six these predictions can be successfully compared with numerical simulations. We will also discuss the limit of large dimension where an exact solution is possible. Some of the results we present here have been already published, but others are original: in particular we improved the discussion of the large dimension limit and we obtained new results on the correlation function and the contact force distribution in three dimensions. We also try here to clarify the main assumptions that are beyond our theory and in particular the relation between our static computation and the dynamical procedures used to construct amorphous packings.Comment: 59 pages, 25 figures. Final version published on Rev.Mod.Phy

Random Sequential Addition of Hard Spheres in High Euclidean Dimensions

Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$ in the infinite-time or saturation limit for the first six space dimensions ($1 \le d \le 6$). Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each =of these dimensions. We find that for $2 \le d \le 6$, the saturation density $\phi_s$ scales with dimension as $\phi_s= c_1/2^d+c_2 d/2^d$, where $c_1=0.202048$ and $c_2=0.973872$. We also show analytically that the same density scaling persists in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any $d$ given by $\phi_s \ge (d+2)(1-S_0)/2^{d+1}$, where $S_0\in [0,1]$ is the structure factor at $k=0$ (i.e., infinite-wavelength number variance) in the high-dimensional limit. Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.Comment: 38 pages, 9 figures, 4 table

Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search

By applying Grover's quantum search algorithm to the lattice algorithms of Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and Stehl\'{e}, we obtain improved asymptotic quantum results for solving the shortest vector problem. With quantum computers we can provably find a shortest vector in time $2^{1.799n + o(n)}$, improving upon the classical time complexity of $2^{2.465n + o(n)}$ of Pujol and Stehl\'{e} and the $2^{2n + o(n)}$ of Micciancio and Voulgaris, while heuristically we expect to find a shortest vector in time $2^{0.312n + o(n)}$, improving upon the classical time complexity of $2^{0.384n + o(n)}$ of Wang et al. These quantum complexities will be an important guide for the selection of parameters for post-quantum cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page

On error correction with errors in both the channel and syndrome

International audienc

Explicit Construction of RIP Matrices Is Ramsey‐Hard

© 2019 Wiley Periodicals, Inc. Matrices Φ ∈ ℝn × p satisfying the restricted isometry property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for n = logO(1)p, the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so-called (Formula presented.) sparsity bottleneck. The notable exception is the work by Bourgain et al. constructing an n × p RIP matrix with sparsity s = Θ(n1/2 + ϵ), but in the regime n = Ω(p1 − δ). In this short note we resolve this open question by showing that an explicit construction of a matrix satisfying the RIP in the regime n = O(log2p) and s = Θ(n1/2) implies an explicit construction of a three-colored Ramsey graph on p nodes with clique sizes bounded by O(log2p) — a question in the field of extremal combinatorics that has been open for decades. © 2019 Wiley Periodicals, Inc