Related papers: Random perfect lattices and the sphere packing pro…
Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant…
Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize…
What particle shape will generate the highest packing fraction when randomly poured into a container? In order to explore and navigate the enormous search space efficiently, we pair molecular dynamics simulations with artificial evolution.…
The study of real-life network modeling has become very popular in recent years. An attractive model is the scale-free percolation model on the lattice $\mathbb{Z}^d$, $d\ge1$, because it fulfills several stylized facts observed in large…
The ideal glass, a disordered system of particles with zero configurational entropy, cannot be realized through thermal processes. Nevertheless, we present a method for constructing ideal jammed packings of soft spheres, and thus the zero…
Using exact continuous quantum Monte Carlo techniques, we study the zero and finite temperature properties of a system of harmonically trapped one dimensional spin 1/2 fermions with short range interactions. Motivated by experimental…
Motivated by a recently identified severe discrepancy between a static and a dynamic theory of glasses, we numerically investigate the behavior of dense hard spheres in spatial dimensions 3 to 12. Our results are consistent with the static…
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the…
We investigate the average number of lattice points within a ball for the $n$th cyclotomic number field, where the lattice is chosen at random from the set of unit determinant ideal lattices of the field. We show that this average is nearly…
In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete…
A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on ${\mathbb Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for…
We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of…
This paper presents the asymptotic analysis of random lattices in high dimensions to clarify the distance properties of the considered lattices. These properties not only indicate the asymptotic value for the distance between any pair of…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy…
We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates…
We present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the…
Imagine we want to split a group of agents into teams in the most \emph{efficient} way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied \textsc{Coalition…
Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…
Many lattice-based crypstosystems employ ideal lattices for high efficiency. However, the additional algebraic structure of ideal lattices usually makes us worry about the security, and it is widely believed that the algebraic structure…