相关论文: Lack of Sphere Packing of Graphs via Non-Linear Po…
The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be extended to higher dimensions. In particular, it is shown that every limit of finite…
We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all…
As a corollary of our main results, we prove that for any connected $\mathbb{Z}^d$-periodic graph, when edge weights and potentials are treated as variables, the corresponding periodic graph operators generically (i.e., outside a proper…
We explore a new way to simulate quantum field theory, without introducing a spatial lattice. As a pilot study we apply this method to the 3d \lambda \phi^4 model. The regularisation consists of a fuzzy sphere with radius R for the two…
In this short note, we exhibit an infinite family of hyperbolic rational homology $3$--spheres which do not admit any fillable contact structures. We also note that most of these manifolds do admit tight contact structures.
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 prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and…
Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_{\alpha}$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and…
We study, via the replica method of disordered systems, the packing problem of hard-spheres with a square-well attractive potential when the space dimensionality, d, becomes infinitely large. The phase diagram of the system exhibits…
As the continuum limit is approached, lattice QCD simulations tend to get trapped in the topological charge sectors of field space and may consequently give biased results in practice. We propose to bypass this problem by imposing open…
We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from $\mathbb Z^2$ via the addition of an extensive number of extra…
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved…
We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges,…
In this paper, we examine covering graphs that are obtained from the $d$-dimensional integer lattice by adding pendant edges. In the case of $d=1$, we show that the Laplacian on the graph has a spectral gap and establish a necessary and…
In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape.…
The aim of this paper is to review and discuss qualitatively some results on the properties of amorphous packings of hard spheres that were recently obtained by means of the replica method. The theory gives predictions for the equation of…
The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of…
A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the…
A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of…
Using non-Abelian Hodge theory for parabolic Higgs bundles, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathrm{SL}_3(\mathbb{Z})$. These give rise to…