Related papers: A Solidification Phenomenon in Random Packings
The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The…
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded…
We build upon previous work on the densities of uniform random walks in higher dimensions, exploring some properties of the even moments of these densities and extending a result about their modularity.
It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation…
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…
A family of m independent identically distributed random variables indexed by a chemical potential \phi\in[0,\gamma] represents piles of particles. As \phi increases to \gamma, the mean number of particles per site converges to a maximal…
In this short note, we make an observation relating symplectic packings of the standard symplectic ball by two sets and Hamiltonian linking.
Let k >= 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree $\Omega(n^{k-1})$…
Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A…
We generalize finite-sample bounds for convex clustering to the setting where affinity weights appearing in the objective correspond to a general connected graph. These bounds and their analysis lead to a better understanding of clustering…
Characterization of real-world complex systems increasingly involves the study of their topological structure using graph theory. Among global network properties, small-world property, consisting in existence of relatively short paths…
Universality theorems (in the sense of N. Mn\"{e}v) claim that the realization space of a combinatorial object (a point configuration, a hyperplane arrangement, a convex polytope, etc.) can be arbitrarily complicated. In the paper, we prove…
We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a…
We study the collective phenomenon in the scattering of a single-photon by one or two layers of two-level atoms. By modeling the photon dispersion with a two-dimensional (2D) coupled cavity array, we analytically derive the scattering…
A collection of spherical particles can be packed tightly together into an amorphous packing known as "random close packing" (RCP). This structure is of interest as a model for the arrangement of molecules in simple liquids and glasses, as…
We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and…
This work provides an example that motivates and illustrates theoretical results related to a combination of small-gain and density propagation conditions. Namely, in case the small-gain fails to hold at certain points or intervals the…
We introduce a model for active transport on inhomogeneous networks embedded in a diffusive environment and investigate the formation of particle clusters. In the presence of a hard-core interaction, cluster sizes exhibit an algebraically…
We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…
We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let $\bf P$ be a unimodular probability measure on rooted networks $(G, o)$ with…