Related papers: Lattice embeddings in percolation
We show that inclusions of $p$-metric spaces always produce genuine linear embeddings at the level of Lipschitz-free $p$-spaces. More precisely, for every $0<p<1$ and every inclusion $ \mathit{N}\subset \mathit{M}$ of $p$-metric spaces, the…
The physics of $k$-core percolation pertains to those systems whose constituents require a minimum number of $k$ connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from…
We consider independent edge percolation models on Z, with edge occupation probabilities p_<x,y> = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We prove that oriented percolation occurs when beta > 1 provided p is chosen…
The dimer problem arose in a thermodynamic study of diatomic molecules, and was abstracted into one of the most basic and natural problems in both statistical mechanics and combinatoric mathematics. Given a rectangular lattice of volume V…
We present the extension of Rosenfeld's fundamental measure theory to lattice models by constructing a density functional for d-dimensional mixtures of parallel hard hypercubes on a simple hypercubic lattice. The one-dimensional case is…
We consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of Cellular Automata defined on the $d$-dimensional lattice $\{1,2,...,L\}^d$…
Representing lattices L by equivalence relations amounts to embed them into the lattice Part(V) of all partitions of a set V, and has a long history. Here we are concerned with MODULAR lattices L and aim for sets V as small as possible,…
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The…
We investigate site and bond percolation in triangular and square lattices subjected to linear distortion. In contrast to previously studied distortion schemes that preserve lattice geometry, linear distortion dislocates regular lattice…
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a…
Let $\Gamma$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1).…
We investigate the formation of an infinite cluster of entangled threads in a (2+1)-dimensional system. We demonstrate that topological percolation belongs to the universality class of the standard 2D bond percolation. We compute the…
We consider the Bernoulli bond percolation process $\mathbb{P}_{p,p'}$ on the nearest-neighbor edges of $\mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which…
We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subseteq…
We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling…
Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase…
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots,…
In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $\mathbb{L}^d$ and the set of non-negative integers…
In this note we prove that a molecule $d(x,y)^{-1}(\delta(x)-\delta(y))$ is an exposed point of the unit ball of a Lispchitz free space $\mathcal F(M)$ if and only if the metric segment $[x,y]=\{z \in M \; : \; d(x,y)=d(z,x)+d(z,y) \}$ is…
We report the critical point for site percolation for the "explosive" type for 2D square lattices using Monte Carlo simulations and compare it to the classical well known percolation. We use similar algorithms as have been recently reported…