Related papers: Lattice embeddings in percolation

We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge…

We consider a dependent percolation model on the square lattice $\mathbb{Z}^2$. The range of dependence is infinite in vertical and horizontal directions. In this context, we prove the existence of a phase transition. The proof exploits a…

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit) for the…

We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…

I calculated the exact site and bond percolation probability on lattice-like graphs for given dimension d: site percolation probability is 1/d and bond percolation probability is 0.5(d-1)^{-1/2}

Let $ \mathbb{L}^{d} = ( \mathbb{Z}^{d},\mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ \mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional…

We test the universal finite-size scaling of the cluster mass order parameter in two-dimensional (2D) isotropic and directed continuum percolation models below the percolation threshold by computer simulations. We found that the simulation…

By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the…

Kalu\v{z}a, Kopeck\'a and the author have shown that the best Lipschitz constant for mappings taking a given $n^{d}$-element set in the integer lattice $\mathbb{Z}^{d}$, with $n\in \mathbb{N}$, surjectively to the regular $n$ times $n$ grid…

Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp…

We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, \textbf{87} 052107 (2013)],…

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair…

In this paper we consider the structure of site percolation models on three-dimensional square lattices with various shapes of (1,d)-neighborhood. For these models, are proposed iso- and anisotropic modifications of the invasion percolation…

The random Lorentz gas (RLG) is a minimal model of both percolation and glassiness, which leads to a paradox in the infinite-dimensional, $d\rightarrow\infty$ limit: the localization transition is then expected to be continuous for the…

We study the problem of constructing bulk and surface embedded modes (EMs) inside the quasi-continuum band of a square lattice, using a potential engineering approach \`a la Wigner and von Neumann. Building on previous results for the…

We simulate directed site percolation on two lattices with 4 spatial and 1 time-like dimensions (simple and body-centered hypercubic in space) with the standard single cluster spreading scheme. For efficiency, the code uses the same…

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex…

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as "percolation of words". We give a positive answer to their Open Problem 2: almost surely, all…

On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+…

We expand the critical point for site percolation on the $d$-dimensional hypercubic lattice in terms of inverse powers of $2d$, and we obtain the first three terms rigorously. This is achieved using the lace expansion.