Related papers: Does Eulerian percolation on $Z^2$ percolate ?
We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than nearest-neighbors. We show that for a percolation range to next-nearest neighbors the percolation threshold…
We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through…
In this article, we consider an anisotropic finite-range bond percolation model on $\mathbb{Z}^2$. On each horizontal layer $\{(x,i): x \in \mathbb{Z}\}$ we have edges $\langle(x, i),(y, i)\rangle$ for $1 \leq |x - y| \leq N$. There are…
We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges…
In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries…
A layered graph $G^\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \in [0,1]$ on $G^\times$ one intuitively would…
In [Watanabe et al., Phys. Rev. Lett. 93 190601 (2004)], the authors show numerically that spanning and percolation probabilities in two-dimensional systems with different aspect ratios obey a form of "superscaling". In this comment, we…
The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It…
In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $\theta_n(p)$ is the probability that there is a path…
We investigate properties of two-dimensional finite-scale percolation systems whose size along the current flow is smaller than the perpendicular size. Successive thresholds of appearing multiple percolation channels in such systems have…
Conduction in bulk polycrystalline high-T$_c$ superconductors with relatively high critical currents has been shown to be percolative. This phenomenon is due to weak links at grain boundaries. These weak links are the major limiting factor…
We study the scaling of the average cluster size and percolation strength of geometrical clusters for the two-dimensional Ising model. By means of Monte Carlo simulations and a finite-size scaling analysis we discuss the appearance of…
We study FK-percolation where the edge parameters are chosen as independent random variables in the near-critical regime. We show that if these parameters satisfy a natural centering condition around the critical point, then the quenched…
We consider percolation properties of the Boolean model generated by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical activity, such that percolation…
Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the…
We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z}^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty]) < p_c(d)$, it is known that the time constant $\mu_G$ exists. We are…
We present structural properties of two-dimensional polymers as far as they can be described by percolation theory. The percolation threshold, critical exponents and fractal dimensions of clusters are determined by computer simulation and…
Percolation plays an important role in fields and phenomena as diverse as the study of social networks, the dynamics of epidemics, the robustness of electricity grids, conduction in disordered media, and geometric properties in statistical…
Following the approach outlined in [18], convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied in [7] is established. This puts the said models in the same universality class as the standard…
For a given tiling of the euclidean plane ${\bf E}^2$, we call the degree of freedom of perturbed edges of prototiles {\it escher degree}. In this paper we consider non-periodic L-tilings by 2 prototiles and obtain the escher degree of…