Related papers: Two-Dimensional Critical Percolation: The Full Sca…
We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For $d=2$ they clearly…
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.
We suggest how versions of Schramm's SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain.
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field…
We study the critical properties of the monopole-percolation transition in U(1) lattice gauge theory coupled to scalars at infinite ($\beta=0$) gauge coupling. We find strong scaling corrections in the critical exponents that must be…
We derive an exact, simple relation between the average number of clusters and the wrapping probabilities for two-dimensional percolation. The relation holds for periodic lattices of any size. It generalizes a classical result of Sykes and…
We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for {\it (i)} the effect of the finite size of the system, and {\it (ii)} the dependence…
The universal behaviour of the directed percolation universality class is well understood, both the critical scaling as well as finite size scaling. This article focuses on the block (finite size) scaling of the order parameter and its…
A 1/L-expansion for percolation problems is proposed, where L is the lattice finite length. The square lattice with 27 different sizes L = 18, 22 ... 1594 is considered. Certain spanning probabilities were determined by Monte Carlo…
Recently, the effective medium approach using 2x2 basic cluster of model lattice sites to predict the conductivity of interacting droplets has been presented by Hattori et al. To make a step aside from pure applications, we have studied…
Making use of a recent complete calculation of a chiral six-point correlation function C(z) in a rectangle we calculate various quantities of interest for percolation (SLE parameter \kappa = 6) and many other two-dimensional critical…
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a…
We consider the isotropic two-dimensional abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from…
We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting…
We consider directed polymers in random environment in the critical dimension $d = 2$, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random…
We study the critical behavior of various geometrical and transport properties of percolation in 6 dimensions. By employing field theory and renormalization group methods we analyze fluctuation induced logarithmic corrections to scaling up…
We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing…
The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid…
We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the…