Related papers: A crossing probability for critical percolation in…
We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points in the scaling limit. This generalizes both Cardy's…
Following H. Tomita and C. Murakami we propose an analytical model to predict critical probability of percolation. It is based on the excursion set theory which allows us to consider N-dimensional bounded regions. Details are given for the…
We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical…
The probabilities of clusters spanning a hypercube of dimensions two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen--Kopelman algorithm combined with Grassberger's…
The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on…
In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical…
We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DCHN11] and [CS12].…
A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the…
We argue the exact universal result for the three-point connectivity of critical percolation in two dimensions. Predictions for Potts clusters and for the scaling limit below p_c are also given.
It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of…
Given a graph $G$, we consider a model for a random cover of $G$ by taking two parallel copies of $G$ and crossing every pair of parallel edges randomly with probability $q$ independently of each other. The resulting graph $G_q$, is a…
A question relating the critical probability for percolation, the critical probability for a unique infinite cluster and graph limits is presented, together with some partial results.
Recently, the authors showed that the critical probability for random Voronoi percolation in the plane is 1/2. A by-product of the method was a short proof of the Harris-Kesten Theorem concerning bond percolation in the planar square…
We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict to the…
Despite great progress in the study of critical percolation on $\mathbb{Z}^d$ for $d$ large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions.…
The logarithmic conformal field theory describing critical percolation is further explored using Watts' determination of the probability that there exists a cluster connecting both horizontal and vertical edges. The boundary condition…
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds,…
2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities…
We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Omega <= 72/91, omega = D Omega <= 3/2, and Delta_1 = nu omega <= 2, based upon Cardy's result for the critical crossing probability on an…
We consider oriented percolation on Z^d times Z_+ whose bond-occupation probability is pD(...), where p is the percolation parameter and D is a probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-\alpha} for some \alpha>0.…