相关论文: Sharp thresholds and percolation in the plane

We consider the model of the Brownian plane, which is a pointed non-compact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar…

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the…

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently the critical polynomial $P_{\rm B}(p,L)$ was introduced for planar-lattice percolation models, where $p$ is the occupation…

We prove a general Russo-Seymour-Welsh result valid for any invariant planar percolation process satisfying positive association. This means that the probability of crossing a rectangle in the long direction is related by a homeomorphism to…

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…

A popular question in Bernoulli percolation models is if the probability of connection between two vertices in a transitive graph decays monotonically with the distance between these two vertices. For example, on the square lattice is an…

In $H$-percolation, we start with an Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We…

In this paper we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in…

We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a 1-dimensional underlying lattice. We find a non-classical critical point in the limit of the number of long-range bonds in the system…

Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a…

The properties of polymer composites with nanofiller particles change drastically above a critical filler density known as the percolation threshold. Real nanofillers, such as graphene flakes and cellulose nanocrystals, are not idealized…

We consider Bernoulli percolation on $\mathbb Z^d$ with $d>6$. We prove an up-to-constant estimate for the critical two-point function restricted to a half-space. This completes previous results of Chatterjee and Hanson (Commun. Pure Appl.…

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the…

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter and the…

We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained…

Consider balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ with probability $1-\exp(-\beta J(x, y) )$ where $J(x, y) \asymp \|…

Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel…

While classical percolation is well understood, percolation effects in randomly packed or jammed structures are much less explored. Here we investigate both experimentally and theoretically the electrical percolation in a binary composite…

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards…

We consider Poisson Boolean percolation on $\mathbb R^d$ with power-law distribution on the radius with a finite $d$-moment for $d\ge 2$. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions.…