Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang’s argument requires only 2-fold (...

Draw planes in IR 3 that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally deene a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of ...

Proof. Consider p-bond percolation on an an infinite dregular tree T rooted at ρ and write CT (ρ) for the component containing ρ. Write CG(v) for the component of v in Gp. It is clear that we can couple such that |CG(v)| ≤ |CT (ρ)|. Recall the exploration process on T : At time t = 0 only ρ = v1 is active, other vertices neutral. At time t ≥ 0 if the number of active vertices Mt > 0, then set t...

We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random w...

We study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0, n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that EH(n) = An + √ 3 4π log(n) + o(log(n)), wher...

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as “lamplighter groups”. We prove that zero speed for random walk on a lamplig...

The vulcanization transition is addressed via a minimal replica-field-theoretic model. The appropriate long-wavelength behavior of the two- and three-point vertex functions is considered diagrammatically, to all orders in perturbation theory, and identified with the corresponding quantities in the Houghton-Reeve-Wallace field-theoretic approach to the percolation critical phenomenon. Hence, it ...

A new formula for continuum percolation on the Euclidean space R (d ≥ 2), which is analogous to Russo’s formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite c...

We develop a general theory for percolation in directed random networks with arbitrary two-point correlations and bidirectional edges--that is, edges pointing in both directions simultaneously. These two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena. Equations for the percolation threshold and the sizes of the giant components are d...

In this paper, the critical behavior of the percolation models is reviewed, with emphasis on the scaling laws present and its relationship with other simple models like the Ising magnet. A liitle bit of the history and some examples of its application are given to illustrate the range of phenomena that can be described with such models.