We study random walks on supercritical percolation clusters on wedges in Z, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z is equivalent to the (a.s.) exis...

1 Executive Summary We have continued with a review on the bond percolation and the introduced the coupling of the bond percolation process, then defined the critical phenomenon and the percolation probability. There is a theorem about the percolation probability is a non decreasing function and we bounded it between 1/3 and 2/3 for the two dimensional case. 2 Bond Percolation 2.1 Preliminaries...

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in ot...

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indic...

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness Ž of the infinite cluster is equivalent to nondecay of connectivity a.k.a. . long-range order . We then derive applications concerning uniqueness in Kazhdan groups and in wreath products ...

We consider critical site percolation on the triangular lattice, that is, we choose X(v) = 0 or 1 with probability 1/2 each, independently for all vertices v of the triangular lattice. We say that a word (ξ1, ξ2, . . . ) ∈ {0, 1}N is seen in the percolation configuration if there exists a selfavoiding path (v1, v2, . . . ) on the triangular lattice with X(vi) = ξi, i ≥ 1. We prove that with pro...

We prove the existence of a (random) Lipschitz function F : Z → Z such that, for every x ∈ Z, the site (x, F (x)) is open in a site percolation process on Z. The Lipschitz constant may be taken to be 1 when the parameter p of the percolation model is sufficiently close to 1.