A lower bound for the critical probability in a certain percolation process

Abstract

Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices and the sides of the unit squares (including endpoints) are called links. Each link of L is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links. To avoid trivial cases, we shall always assume 0 < p < 1. The lattice L, with the designations active or passive attached to the links, is called a random maze. A set of links is called connected if the points comprising the links (including endpoints) form a connected point set in the plane.

Sharp thresholds and percolation in the plane

“…In other contexts, Lemma 2 is often known as Kleitman's Lemma [18]. The present context is exactly that of Harris' original paper [14]: X will be a set of edges or vertices in the lattice (according to whether we are considering site or bond percolation), and X p will be the subset of X consisting of the open edges/vertices. Thus an event is increasing if it is preserved by changing the states of one or more edges/vertices from closed to open, and Harris' Lemma states that increasing events are positively correlated.…”

“…From (14), the event that a bond is open is increasing in the product probability space defined by the v x , and condition (i) follows. Condition (ii) follows from our symmetry assumptions on w, and (iv) is immediate as for independent bond percolation.…”

“…Let L * be the dual lattice to L = Z 2 ; we may realize L * so that the dual edge e * of each edge e of L has the same midpoint as e. As in the independent case (see Lemma has no closed vertical crossing; indeed, the probability measure is irrelevant to this observation. In our set-up, the state of e * is also defined by (14). Hence, e * is closed if and only if…”

“…In [5], a short proof was given of the fundamental result of Harris [14] and Kesten [16] that the critical probability p H = p H (Z 2 , bond) for bond percolation in the planar square lattice Z 2 is equal to 1/2, where p H is the critical probability for the occurrence of percolation (see below), and Z 2 is the graph with vertex set Z 2 in which vertices are adjacent if and only if they are at Euclidean distance 1. The methods used in [5] were developed in [4] to prove the new result that the critical probability for percolation in random plane Voronoi tilings is also 1/2.…”

Percolation and disordered systems

“…They proved in 81,174,175] that the percolation model has a phase transition, and they developed some technology for studying the two phases of the process. These early papers were followed swiftly by a small number of high quality articles by others, particularly 138,181,339], but interest agged for a period beginning around 1964. Despite certain appearances to the contrary, some individuals realised 8 that a certain famous conjecture remained unproven, namely that the critical probability of bond percolation on the square lattice equals 1 2 .…”