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

Source 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.