Poisson percolation on the square lattice

Suppose that on the square lattice the edge with midpoint x becomes open at rate parallel to x parallel to(-alpha)(infinity). Let rho(x, t) be the probability that the corresponding edge is open at time t and let n(p, t) be the distance at which edges are open with probability p at time t. We show that with probability tending to 1 as t -> infinity: (i) the open cluster containing the origin C-0 (t) is contained in the square of radius n(p(c) - epsilon, t), and (ii) the cluster fills the square of radius n(p(c) + epsilon, t) with the density of points near x being close to theta(rho(x, t)) where theta(p) is the percolation probability when bonds are open with probability p on Z(2). Results of Nolin suggest that if N = n(p(c), t) then the boundary fluctuations of C-0(t) are of size N-4/7.