Poisson percolation on the square lattice

Suppose that on the square lattice the edge with midpoint x becomes open at rate ∥x∥σ^-1 . Let ρ(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 → σ: (i) the open cluster containing the origin ℂ0(t) is contained in the square of radius n(pc-∈, t), and (ii) the cluster fills the square of radius n(pc+∈, t) with the density of points near x being close to θ(ρ(x, t)) where θ(p) is the percolation probability when bonds are open with probability p on ℤ². Results of Nolin suggest that if N = n(pc, t) then the boundary uctuations of ℂ0(t) are of size N^4/7.

Duke Scholars

Author Richard Timothy Durrett Mathematics