Poisson percolation on the square lattice

Abstract

On the square lattice raindrops fall on an edge with midpoint $x$ at rate $\|x\|_\infty^{-α}$. The edge becomes open when the first drop falls on it. Let $ρ(x,t)$ be the probability that the edge with midpoint $x=(x_1,x_2)$ 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 $t \to \infty$: (i) the cluster containing the origin $\mathbb C_0(t)$ is contained in the square of radius $n(p_c-ε,t)$, and (ii) the cluster fills the square of radius $n(p_c+ε,t)$ 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 $\mathbb Z^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $\mathbb C_0(t)$ are of size $N^{4/7}$.