Poisson percolation on the square lattice

On the square lattice raindrops fall on an edge with midpoint $x$ at rate $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}^{-\alpha }$. The edge becomes open when the first drop falls on it. Let $\rho (x,t)$ 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 $t\to \mathrm{\infty }$: (i) the cluster containing the origin $\mathbb C_0(t)$iscontainedinthesquareofradius$n(p_c-\epsilon,t)$, and (ii) the cluster fills the square of radius $n(p_c+ϵ,t)$ with the density of points near $x$ being close to $\theta (\rho (x,t))$ where $\theta (p)$ is the percolation probability when bonds are open with probability $p$ on $\mathbb Z^2$\mathrm{.}ResultsofNolinsuggestthatif$N=n(p_c,t)$ then the boundary fluctuations of ${\mathbb{C}}_0(t)$ are of size $N^{4\mathrm{/}7}$.