Poisson Percolation on the Square Lattice

Abstract

In this paper, we examine two versions of inhomogeneous percolation on the 2D lattice, which we will refer to as non-oriented and oriented percolation, and describe the limiting shape of the component containing the origin in both cases. To define the nonoriented percolation process that we study, we consider the square lattice where raindrops fall on an edge with midpoint $x$ at rate $|x|_\infty^{-\alpha}$. The edge becomes open when the first drop falls on it. We call this process "nonoriented Poisson percolation". 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 $t \to \infty$: (i) the cluster containing the origin $\CC_0(t)$ is contained in the square of radius $n(p_c-\ep,t)$, and (ii) the cluster fills the square of radius $n(p_c+\ep,t)$ 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 $\ZZ^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $\CC_0(t)$ are of size $N^{4/7}$. In the second part of the paper, we prove similar, yet not-studied-before, results for the asymptotic shape of the cluster containing the origin in the oriented case of Poisson percolation. We show that the density of occupied sites at height $y$ in the open cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.