Poisson percolation on the square lattice

TL;DR

This paper studies a Poisson percolation model on the square lattice where edges open over time with rates depending on their position, revealing asymptotic cluster shapes and boundary fluctuations.

Contribution

It introduces a time-dependent percolation model with position-based rates and characterizes the asymptotic shape and density of the resulting clusters.

Findings

Cluster containing the origin is contained within a square of radius n(p_c−ε,t).

Cluster fills a square of radius n(p_c+ε,t) with density close to θ(ρ(x,t)).

Boundary fluctuations are of size N^{4/7} as per Nolin's results.

Abstract

On the square lattice raindrops fall on an edge with midpoint $x$ at rate $\|x\|_\infty^{-\alpha}$. 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 $t \to \infty$: (i) the cluster containing the origin $\mathbb C_0(t)$ 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 $\mathbb Z^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $\mathbb C_0(t)$ are of size $N^{4/7}$.