# Poisson Percolation on the Square Lattice

Date

2019-04-01
Author

Advisors

Durrett, Richard

Junge, Matthew

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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.

Type

Honors thesisDepartment

MathematicsPermalink

https://hdl.handle.net/10161/19074Citation

Cristali, Irina (2019). *Poisson Percolation on the Square Lattice.*Honors thesis, Duke University. Retrieved from https://hdl.handle.net/10161/19074.

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