\section{Percolation Theory}

\subsection{Bernoulli Percolation}

A set $A \subseteq \mathbb{Z}^{d}$ is called {\em symmetric} if $-x \in A$ for every $x \in A$. Let $\sim_{A}$ denote the relation on $\mathbb{Z}^{d}$ where $A$ is a finite symmetric set and $x \sim_{A} y$ if and only if $x-y \in A$. The graph $\left( \mathbb{Z}^{d},\left\{ \left\{ x,y\right\} \mid x \sim_{A} y  \right\} \right)$ is denoted as $\left( \mathbb{Z}^{d},\sim_{A}\right)$. We say that the subset $S$ of $\mathbb{Z}^{d}$ is {\em$A$-connected} if it induces a connected subgraph of $\left( \mathbb{Z}^{d},\sim_{A}\right)$. For example, if $A=\left\{ z \in \mathbb{Z}^{d} \mid \left\|  z\right\|_{1} = 1 \right\}$, then the graph $\left( \mathbb{Z}^{d},\sim_{A}\right)$ corresponds to the integer lattice.

{\em Percolation theory} is a study of connectedness properties of random sets in space. In particular, lattice percolation models are concerned with properties of geometric graphs on subsets of $\mathbb{Z}^{d}$, embedded in $\mathbb{R}^{d}$. Let $p \in \left[ 0,1\right]$, then for every $x \in \mathbb{Z}^{d}$ we toss a coin and with the probability $p$ we mark the site $x$ as {\em open} and with the probability $1-p$ we mark $x$ as {\em closed}. Let $C_{p}$ denote the random set of open sites.

The main question is: Is there any $p \in \left( 0,1 \right)$ such that the component in $C_{p}$ containing origin is infinite? Note that if the origin is closed, then this set is empty. If we denote the probability that this component is infinite as $\Theta_{z}\left(p\right)$, then $\Theta_{z}\left(p\right)$ is increasing in $p$ and there is a critical value $p_{c}$ such that $\Theta_{z}\left(p\right)=0$ for $p < p_{c}$ (we say that the Bernoulli process $C_{p}$ is {\em subcritical} in such a case) and $\Theta_{z}\left(p\right)>0$ for $p>p_{c}$ (the Bernoulli process $C_{p}$ is {\em supercritical}). It is known that $p_{c} \in \left( 0,1\right)$.


\begin{thm}
Suppose $p < p_{c}$. Then if $C_{0}$ denotes the cluster at the origin for $C_{p}$ and $\left|C_{0}\right|$ denotes its order, \[\ \limsup_{n \rightarrow \infty}{ \left(n^{-1}\log{Pr{\left[ \left| C_{0}\right|\geq n\right]}}\right)}<0.\]
\end{thm}

\subsection{Continuum Percolation}

Let $H_{\lambda}$ denote a homogeneous Poisson process of intensity $\lambda$ on $\mathbb{R}^{d}$. Continuum percolation can be characterized as the study of large components of the infinite graph $G \left( H_{\lambda};1\right)$.

For $s>0$ let the box $B\left( s\right) = \left[ -{{s} \over {2}}, {{s} \over {2}}\right]^{d}$ and let $H_{\lambda,s}$ be the restriction $H_{\lambda} \cap B\left(s\right)$ of the homogeneous Poisson process $H_{\lambda}$ on the box $B\left( s \right)$. The random geometric graphs are large but finite vertex sets and thus we can describe them using the graphs $G\left( H_{\lambda,s};1\right)$ for large $s$. Using the Scaling Theorem we see that the random geometric graph $G \left( P_{n};r_{n}\right)$ is isomorphic to a copy of the graph $G\left( H_{\lambda,s};1\right)$ for appropriate $\lambda$ and $s$.

Let $H_{\lambda,0}$ denote the point process $H_{\lambda} \cup \left\{ 0\right\}$ where $0$ is the origin in $\mathbb{R}^{d}$. Suppose that $k$ is a positive integer and $C_{0}$ is the component of the graph $G\left(H_{\lambda,0};1 \right)$ containing the origin. Denote the probability that $\left| C_{0} \right|=k$ as $p_{k}\left(\lambda\right)$. The {\em percolation probability} $p_{\infty}\left(\lambda\right)$ is the probability that the origin lies in an infinite component of  $G\left(H_{\lambda,0};1 \right)$ and it is defined as
\[
p_{\infty}\left(\lambda\right)=1-\sum_{k=1}^{\infty}{p_{k}\left(\lambda\right)}.
\]

The {\em critical value} $\lambda_{c}$ is defined as 
\[
\lambda_{c}=\inf{\left\{\lambda > 0 \mid p_{\infty}\left(\lambda\right)>0\right\}},
\]
and it depends on the dimension $d$ and the choice of norm. For $d \geq 2$ we know that $0<\lambda_{c}<\infty$, but the exact values for $\lambda_{c}$ or $p_{\infty}\left(\lambda\right)$ are not known. For $d=2$ and the Euclidean norm $l_{2}$ simulation studies indicate that $\lambda_{c} \approx 1.44$ and the rigorous bounds say that $0.696 < \lambda_{c} <3.372$.