\section{Existence of a giant component}
In this lecture we work in the \textit{supercritical regime}, i.e. the density $\lambda$ is always greater than $\lambda_c$. The Poisson Point Process (P.P.P.) is done on a box $B(s) = \left[-\frac{s}{2}, \frac{s}{2}\right]^d \subseteq \mathbb{R}^d$.

Let $B = \prod^d_{k = 1} [a_k, b_k]$ and let $\pi_k: \mathbb{R}^d \rightarrow \mathbb{R}$ be a projection onto the $k$-th coordinate. The box $B$ is \textit{$k$-crossing} for random geometric graph $G(X, r)$ if there are vertices $x'$ and $x''$ such that $|\pi_k(x') - a_k| < \frac{r}{2}$, $|\pi_k(x'') - b_k| < \frac{r}{2}$ and both vertices lie in the same component of $G$.

The box is \textit{crossing} if for every $k$ the box is $k$-crossing.

Here we restrict the analysis only to the case $d = 2$. Define a rectangular box for a given height $a$: $B(a, j) = [0, ja] \times [0,a]$ (for $j = 1, 2, 4$). Let $\mathrm{LR}_a$ denote the event that there is a 1-crossing component for $B(a,2)$ on $G(H_{\lambda,s} \cap B(a,2), 1)$, i.e. the graph is crossing the rectangle from left to right along the longer side. Similarly let $\mathrm{LLR}_a$ be the event that there is a 1-crossing component for $B(a,4)$ on $G(H_{\lambda,s} \cap B(a,4), 1)$ and let $\mathrm{SLR}_a$ be the event that there is a 1-crossing component for $B(a,1)$ on $G(H_{\lambda,s} \cap [0, a] \times B(a,1), 1)$.

\begin{lemma}[Lemma 10.4 in~\cite{PenroseBook}]
For $d = 2$ $\Pr(\mathrm{SLR}_a)\uparrow 1$ and $\Pr(\mathrm{LR}_a)\uparrow 1$ as $a \uparrow \infty$.
\end{lemma}