\begin{proof}[Proof of the first part of Theorem 10.9]
We now consider the space $\mathbb{R}^d$ for $d \geq 2$. Let us denote $C_{\infty}$ the only infinite component of the graph (we are still in supercritical regime).

For nonnegative $s$ let $c(B(s))$ be the size of $C_{\infty} \cap B(s)$.  We set $\Phi_s = \log^2 s$. The following holds by an ergodic theorem (10.34):%TODO

$$\frac{1}{s^d} c(B(s)) \rightarrow \lambda p_{\infty} (\lambda) \quad \mathrm{as} \quad s \rightarrow \infty$$

Component $C_{\infty}$ can be divided into more than one component in the box $B(s)$. Let us denote the components $C_1, C_2, \dots, C_n$ (in decreasing order). We can see that on the margin of the box $B(s)$, precisely inside $B(s) \setminus B(s - 2)$, there is at least one point $x_i$ belonging to $C_i$ for every $i$ from $1$ to $n$ (see Figure \ref{fig:intersectionWithBox}). The balls $\mathrm{Ball}(x_i, 1/2)$ are disjoint. It holds
\begin{figure}
	\centering
	\includegraphics[width=7cm]{figIntersectionWithBox.pdf}
	\caption{Infinite component divided into parts inside the box $B(s)$}
	\label{fig:intersectionWithBox}
\end{figure}

$$n \cdot \mathrm{Vol}(\text{Ball of radius $1/2$}) \leq \mathrm{Vol}(B(s) \setminus B(s - 2)) = k_1 s^{d - 1}$$

for some constant $k_1$. Thus the number of components satisfies $n \leq k_2 s^{d - 1}$ for some constant $k_2$.

We want to prove

$$\frac{1}{s^d} |C_1| \stackrel{p}{\rightarrow} \lambda p_{\infty} (\lambda).$$

The diameter of each component inside $B(s)$ is almost surely at most $\Phi_s$, so we have

$$\lim_{s \uparrow \infty} \Pr\left( \sum^{n}_{i = 2} |C_i| \leq n \cdot \log^{2d} s \right) = 1$$

$$\lim_{s \uparrow \infty} \Pr\left(\frac{1}{s^d} \sum^{n}_{i = 2} |C_i| \leq \frac{k_2 s^{d - 1} \cdot \log^{2d} s}{s^d} \right) = 1$$

$$\frac{1}{s^d} \sum^{n}_{i = 2} |C_i| \stackrel{p}{\rightarrow} 0 $$
%$$\lim_{s \uparrow \infty} \Pr\left(\frac{1}{s^d} \sum^{n}_{i = 2} |C_i| = 0\right) = 1$$

Thus the sum of sizes of components in the box $B(s)$ excluding the largest one converges to zero with probability one. By 10.34 we obtain

$$\frac{1}{s^d} |C_1| \stackrel{p}{\rightarrow} p_{\infty} (\lambda)$$

%TODO by proposition 10.13
%Thus largest component $C_1$ is crossing for $B(s)$ and all other components have metric diameter smaller than $\Phi_s = \log^2 s$.

\end{proof}

The intersection of the infinite component $C_{\infty}$ and a box split $C_{\infty}$ to smaller components, but with probability tending to one there will be only one large component crossing for the box and possibly a lot of small components.

\section{Critical Thresholds}
Finally we show what is known about critical thresholds for different properties of a random geometric graph $\mathrm{RGG}(n,r)$ on the box $[0,1]^d$ ($n$ is the number of points and $r$ is the radius, i.e. the maximal distance for adjacent points). The degree distribution in RGG follows $\mathrm{Bin}(n - 1, \pi r^2)$.

It is known that every monotone graph property has a sharp threshold $r_c$, i.e. when $r < r_c - \epsilon$ then the property holds for RGG with probability zero and when $r > r_c + \epsilon$ the property holds for RGG with probability tending to one.

For the existence of a giant component it is known that the critical threshold $r_c$ is $\sqrt{\frac{\lambda_c}{n}}$ ($\lambda_c$ is approximately $1.44$ from the experiments). For the graph connectivity, $r_c = \sqrt{ \frac{\log n}{\pi n}}$.

What remains open is the exact value of $\lambda_c$ for the existence of a giant component.
