\section{Size of the largest component}
%\newcommand{\Pr}{\text{PR}}
\newcommand{\dx}{\text{dx}}

Let $L_1(G(H_{\lambda, s}, 1))$ be the size of the largest connected
component in the random geometric graph on $[-s/2, s/2]^d$ with density
$\lambda$ and the edges between points at distance at most $1$.

We use $\xi(\lambda) = -\log(\lim_{n\rightarrow \infty} p_n(\lambda)^{1/n})$ from the previous chapter.

\begin{thm}
If $\lambda < \lambda_c$, then
\[
\frac{L_1(G(H_{\lambda, s}, 1))}{\log s} \stackrel{p}{\rightarrow} \frac{d}{\xi(\lambda)} \quad \text{as } s \rightarrow \infty.
\]

The notation $\stackrel{p}{\rightarrow}$ stands for the convergence in probability and it means
\[
\Pr\left[\lim_{s \rightarrow \infty} \frac{L_1(G(H_{\lambda, s}, 1))}{\log s} = \frac{d}{\xi(\lambda)}\right] = 1.
\]
\end{thm}

\begin{proof}
First, we prove that if $\alpha > d/\xi(\lambda)$,
then $\Pr[L_1(G(H_{\lambda, s}, 1)) \ge \alpha \log s] \rightarrow 0$ as $s \rightarrow \infty$.

Let $N_s(\alpha)$ be the number of vertices in components of size at least $\alpha \log s$.
Recall that $V_x$ is the size of the component containing the point $x$.
If a graph has a component of size at least $\alpha \log(s)$, then $N_s(\alpha) \ge 1$ and thus
\[
\Pr[L_1(G(H_{\lambda, s}, 1)) \ge \alpha \log s] \le E[N_s(\alpha)] = \lambda \int_{B(s)}\Pr[V_x \ge \alpha \log s] \dx.
\]

For a given point $x$, we have
\[
\Pr[V_x \ge \alpha \log s] = \sum_{k = \alpha \log s}^{\infty} P_k(\lambda) \le e^{-\alpha \log s \xi'}
\]
for some $\xi' < \xi(\lambda)$.

Thus
\[
\Pr[L_1(G(H_{\lambda, s}, 1)) \ge \alpha \log s] \le E[N_s(\alpha)]
\le \lambda s^d e^{-\lambda \log s \xi'} = \lambda s^{d - \alpha \xi'}.
\]
Since $d < \alpha \xi'$, this expression tends to $0$ as $s$ goes to infinity.


Second, we prove that if $\beta < d/\xi(\lambda)$,
then $\Pr[L_1(G(H_{\lambda, s}, 1)) > \beta \log s] \rightarrow 1$ as $s \rightarrow \infty$.

Let $m(s)$ be the maximum number of balls of radius $\beta \log s$ that can be packed into $B(s)$.
Then
\begin{equation}
\label{eq:msineq}
m(s) C \log^d(s) \le s^d,
\end{equation}
where $C$ is a universal constant.
Let $B_{1,s} \dots B_{m(s), s}$ be the sequence of the balls of the largest packing
and let $x_{1,s} \dots x_{m(s), s}$ be the sequence of their centers.

We choose $\lambda' \in (0, \lambda)$ such that $\beta < d/\xi(\lambda')$.
This is possible by the continuity of $\xi$ and because $\beta < d/\xi(\lambda)$.
Using the superposition theorem, we split $H_{\lambda, s}$
into two independent processes $H_{\lambda', s}$ and $H_{\lambda'-\lambda, s}$.

We know how to estimate the probability
that the point $0$ is in a component of a given size in $H_{\lambda', s} \cup \{0\}$.
We will use this estimate for each ball $B_{i,s}$
such that there is exactly one point of $H_{\lambda-\lambda', s}$ at distance at most one from $x_{i,s}$
with this point having the role of $0$.

If there is exactly one point of $H_{\lambda-\lambda', s}$ at distance at most one from $x_{i,s}$,
then let $X_{i,s}$ be this point
and let $V_{i,s}$ be the size of the component of
$G((H_{\lambda', s}\cap B_{i,s}) \cup \{X_{i,s}\}; 1)$ containing $X_{i,s}$.
If $X_{i,s}$ is undefined, then let $V_{i,s}$ be zero.
Thus $V_{i,s}$ is a lower bound on the size of the largest component of $H_{\lambda, s} \cap B_{i,s}$.

Let $\Theta_d$ be the volume of the unit ball.
The probability that $|H_{\lambda-\lambda', s} \cap B(x_{i,s}, 1)| = 1$
is $\Theta_d \cdot (\lambda - \lambda') e^{\Theta_d (\lambda - \lambda')}$.
If $X_{i,s}$ is defined, the probability that $V_{i,s} = \beta \log s$ is $p_{\beta \log s}(\lambda)$.


By the independence of the processes $H_{\lambda', s}$ and $H_{\lambda'-\lambda, s}$,
\[
\Pr[V_{i,s} \ge \beta \log s]
\ge \Theta_d (\lambda - \lambda') e^{\Theta_d (\lambda - \lambda')} p_{\beta \log s}(\lambda)
\ge e^{-\xi'' \beta \log s} = s^{\xi'' \beta}.
\]
for some $\zeta'' > \zeta(\lambda')$ satisfying $\zeta'' \beta < d$.

Since the balls $B_{1,s} \dots B_{m(s), s}$ are disjoint, we can estimate the probability that
none of them contains a component of size at least $\beta \log s$ and so
\[
\Pr[L_1(G(H_{\lambda, s}, 1)) < \beta \log s]
\le (1-s^{\xi'' \beta})^m{s}
\le \exp(-s^{\xi'' \beta} m(s)),
\]
which tends to zero as $s$ goes to infinity by~\eqref{eq:msineq} and because $\xi'' \beta < d$.


\end{proof}