\begin{thm}[Theorem 10.9 in~\cite{PenroseBook}]
In a supercritical setting, as $s$ goes to infinity, with high probability,
$$ s^{-d} L_1(H_{\lambda,s},1) \rightarrow \lambda P_\infty(\lambda)$$
$$ s^{-d} L_2(H_{\lambda,s},1) \rightarrow 0.$$

That means, there will be an infinite component and such component
will be unique, other components will be very small.
\end{thm}

\begin{proof}[Proof of the second part]
We prove the second part for $d=2$ (that is, that the second largest component is small),
although the theorem holds for higher dimensions as well.

Consider a function $\Phi_s, s \ge 0$ such that $\frac{\Phi_s}{\log s} \rightarrow \infty$.
If we define the metric diameter of a component as:
$$ {\rm diam}(S) \equiv \sup_{x,y \in S} ||x -y||,$$
then we want to show that no other component has metric diameter greater than $\Phi_s$.

Consider a partition of a box $s \times s$ into $m_s$ rows and $m_s$
columns, where $m_s \equiv \frac{s}{\Phi_s}$. Let us also define a horizontal
and vertical ``domino'', that is a piece of our grid of size $1 \times 2$ or $2
\times 1$, respectively.

The crucial step is to use Lemma~\ref{l:expon} to calculate the probability that the largest component
traverses a horizontal (vertical) domino piece from its left (top) boundary to its right (bottom) boundary:

$$P[{\rm traversal}] \ge 1 - \exp(-c \Phi_s) = 1 - \exp(-c s/m_s).$$

The probability is high enough to guarantee that with high probability, all dominos in the
$m_s \times m_s$ grid will be crossed.
Indeed, the probability, that not all the dominos are crossed is at most
\[
2 m_s^2 \exp(-cs/m_s) \le \exp(-\mathrm{const}s).
\]

We can easily see that if a $1 \times 1$
piece is traversed horizontally and vertically, both paths have to belong to
the same, giant component.

$$P[{\rm crossing\ everywhere}] \ge 1 - m^2_s \exp(-c s/m_s) \ge 1 - \exp(-{\rm const} \cdot s)
%= 1 - \exp(-c \Phi_s)
$$

As our infinite component is crossing in every piece of the grid, it follows that any other component
must be avoiding all the crossing paths, and therefore its diameter is at most $O(\Phi_s)$.
\end{proof}
