On the size of the largest cluster in $2D$ critical percolation
We consider (near-)critical percolation on the square lattice. Let $\mathcal{M}_{n}$ be the size of the largest open cluster contained in the box $[-n,n]^2$, and let $\pi(n)$ be the probability that there is an open path from $O$ to the boundary of the box. It is well-known (see \cite{BCKS01}) that for all $0< a < b$ the probability that $\mathcal{M}_{n}$ is smaller than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are bounded away from $0$ as $n \rightarrow \infty$. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that $\mathcal{M}_{n}$ is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. %It seems that this question has not been addressed implicitly or explicitly in the literature. By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of $1/\pi(n)$ appears to be essential for the argument.
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\begin{document}
\title{On the size of the largest cluster in $2D$ critical percolation}
\author{J. van den Berg\footnote{CWI and VU University Amsterdam}
\, and  R. Conijn\footnote{VU University Amsterdam} \\
{\footnotesize email: J.van.den.Berg@cwi.nl, R.P.Conijn@vu.nl}
}
\date{}

\maketitle

\begin{abstract}
We consider (near-)critical percolation on the square lattice.
Let $\mathcal{M}_{n}$ be the size of the largest open cluster contained in the box $[-n,n]^2$,
and let $\pi(n)$ be the probability that there is an open path from $O$ to the boundary of the box.
It is well-known (see \cite{BCKS01}) that for all $0< a < b$ the probability that $\mathcal{M}_{n}$ is smaller
than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are
bounded away from $0$ as $n \rightarrow \infty$. 
It is a natural question, which arises for instance in the study of so-called frozen-percolation processes,
if a similar result holds for the probability that $\mathcal{M}_{n}$
is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. 
%It seems that this question has not been addressed implicitly or explicitly in the literature.
By a suitable partition of the box, and a careful construction involving the building blocks, we show that the
answer to this question is affirmative. 
The `sublinearity' of $1/\pi(n)$ appears to be essential for the argument.
\end{abstract}

\section{Introduction and main result}\label{sec:introduction}
Consider bond percolation on $\mathbb{Z}^{2}$ with parameter $p$.
(See \cite{G99} for a general introduction to percolation theory.) Let $\Lambda_{n} = [-n,n]^2 \cap \mathbb{Z}^2$
and let, for $v \in \Lambda_{n}$,  $\mathcal{C}_{n}(v)$ denote the size of the open cluster of $v$ inside the box $\Lambda_{n}$:
\begin{equation*}
 \mathcal{C}_{n}(v) = |\{ w \in \Lambda_{n}: v \lra w, \quad \textrm{inside}\; \Lambda_{n}\}|,
\end{equation*}
where we use the standard notation $v \lra w$ for the existence of an open path from $v$ to $w$, and
where the addition `inside $\La_n$' means that we require the existence of such a path which is located entirely in
$\La_n$. 
For a set $A \subset \mathbb{Z}^{2}$ we denote by $\partial A$ the (internal) boundary of $A$:
\begin{equation*}
 \partial A = \{ v \in A: \exists w \not\in A: (v,w)\textrm{ is an edge } \}.
\end{equation*}
The remaining part of $A$ will be called the interior of $A$.
Let $\pi_{p}(n)$ be the probability $\prob_{p}(O \lra \partial \Lambda_{n})$.
For simplicity we write $\pi(n)$ for $\pi_{\frac{1}{2}}(n)$.

We are interested in the size of `large' open clusters in $\La_n$ for the case where $p$ is equal (or close) to the
critical value $1/2$.
It is known in the literature that, informally speaking, the size of the largest open cluster is typically
of order $n^2 \pi(n)$: For any $c > 0$, there is a `reasonable' probability that it is larger (smaller)
than $c n^2 \pi(n)$, and this probability goes to $0$ uniformly in $n$ as $c \rightarrow \infty$
($c \rightarrow 0$). (See \cite{BCKS99}, \cite{BCKS01}; see also \cite{J03} Section 3.)
However, the question whether for all $0 < a < b$ there is a `reasonable' probability that there is an open
cluster with size {\em between} $a n^2 \pi(n)$ and
~$b n^2 \pi(n)$ has not been investigated in the literature.
 
This question, which is also natural by itself, arises e.g. in the study
of finite-parameter frozen-percolation models.
In these models each edge is closed at time $0$ and `tries' to become open
at some random time, independently of the other edges. However, an open cluster stops growing as soon as its
size has reached a certain
(large) value $M$, the parameter of the model. (See \cite{BLN12} where this was studied for the case where
the `size' of a cluster is defined as its diameter instead of its volume.)
The investigation of such processes leads to the question
how two open clusters which both have size of order $M$ but smaller than $M$, merge to a cluster of size bigger than $M$,
which in turn leads to the question at the end of the previous paragraph.
To state our main result, an affirmative answer to that question, we first need a few more definitions.

For $k,l \in \mathbb{N}$, we denote by $HC(k,l)$ the event that there is an open horizontal crossing in the box
$[0,k]\times [0,l]$. (This is an open path from the left side to the right side of the box, of which all vertices, except
the starting and end point, are in the interior of the box). Let the ``characteristic length'' be as defined in e.g.
\cite{N08} and \cite{K87}:
For a fixed  $\epsilon \in (0,\frac{1}{2})$:
\begin{equation}
 L(p) = L_{\epsilon}(p) = \left\{ \begin{array}{ll} \min{\{ n\in \mathbb{N}: \prob_{p}(HC(n,n)) \le \epsilon\} } & \textrm{if } p < \frac{1}{2},\\
                          \min{\{ n \in \mathbb{N}: \prob_{p}(HC(n,n)) > 1 - \epsilon\} } & \textrm{if } p >\frac{1}{2},
                         \end{array}\right.
\end{equation}
and $L(\frac{1}{2}) = \infty$.
The precise value of $\epsilon$ is not essential. Throughout this paper we will consider it as being fixed, and therefore
we omit it from our notation.

As said before, our main question concerns the existence of {\em some} open cluster in $\La_n$ with size in some
specific interval.
The proof we obtained gives, with only a tiny bit of extra work, something stronger;
it shows that 
with `reasonable' probability the {\em maximal} open cluster has this property. 
Therefore we state our main result in this
stronger form (and remark that we do not know an essentially simpler proof of the original weaker form): \\
%\begin{theorem}\label{thm:existClustInInt}
% Let $0 < a < b < \infty$. There exist $\delta > 0$ and $N \in \mathbb{N}$
%such that, for all $p\in (0,1)$ and $n: N \le n \le L_{\epsilon}(p)$:
%\begin{equation*}
% \prob_{p}\left(\exists v \in \Lambda_{n}: \mathcal{C}_{n}(v) \in (an^2 \pi(n), bn^2 \pi(n))\right) > \delta.
%\end{equation*}
%\end{theorem}
Denote by $\mathcal{M}_{n}$ the size of the maximal open cluster in $\Lambda_{n}$. More precisely,
\begin{equation*}
\mathcal{M}_{n} = \max_{v \in \Lambda_{n}}{ \mathcal{C}_{n}(v) }.
\end{equation*}

\begin{theorem}\label{thm:maxClusterInInt}
 Let $0 < a < b$. There exist $\delta > 0$ and $N \in \mathbb{N}$ such that, for all $n \geq N$ and all $p$ with
$L(p) \geq n$,
\begin{equation*}
 \prob_{p}\left(\mathcal{M}_{n} \in (an^2 \pi(n), bn^2 \pi(n))\right) > \delta.
\end{equation*}
\end{theorem}
\smallskip
The proof is given in Section \ref{sect-proof}. Section \ref{sect-notation} will list the main ingredients 
used in the
proof. The proof involves a suitable partition of $\La_n$ in smaller boxes and annuli. A brief and informal summary
is given in the beginning of Section \ref{sect-proof}, after the description of these objects. 

\section{Ingredients for the proof of Theorem \ref{thm:maxClusterInInt}}\label{sect-notation}
We will make amply use of standard RSW results of the following form: 
For all $l > 0$ there exists $\delta(l) > 0$ such that, for all $k$ and all $p$ with
$L(p) \geq k$,
$\prob_{p}(HC(k,\lceil l k \rceil)) \ge \delta(l)$.
%And for all $m \in \mathbb{N}$ there exist $\delta(\frac{1}{m}) > 0$ such that, uniform in $p$,
%$\prob_{p}(HC(k,\lfloor \frac{k}{m}\rfloor)) \ge \delta(\frac{1}{m})$ if $k (\le L_{\epsilon}(p))$ is sufficiently large.
%
%For a circuit $\gamma$ in $\mathbb{Z}^{2}$ we denote by Int$(\gamma)$
%the bounded connected component of $\mathbb{Z}^{2} \setminus \gamma$.
%
%We need some notation for the `restriction' of an event to a subset of $\mathbb{Z}^{2}$.
%But to introduce this notation we first need some other notations.
%Let $E$ denote the set of edges of $\mathbb{Z}^{2}$ and $\Omega = \{0,1\}^{E}$.
%For $\omega \in \Omega$ and $F \subset E$ we will write $\omega_{F} \in \{0,1\}^{F}$ for the `restriction' 
%$(\omega_e, e \in F)$ of $\omega$ to $F$.
%Let $A \subset \Omega$ and $W \subset \mathbb{Z}^{2}$. We write $E(W)$ for the set of all edges of
%which both endpoints are in $W$. 
%Informally, we use the notation $A(W)$ for the set of all configurations $\omega \in \Omega$ that belong to $A$
%or can be turned to an element of $A$ by modifying $\omega$ outside $E(W)$.
%More precisely,
%\begin{equation}\label{eq:notationRestrOfIncrEvent}
% A(W) = \{\omega_{E(W)}: \omega \in A\} \times \{ 0,1\}^{E\setminus E(W)}.
%\end{equation}
For a set $W$ of vertices define 
\begin{equation}\label{eq:defCtilde}
 \tilde{\mathcal{C}}(W) = |\{ v \in W: v \lra \partial W\}|.
\end{equation}
For $k, n \in \N$, we use the notation $\La_{k,n}$ for the rectangle $[-k,k] \times [-n,n]$.
We will use the following properties of $\pi_{p}(n)$ from the literature. 
\begin{theorem}\label{lem:PiBounds} There exist $\alpha, C_{1}, \cdots, C_6 >0$ such that: 
 \begin{enumerate}[(i)]
  \item For all $m \le n$:
   \begin{equation*}
    C_{1}(\frac{n}{m})^{\alpha} \le \frac{\pi(m)}{\pi(n)} \le  C_{2}(\frac{n}{m})^{\frac{1}{2}}.
   \end{equation*}
  \item For all $n \in \mathbb{N}$:
   \begin{equation*}
    \sum_{k=0}^{n} \pi(k) \le C_{3}\cdot n\pi(n).
   \end{equation*}
  \item For all $p\in (0,1)$ and all $n \le L(p)$,
   \begin{equation*}
    C_{4}\pi(n) \le \pi_{p}(n) \le C_{5}\pi(n).
   \end{equation*}
   \item 
For all $k, n \in \mathbb{N}$ and $p$ with $L(p) \ge k \wedge n$, 
   \begin{equation*}
   \mathbb{E}_p[\tilde{\mathcal{C}}(\Lambda_{k,n})] \leq C_6 k n \, \pi( k\wedge n).
   \end{equation*}
 \end{enumerate}
\end{theorem}
\begin{proof}
The inequalities in (i) are well-known. (The first follows easily from RSW arguments, and the second 
goes back to \cite{BK85}; see also for example \cite{BCKS01}).
%The lower bound is Lemma 4.5 (iii) (for a proof see:
%\cite{BCKS99}) and the upper bound is (6.22).
Part (ii) follows from (7) in \cite{K86}.
Part (iii) is Theorem 1 in \cite{K87}.
Part (iv), of which versions  are explicitly in the literature (see e.g. \cite{K86}, \cite{K87} and \cite{N08}),
is proved as follows (where we assume that $k \leq n$):
\begin{eqnarray}
 \, & \, &  \mathbb{E}_p[\tilde{\mathcal{C}}(\Lambda_{k,n})] = \sum_{v \in \Lambda_{k,n}}
\prob_p(v \lra \partial \Lambda_{k,n}) \nonumber\\
\,  & \le & \sum_{v \in \Lambda_{k,n}} \pi_p(d(v, \partial \Lambda_{k,n})) \nonumber
\,\, \le \,\, 8n\sum_{l=0}^{k} \pi_{p}(l) \le C_{6} n k \pi(k),
\end{eqnarray}
where the last inequality uses part (ii) and (iii). \qed
\end{proof}

%By the standard RSW arguments, for all $l \in \mathbb{N}$ there exist $\delta(l) > 0$ such that, uniform in $p$,
%$\prob_{p}(HC(k,lk)) \ge \delta(l)>0$ for all $k \le L_{\epsilon}(p)$.
%And for all $m \in \mathbb{N}$ there exist $\delta(\frac{1}{m}) > 0$ such that, uniform in $p$,
%$\prob_{p}(HC(k,\lfloor \frac{k}{m}\rfloor)) \ge \delta(\frac{1}{m})$ if $k (\le L_{\epsilon}(p))$ is sufficiently large.

\medskip
Define
\begin{equation*}
 Y(m) = |\{ v \in \Lambda_{m}: v \lra \partial\Lambda_{2m} \}|.
\end{equation*}
We need the following result for the distribution of $Y(m)$, which is essentially in
\cite{BCKS99} and (for the special case $p = 1/2$) \cite{K86}.

\begin{theorem} \label{lem:momentBoundExpConnBoundary}
There exist $\delta_{1}, C_{7} > 0$ such that, for all $p\in (0,1)$ and all $m \le L(p)$:
\begin{equation}\label{eq:posProbEnoughVertConn}
 \prob_{p}(Y(m) \ge C_{7}m^{2}\pi(m)) \ge \delta_{1}.
\end{equation}
\end{theorem}
\begin{proof}
By Lemma 6.1 in \cite{BCKS99} there
exists $C_{8} > 0$ such that for all $p \in (0,1)$ and $m \le L(p)$,
% \begin{equation} \label{eq:power-Y}
$\mathbb{E}_{p}[(Y(m))^{2}] \le C_{8} (m^{2}\pi(m))^{2}$.
 %\end{equation}
%(The definition of $L_{0}(p)$ in \cite{BCKS99} is different from $L_{\epsilon}(p)$.
%But as noted in \cite{BCKS99} remark (vi), they only need properties of $L_{0}(p)$)
%which also hold for $L_{\epsilon}(p)$, by standard RSW arguments).
Further, by the definition of $\pi(n)$ and parts (i) and (iii) of Theorem \ref{lem:PiBounds},
there exists $C_{9} > 0$ such that
%\begin{equation}\label{eq:lowerBoundExpConnBoundary}
$\mathbb{E}_{p}[Y(m)] \ge C_{9}m^{2}\pi(m)$.
%\end{equation}
These two inequalities, and the one-sided Chebyshev's inequality, give
Theorem \ref{lem:momentBoundExpConnBoundary}. \qed
%\begin{eqnarray*}
% \prob_{p}\left(Y(m) \ge \frac{1}{2}C_{8}m^{2}\pi(m)\right) & \ge & \prob_{p}\left(Y(m) \ge \frac{1}{2}\mathbb{E}_{p}[Y(m)]\right)\\
% & \ge & \frac{\frac{1}{4}\mathbb{E}_{p}[Y(m)]^{2}}{\frac{1}{4}\mathbb{E}_{p}[Y(m)]^{2} + \textrm{Var}_{p}(Y(m))}.
%\end{eqnarray*}
%Now apply \eqref{eq:power-Y} (with $k =2$) and \eqref{eq:lowerBoundExpConnBoundary}.
\end{proof}

\medskip
It was shown in \cite{K86} (and extended/generalized in \cite{BCKS99}
and \cite{BCKS01}) that $\mathcal{M}_{n}$, the size of the largest open cluster in $\La_n$, is typically of
order $n^2 \pi(n)$. In particular, its expectation has an upper and a lower bound which are linear in
$n^2 \pi(n)$. In the proof of Theorem \ref{thm:maxClusterInInt} we use the following result from \cite{BCKS01}.
\begin{theorem}\label{thm:maxClustCanBeSmall} {\em \bf (\cite{BCKS01} Thm. 3.1 (i), Thm. 3.3 (ii))} \\
Let $p_{n}$ be a sequence, such that $n \le L(p_{n})$ for all $n$. Then 
for all $K > 0$,
 \begin{equation*}
  \liminf_{n \to \infty} \prob_{p_{n}}\left(\frac{\mathcal{M}_{n}}{n^2 \pi(n)} < K\right) > 0.
 \end{equation*}
\end{theorem}

Finally, to streamline the arguments in Section \ref{sect-compl} at the end of the proof of Theorem \ref{thm:maxClusterInInt},
we state here the following fact about `steering' the outcome of the sum of independent random variables.
It is a simple observation rather than a lemma, and versions of it have without doubt been
used in the probability literature in various contexts.
\begin{lemma}\label{lem:sumOfRandomVars} Let $0 < \al < \be$,
and let $k \in \N$ be such that $\al/k < (\be - \al)/2$.
Further, let $\eta_{1}, \eta_{2} > 0$ and let $(X_{i})_{1 \leq i \le k}$ be independent random variables,
(not necesarrily identically distributed) which satisfy the following:
\begin{eqnarray*}
 \prob\left(X_{i} \in (\frac{\al}{k}, \frac{\be-\al}{2})\right) & \ge & \eta_{1};\\
 \prob\left(X_{i} \le \frac{\be-\al}{2k} \right) & \ge & \eta_{2}.
\end{eqnarray*}
Then
\begin{equation*}
 \prob\left(\sum_{i=1}^{k} X_{i} \in (\al, \be)\right) \ge (\eta_{1}\wedge \eta_{2})^{k}.
\end{equation*}
\end{lemma}
\begin{proof}
For $1 \leq i \leq k$ we say that `step $i$ is proper' if
$$
X_i  \begin{cases} \in (\frac{\al}{k}, \frac{\be - \al}{2}) & \mbox{ if } \sum_{j=1}^{i-1} X_j < \al \\
\leq \frac{\be - \al}{2 k} & \mbox{ otherwise. } \end{cases}
$$ 
It is clear that if all steps $i =1, \cdots, k$ are proper, then $\sum_{i=1}^k X_i \in (\al,\be)$.
It is also easy to see that, for each $i$, the
conditional probability that step $i$ is proper, given that all steps $1, \cdots, i-1$ are proper,
is at least $\min(\eta_1, \eta_2$). \qed
\end{proof}

\section{Proof of Theorem \ref{thm:maxClusterInInt}} \label{sect-proof}
We first give a proof for the special case $p = 1/2$ and therefore drop the subscript $p$ from the notation $\P_p$ and
$\E_p$. At the end of Section \ref{sect-compl} we point out that (due to the `uniformity' of the ingredients stated
in Section \ref{sect-notation}) the proof for the general case is essentially the same.

\subsection{More definitions, and brief outline of the proof}\label{sec:proofDefi}

Let $s, t \in \mathbb{N}$ with $t \le \frac{1}{3}s$.
The proof involves a construction using the following boxes and annuli.
\begin{eqnarray*}
 B_{0,0} & = & \Lambda_{s}.\\
 A_{0,0}^{I} & = & \Lambda_{s} \setminus \Lambda_{s - t}; \,\,\,\,
A_{0,0}^{II}  =  \Lambda_{s -t} \setminus \Lambda_{s - 2t}; \,\,\,\,
 A_{0,0}^{III}  =  \Lambda_{s -t} \setminus \Lambda_{s - 2t}. \\
 A_{0,0}' & = & A_{0,0}^{I} \cup A_{0,0}^{II} \cup A_{0,0}^{III}.\\
 H_{0,0} & = & ([0,4t]\times [0,t] + (s-2t, 0)) \cap \mathbb{Z}^{2}.\\
 V_{0,0} & = & ([0,t]\times [0,4t] + (0, s-2t)) \cap \mathbb{Z}^{2}.
\end{eqnarray*}
More generally, for all $i,j \in \mathbb{Z}$ we define $B_{i,j} = B_{0,0} + (2is,2js)$, 
$A_{i,j}^{I} = A_{0,0}^{I} + (2is,2js)$, etcetera.

\begin{figure}
 \centering
 \scalebox{1.5}{\includegraphics{smallBoxes.pdf}}
 \caption{Partition of the box $\Lambda_{n}$. Here $n=40, m = 3, s=13, t=2$.}
 \label{fig:smallboxes}
\end{figure}
\smallskip
Before we go on, we give a very brief and informal summary of the proof (see Figure \ref{fig:smallboxes}):
The box $\La_n$ in the statement of the theorem will
be (roughly) partitioned in $m^2$ boxes $B_{i,j}$ defined above, where the $s$ (and hence $m$) and $t$ will be
chosen appropriately, depending on $n$, $a$ and $b$. (For elegance/symmetry we take the number $m$ odd).
We will `construct' an open cluster of which the `skeleton' consists of circuits in the annuli $A_{i,j}^{II}$,
`glued' together by connections in the `corridors' $V_{i,j}$ and $H_{i,j}$. (The other annuli defined
above will be used for technical reasons in the proof). The setup is such that
the contributions from the different $B_{i,j}$'s to the total cluster size are roughly independent,
and that these contributions can be `steered' to get the total sum inside the desired interval.
In some sense this replaces the original problem for
the box $\La_n$ by a similar problem, but now for the smaller boxes $B_{i,j}$.
Apart from the technicalities involving the
control of local dependencies, there is a subtle aspect in the proof related to the
asymptotic behaviour of $\pi(n)$: Although the precise power-law behaviour of $\pi(n)$ is not
important, it seems to be essential for the arguments that the exponent in a power-law upper
bound is strictly smaller than $1$ (see the note at the
end of the proof of Lemma \ref{lem:chooseNxEps})).

\medskip
Now we continue with the precise constructions mentioned above. First we give some more notation and definitions.
Let $E$ denote the set of edges of $\mathbb{Z}^{2}$ and $\Omega = \{0,1\}^{E}$.
For $\omega \in \Omega$ and $F \subset E$ we will write $\omega_{F} \in \{0,1\}^{F}$ for the `restriction'
$(\omega_e, e \in F)$ of $\omega$ to $F$.
Let $A \subset \Omega$ and $W \subset \mathbb{Z}^{2}$. We write $E(W)$ for the set of all edges of
which both endpoints are in $W$.
Informally, we use the notation $A(W)$ for the set of all configurations $\omega \in \Omega$ that belong to $A$
or can be turned to an element of $A$ by modifying $\omega$ outside $E(W)$.
More precisely,
\begin{equation}\label{eq:notationRestrOfIncrEvent}
 A(W) = \{\omega_{E(W)}: \omega \in A\} \times \{ 0,1\}^{E\setminus E(W)}.
\end{equation}

\begin{figure}
 \centering
 \scalebox{0.4}{\includegraphics{evOij.pdf}}
 \caption{Illustration of the event $\tilde{O}^{s,t}$.}
 \label{fig:eventO}
\end{figure}
We denote by $\tilde{O}^{s,t}$ the event that (i) - (iii) below occur (see Figure \ref{fig:eventO}):
\begin{enumerate}[(i)]
 \item $\forall i,j\in \mathbb{Z}$: the annulus $A_{i,j}^{II}$ contains an open circuit;
 \item $\forall i,j\in \mathbb{Z}$: $H_{i,j}$ contains an open connection between the two widest open circuits in the annuli $A_{i,j}^{II}$
and $A_{i+1,j}^{II}$;
 \item  $\forall i,j\in \mathbb{Z}:$ $V_{i,j}$ contains an open connection between the two widest open circuits in the annuli $A_{i,j}^{II}$
and $A_{i,j+1}^{II}$.
\end{enumerate}
The introduction of this event looks meaningless since it has probability $0$. It will only be used
to give a `compact' description of the following events (which do play a key role in the proof). 
\begin{definition}\label{def:O} Let $m, s, t \in \mathbb{N}$, with $t \le \frac{1}{3}s$ and $m$ odd. Let $i, j \in \Z$.
We define, using notation \eqref{eq:notationRestrOfIncrEvent}, the following events:
 \begin{eqnarray*}
  O^{m,s,t} & = & \tilde{O}^{s,t}(\Lambda_{ms}).\\
  O_{i,j}^{s,t} & = & \tilde{O}^{s,t}(B_{i,j}).
 \end{eqnarray*}
\end{definition}
\textbf{Remark:} From now on, for given $m,s,t$, the indices $i,j$ under consideration will always be assumed to
be in the set $\{-\frac{1}{2}(m-1), \cdots , 0, \cdots, \frac{1}{2}(m-1)\}$.

\subsection{Expected cluster size in a narrow annulus}\label{sect-czann}
For a circuit $\gamma$ in $\mathbb{Z}^{2}$ we denote by Int$(\gamma)$
the bounded connected component of $\mathbb{Z}^{2} \setminus \gamma$, and define
\begin{equation}\label{eq:defCijGamma}
\mathcal{C}^{\gamma} = |\{ v \in \mathrm{Int}(\gamma): v \lra \gamma \}|.
\end{equation}
Further, for all $i, j$, let $\gamma_{i,j}$ denote the widest open circuit in the annulus $A_{i,j}^{II}$, and
define, for $W \subset \La_n$,
\begin{equation} \label{eq:defCij}
 \mathcal{C}_{i,j}(W) =  |\{ v \in W: v \lra \gamma_{i,j} \}|.
\end{equation}
If there is no  open circuit in $A_{i,j}^{II}$, then $\mathcal{C}_{i,j}(W) = 0$.

\smallskip
Recall the definition of $\tilde{\mathcal{C}}$ in \eqref{eq:defCtilde}.
\begin{lemma}\label{lem:ExpSizeEpsxX} There exists a constant $C_{10} > 0 $ such that for all
$s \in \mathbb{N}$, $t \le \frac{1}{3}s$ and all $i,j$:
\begin{equation*}
 \mathbb{E}[\mathcal{C}_{i,j}(A_{i,j}') | O_{i,j}^{s,t}] \le \mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}') | O_{i,j}^{s,t}]
\le C_{10} st \pi(t).
\end{equation*}
\end{lemma}
\begin{proof}
The first inequality follows immediately from the obvious fact that,
on the event $O_{i,j}^{s,t}$, $\mathcal{C}_{i,j}(A_{i,j}')$
is smaller than or equal to $\tilde{\mathcal{C}}(A_{i,j}')$. We prove the second inequality.
Without loss of generality we take $i=j=0$.
\begin{figure}
 \centering
 \scalebox{0.3}{\includegraphics{PartAPri.pdf}}
 \caption{The subdivision of $A_{0,0}'$ in $A_{1}', \cdots, A_{8}'$.}
 \label{fig:PartAPri}
\end{figure}
We subdivide the annulus
$A' = A_{0,0}' = \bigcup_{l=1}^{8} A_{l}'$, where $A_{1}', A_{2}', A_{3}', A_{4}'$ are the $(3t \times 3t)$-squares in
the four corners,
and $A_{5}', A_{6}', A_{7}', A_{8}'$ the remaining rectangles (see Figure \ref{fig:PartAPri}).
%e.g.: $A_{1}' = [-s,-s+3t] \times [-s,-s+3t] \cap \mathbb{Z}^{2}$
%and $A_{5}' = [-s+3t,s-3t]\times [-s,-s+3t] \cap \mathbb{Z}^{2}$.
Note that
$\tilde{\mathcal{C}}(A') \le \sum_{l=1}^{8} \tilde{\mathcal{C}}(A_{l}').$
Hence it is sufficient to show that there is a constant $C_{11}$ such that for each $l =1, \cdots,8$,
\begin{equation} \label{eq-r-tCbd}
\E[\tilde{\mathcal{C}}(A_{l}') | O_{0,0}^{s,t}] \leq C_{11} s t \pi(t).
\end{equation}
By symmetry we only have to handle the cases $l=1$ and $l=5$.
%First notice that $\mathbb{E}[\tilde{\mathcal{C}}(A_{l}') | O_{0,0}^{m,s,t}] =
%\mathbb{E}[\tilde{\mathcal{C}}(A_{1}') | O_{0,0}^{m,s,t}]$
%for $l\le 4$ and 
%$\mathbb{E}[\tilde{\mathcal{C}}(A_{l}') | O_{0,0}^{m,s,t}] = \mathbb{E}[\tilde{\mathcal{C}}(A_{5}') | O_{0,0}^{m,s,t}]$
%for $l > 4$. For the conditional expectation of $\tilde{\mathcal{C}}(A_{l}')$, for all $l$, we get
For each $l$ the l.h.s. of \eqref{eq-r-tCbd} is
\begin{equation} \label{eq-r-help1}
\mathbb{E}[\tilde{\mathcal{C}}(A_{l}') | O_{0,0}^{s,t}] = \frac{1}{\prob(O_{0,0}^{s,t})}\sum_{v \in A_{l}'}
\prob(v \lra \partial A_{l}'; O_{0,0}^{s,t}).
\end{equation}
Recall the notation \eqref{eq:notationRestrOfIncrEvent}. For each $v \in A_{1}'$, obviously,
\begin{equation} \label{eq-r-help2}
 \prob(v \lra \partial A_{1}'; O_{0,0}^{s,t}) \le
\prob(v \lra \partial A'_1) \prob(O_{0,0}^{s,t}(A' \setminus A_{1}')).
%\pi(t \wedge d(v, \partial A_{1}'))\prob(O_{0,0}^{s,t}(A' \setminus A_{1}')).
\end{equation}
Further, informally speaking, the event $O_{0,0}^{s,t}(A' \setminus A_{1}')$ can, with a `local surgery
involving a bounded cost in terms of probability', be turned into the event $O_{0,0}^{s,t}$.
More precisely, if $O_{0,0}^{s,t}(A' \setminus A_{1}')$ holds, and there is a horizontal open crossing
of the rectangle $[-s, -s + 6t] \times [-s + t, -s + 2t]$ and of the square $[-s, -s + 3 t] \times 
[-s + 3t, -s + 6 t]$, and a vertical open crossing of the rectangle
$[-s +t, -s + 2t] \times [-s, -s + 6t]$ and of the square $[-s + 3 t, -s + 6 t] \times [-s, -s + 3 t]$, then
the event $O_{0,0}^{s,t}$ holds. Hence, by RSW (and FKG) we have a positive constant $C_{12}$ such that
$\prob(O_{0,0}^{s,t}(A'\setminus A_{1}')) \le C_{12} \prob(O_{0,0}^{s,t})$.
Combining this with \eqref{eq-r-help1} and \eqref{eq-r-help2} gives
\begin{equation}\label{eq-r-case1}
\mathbb{E}[\tilde{\mathcal{C}}(A_{1}') | O_{0,0}^{s,t}]  \le C_{12} \sum_{v \in A'_1} \prob(v \lra \partial A'_1).
\end{equation}
For the case $l=5$ a similar argument (now the `surgery' can be done on the union of the region $V_{0,-1}$ and the rectangle
$[v_{1} - t, v_{1} + t]\times [-s,-s+3t]$)) gives a constant $C_{13}> 0$ such that
\begin{equation}\label{eq-r-case5}
\mathbb{E}[\tilde{\mathcal{C}}(A_{5}') | O_{0,0}^{s,t}]  \le 
C_{13} \sum_{v \in A'_5}  \prob(v \lra \partial A'_5).
\end{equation}
Application of part (iv) of Theorem \ref{lem:PiBounds} to the right-hand sides of \eqref{eq-r-case1} and
\eqref{eq-r-case5} gives \eqref{eq-r-tCbd}. \qed
%If $v \in A_{5}'$:
%\begin{equation*}
% \prob(v \lra \partial A_{5}'; O_{0,0}^{m,s,t}) \le \pi(t \wedge d(v, \partial A_{5}'))\prob(O_{0,0}^{m,s,t}(A' \setminus (G_{1}\cup G(v)))),
%\end{equation*}
%where $G_{1} = V_{0,-1}$ and
%\begin{equation*}
% G(v) =  [v_{1} - t, v_{1} + t]\times [-s,-s+3t].
%\end{equation*}
%Again by standard RSW arguments we get
%$\prob(O_{i,j}^{m,s,t}(A'\setminus (G_{1}\cup G(v)))) \le C_{12,2}\prob(O_{i,j}^{m,s,t})$ for all $v \in A_{5}'$.
%Hence, for all $l \le 8$
%\begin{eqnarray}
% \mathbb{E}[\tilde{\mathcal{C}}(A_{l}') | O_{0,0}^{m,s,t}] & \le & (C_{12,1} \vee C_{12,2})\sum_{v \in A_{l}'} \pi(t \wedge d(v, \partial A_{l}')) \nonumber\\
% & \le & 4(C_{12,1} \vee C_{12,2}) s\left(t \pi(t) + \sum_{k=0}^{t} \pi(k)\right)\nonumber\\
% & \le & \tilde{C}_{12} s t \pi(t),\label{eq:comAlPrime}
%\end{eqnarray}
%where the last step follows from Theorem \ref{lem:PiBounds} (ii).\newline
%Thus $\mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}') | O_{i,j}^{m,s,t}] \le 8\tilde{C}_{12}st \pi(t)$
%which completes the proof of the lemma,  since $\tilde{C}_{12}$ doesn't depend on $m,s$ and $t$. \qed
\end{proof}

\subsection{Properties of nice circuits}
Let $m,s,t$ be as in Definition \ref{def:O}, and recall the Remark about the values of the indices
$i, j$ at the end of Section \ref{sec:proofDefi}. Let, for each $i, j$, $\ga_{i,j}$ be as in the beginning
of Section \ref{sect-czann},
and let $\tilde{\ga}_{i,j}$ be a deterministic circuit in the annulus $A_{i,j}^{II}$.
%As in the remark at the end of section \ref{sec:proofDefi}, the $i,j$
%will be such that $-\frac{1}{2}(m-1)\le i,j \le \frac{1}{2}(m-1)$.
Further we will denote the collection of all $\ga_{i,j}$'s by $(\ga)$, and the collection
of al $\tilde{\gamma}_{i,j}$'s by $(\tilde \gamma)$.
% the collection
%$\gamma =(\gamma_{i,j})$; similarly we write
%$\tilde{\gamma} = (\tilde{\gamma}_{i,j})$.
\begin{definition}\label{def:nice}
 We say that $\tilde{\gamma}_{i,j}$ is \emph{$(s,t)$-nice} if
\begin{equation}\label{eq:defGammaij}
 \mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}')\: | \; O_{i,j}^{s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j}] \le 2 C_{10} s t \pi(t),
\end{equation}
with $C_{10}$ as in Lemma \ref{lem:ExpSizeEpsxX}.
Further, the collection $(\tilde{\gamma})$ is called $(m,s,t)$-nice if 
each circuit in the collection is $(s,t)$-nice.
%if for all $i,j:$ $\tilde{\gamma}_{i,j}$ is $(s,t)$-nice.
\end{definition}
We define $\Gamma_{i,j}^{s,t}$
as the event that $\ga_{i,j}$ is $(s,t)$-nice, and define

%\begin{equation*}
% \Gamma_{i,j}^{s,t} = \bigcup_{\tilde{\gamma}_{i,j}: (s,t)\mathrm{-nice}} \{ \gamma_{i,j} = \tilde{\gamma}_{i,j}\}.
%\end{equation*}
%With this we can define the event
\begin{equation}\label{eq:defGammaNxEps}
 \Gamma^{m,s,t} = \bigcap_{i,j} \Gamma_{i,j}^{s,t}.
\end{equation}
Recall the notation \eqref{eq:defCij}. We will study the open cluster $\mathcal{C}_{0,0}(Q)$, where
\begin{equation*}
 Q = (\Lambda_{n} \setminus \Lambda_{ms}) \cup (\bigcup_{i,j} A_{i,j}').
\end{equation*}
%About the event $\Gamma^{m,s,t}$ we have the following lemma.
\begin{lemma}\label{lem:probGamma}
There exist positive constants $C_{14}$ and $C_{15}$ such that, for all $m, s$ and $t$,
\begin{enumerate}[(i)]
\item 
\begin{equation*}
 \prob(\Gamma^{m,s,t} | O^{m,s,t}) \ge (C_{14})^{m^{2}}.
\end{equation*}
\item For all $(m,s,t)$-nice $(\tilde{\gamma})$ and all $n$ with $n - ms \le t$,
\begin{equation*}
 \prob\left(\mathcal{C}_{0,0}(Q) \le C_{15} m^{2} st \pi(t)\; | \; O^{m,s,t}; (\gamma) = (\tilde{\gamma}) \right)
\ge \frac{1}{2}.
%\delta_{2}(x,\varepsilon).
\end{equation*}
\end{enumerate}
\end{lemma}
%\begin{equation} \label{eq-r-disp}
%\prob(O^{m,s,t}) \le \prod_{i,j} \prob(O_{i,j}^{m,s,t}).
%\end{equation}
\begin{proof} We claim that there is a constant $C_{16} > 0$ such that for all $i,j$:
\begin{equation}\label{eq:SplitOnxEpsWithGamma}
 \prob(O^{m,s,t}; (\gamma) = (\tilde{\gamma})) \ge C_{16}\prob(D)\prob(O_{i,j}^{s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j}),
\end{equation}
where (with the notation \eqref{eq:notationRestrOfIncrEvent})
\begin{equation*}
 D = O^{m,s,t}(\Lambda_{n} \setminus B_{i,j})\cap \bigcap_{\tilde{i},\tilde{j}:(\tilde{i},\tilde{j}) \neq (i,j)} \{ \gamma_{\tilde{i},\tilde{j}} = \tilde{\gamma}_{\tilde{i},\tilde{j}} \}.
\end{equation*}
To prove this claim we write
\begin{equation*}
 G = B_{i,j} \setminus A_{i,j}^{I},\qquad 
 J = A_{i,j}^{I} \cup \bigcup_{(\tilde{i},\tilde{j}) \in M_{i,j}} A_{\tilde{i},\tilde{j}}^{I},\qquad
 K = \Lambda_{n} \setminus (G \cup J),
\end{equation*}
where
$M_{i,j} =  \{(i-1,j),(i,j-1),(i+1,j),(i,j+1) \} \cap \{ -(m-1)/2, \cdots,(m-1)/2 \}^{2}$.
Let $D_{1} = O_{i,j}^{s,t} \cap \{ \gamma_{i,j} = \tilde{\gamma}_{i,j} \}$. We also need an event $D_2$ which, informally 
speaking, connects the structures in the definition of $D_1$ with those in $D$. More precisely,
$$D_{2} = \bigcap_{(\tilde{i},\tilde{j}) \in M_{i,j}} D_{2}^{\tilde{i},\tilde{j}},$$
where $D_{2}^{i+1,j}$ is the event that (i) $H_{i,j} \cap J$ contains a horizontal crossing and (ii) $H_{i,j} \cap A_{i,j}^{I}$
and $H_{i,j} \cap A_{i+1,j}^{I}$ both contain a vertical crossing. The other $D_{2}^{\tilde{i},\tilde{j}}$'s are 
defined similarly.
By RSW (and FKG) there is a positive constant $C_{16}$ such that $\prob(D_{2}) >  C_{16}$.
%Recall the notation for the set $B_{E}$ of edges corresponding
%to a set $B\subset \mathbb{Z}^{2}$ of vertices. 
Note that (with the notation in Section \ref{sec:proofDefi}),
%with the states of the edges in $E(K)$ fixed,
%conditionally on $\omega_{1}  \in \{ 0,1\}^{K_{E}}$ the
the event $D$ is increasing with respect to the edges outside $E(K)$, and that
%yand conditionally on $\omega_{2} \in \{ 0,1\}^{G_{E}}$ the event $D_{1}$ is increasing.
$D_1$ is increasing with respect to the edges outside $E(G)$.
We get
\begin{eqnarray*}
 \lefteqn{\prob(O^{m,s,t}; (\gamma) = (\tilde{\gamma})) \ge \prob(D \cap D_{1} \cap D_{2})}\\
 & = & \sum_{\omega_{1} \in \{  0,1\}^{E(K)}} \prob(\omega_{1}) \sum_{\omega_{2} \in \{0,1\}^{E(G)}}
\prob(D \cap D_{1} \cap D_{2} | \omega_{1}, \omega_{2})\prob(\omega_{2})\\
 & \ge & \sum_{\omega_{1} \in \{  0,1\}^{E(K)}} \prob(\omega_{1}) \sum_{\omega_{2} \in \{0,1\}^{E(G)}} \prob(D | \omega_{1},\omega_{2})\prob(D_{1} | \omega_{1},\omega_{2})\prob(D_{2} | \omega_{1},\omega_{2})\prob(\omega_{2})\\
 & = & \sum_{\omega_{1} \in \{  0,1\}^{E(K)}} \prob(\omega_{1})\prob(D | \omega_{1})\sum_{\omega_{2} \in \{0,1\}^{E(G)}} \prob(D_{1} | \omega_{2})\prob(D_{2})\prob(\omega_{2})\\
 & \ge & C_{16} \prob(D)\prob(D_{1}),
\end{eqnarray*}
where we used FKG in the third line, and in the fourth line we used that
$D$ doesn't depend on the configuration on $G$,
$D_{1}$ doesn't depend on the configuration on $K$, and $D_{2}$ doesn't depend on the configuration on $G \cup K$.
This proves the claim.

By repeating the same arguments for each $B_{i,j}$, we eventually get the following `extension' of \eqref{eq:SplitOnxEpsWithGamma}: 

%\eqref{eq:SplitOnxEpsEasy} leads to $\prob(O^{m,s,t}) \ge C_{13}^{m^{2}} \prod_{i,j} \prob(O_{i,j}^{m,s,t})$
%and \eqref{eq:SplitOnxEpsWithGamma} leads to

\begin{equation}\label{eq:SplitTotal}
 \prob\left((\gamma) = (\tilde{\gamma}) ; O^{m,s,t}\right) \ge
C_{16}^{m^{2}} \prod_{i,j} \prob(\gamma_{i,j} = \tilde{\gamma}_{i,j}; O_{i,j}^{s,t}).
\end{equation}

Now we are ready to prove part (i):
\begin{eqnarray}
 \prob(\Gamma^{m,s,t} | O^{m,s,t}) & = &
\frac{1}{\prob(O^{m,s,t})}\sum_{\tilde{\gamma}:(m,s,t)\textrm{-nice}} \prob\left((\gamma) = (\tilde{\gamma}) ; O^{m,s,t}\right)
\nonumber\\
& \ge & C_{16}^{m^{2}} \prod_{i,j} \prob(\Gamma_{i,j}^{s,t} | O_{i,j}^{s,t}),
\label{eq:theGammasAlmInd}
\end{eqnarray}
where the inequality follows from \eqref{eq:SplitTotal} and the obvious inequality
$\prob(O^{m,s,t}) \le \prod_{i,j} \prob(O_{i,j}^{s,t})$.
This gives part (i) of the lemma because
for each factor in the product of the last expression in \eqref{eq:theGammasAlmInd} we have,
by Definition \ref{def:nice}, Markov's inequality and Lemma \ref{lem:ExpSizeEpsxX},
\begin{eqnarray*}
 \prob(\Gamma_{i,j}^{s,t} | O_{i,j}^{s,t}) & = & \prob\left(\mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}')\: | \; O_{i,j}^{s,t};
\gamma_{i,j} ] \le 2 C_{10} st \pi(t) \mid O_{i,j}^{s,t}\right)\\
 & \ge & 1 - \frac{\mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}')\: | \; O_{i,j}^{s,t}]}{ 2 C_{10} st \pi(t)} \ge \frac{1}{2}.
\end{eqnarray*}

To prove part (ii) first note that
\begin{eqnarray*}
 \mathbb{E}\left[\tilde{\mathcal{C}}(A_{i,j}') | O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right] & = &
\frac{\sum_{v \in A_{i,j}'} \prob\left(v \lra \partial A_{i,j}'; O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right)}
{\prob(O^{m,s,t}; (\gamma) = (\tilde{\gamma}))}\\
 & \le & \frac{\sum_{v \in A_{i,j}'} \prob(v \lra \partial A_{i,j}'; O_{i,j}^{s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j})\prob(D)}{C_{16}\prob(D)\prob(O_{i,j}^{s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j})},
\end{eqnarray*}
where we used \eqref{eq:SplitOnxEpsWithGamma} in the denominator.
Hence
\begin{equation}\label{eq:expCAOleExpCAOij}
 \mathbb{E}\left[\tilde{\mathcal{C}}(A_{i,j}') | O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right] \le
\frac{1}{C_{16}}\mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}') | O_{i,j}^{s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j}]
\le \frac{2 C_{10}}{C_{16}} s t \pi(t),
\end{equation}
where the last inequality is just the `niceness' property (Definition \ref{def:nice}) of $(\tilde \ga)$.
%Let $\bar{\mathcal{C}}(\Lambda_{n}\setminus \Lambda_{ms})$ denote the number of vertices in $\Lambda_{n}\setminus \Lambda_{ms}$
%which are connected to $\partial \Lambda_{ms}$. Then, since $n$ is such that $n - ms \le t$,
%\begin{eqnarray}
% \mathbb{E}[\bar{\mathcal{C}}(\Lambda_{n}\setminus \Lambda_{ms})]
%& \le & \sum_{v \in \Lambda_{n}\setminus \Lambda_{ms}}\prob(v \lra \partial \Lambda_{ms})\nonumber\\
% & \le & 8n \sum_{k=0}^{t} \pi(k)\nonumber\\
% & \le & C_{15} mst \pi(t),\label{eq:expCR}
%\end{eqnarray}
%where we used Theorem \ref{lem:PiBounds} (ii) in the last inequality.
%Let
%\begin{equation*}
% C_{14} = \max{(C_{12},C_{15})}.
%\end{equation*}
%
%Let $k = \frac{6C_{14}}{C_{13}}m^{2} st \pi(t)$. Then
To finish the proof of part (ii), note that, for each $K >0$,
\begin{eqnarray*} 
 \lefteqn{\prob\left(\mathcal{C}_{0,0}(Q) \le K m^2 s t \pi(t) | O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right) \ge
1 - \frac{\mathbb{E}\left[\mathcal{C}_{0,0}(Q) | O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right]}{K m^2 s t \pi(t)}}\\ 
 & \ge & 1 - \frac{\mathbb{E}\left[\tilde{\mathcal{C}}(\Lambda_{n}\setminus \Lambda_{ms})\right] +
\sum_{i,j} \mathbb{E}\left[\tilde{\mathcal{C}}(A_{i,j}') | O^{m,s,t}; (\gamma) = (\tilde{\gamma})\right]}{K m^2 s t \pi(t)}.
\end{eqnarray*}
Applying part (iv) of Theorem \ref{lem:PiBounds} to the first expectation in the r.h.s. of the last expression,
 and \eqref{eq:expCAOleExpCAOij} to each of the other expectations gives, by choosing $K$ sufficiently
large, the desired result. This completes the proof of part (ii) of Lemma \ref{lem:probGamma}.\qed
\end{proof}
% & \stackrel{\eqref{eq:expCR}, \eqref{eq:expCAOleExpCAOij}}{\ge}
%& 1 - \frac{1}{k}\left(C_{15} m st \pi(t) + \frac{1}{C_{13}}
%\sum_{i,j} \mathbb{E}[\tilde{\mathcal{C}}(A_{i,j}') | O_{i,j}^{m,s,t}; \gamma_{i,j} = \tilde{\gamma}_{i,j}] \right)\\
% & \stackrel{\textrm{Def. }\ref{def:nice}}{\ge} & 1 - \frac{1}{k}\left(C_{15} m st \pi(t) +
%\frac{2C_{12}}{C_{13}} m^{2} st \pi(t) \right)\\
% & \ge & 1 - \frac{1}{k}\left(\frac{3C_{14}}{C_{13}}  m^{2} st \pi(t)\right) = \frac{1}{2},
%\end{eqnarray*}
%since $C_{15} \le C_{14} \le \frac{C_{14}}{C_{13}}$. This completes the proof of (ii). \qed

\subsection{Cluster-size contributions inside the circuits}
In this section we write the value $t$ (the width of the relevant annuli and `corridors' in the construction) 
as $\lfloor \varepsilon s \rfloor$. A suitable value for $\varepsilon$ (depending on the values of $a$ and $b$ in
the statement of Theorem \ref{thm:maxClusterInInt}) will be determined in the next section. The main result in the
current section concerns the contribution from the interior of a nice circuit to
the cluster of that circuit.
Recall the notation \eqref{eq:defCijGamma}.
\begin{lemma}\label{lem:sizeIntGam}
There exist constants $C_{17}, C_{18}, \delta_{2} > 0$, and for every $\varepsilon < \frac{1}{12}$ there exists
$\delta_{3}(\varepsilon) > 0$, such that for all $s \in \mathbb{N}$ and all
$(s,\lfloor \varepsilon s\rfloor)$-nice circuits $\tilde{\gamma}_{0,0}$ in $A_{0,0}^{II}$,
\begin{enumerate}[(i)]
 \item \begin{equation} \label{eq-r-leminside-i}
 \prob\left(\mathcal{C}^{\tilde{\gamma}_{0,0}} \in
(C_{17}s^2 \pi(s), C_{18}s^2 \pi(s))\; | \; \gamma_{0,0} = \tilde{\gamma}_{0,0}\right) \ge \delta_{2}.
\end{equation}
\item \begin{equation} \label{eq-r-leminside-ii}
\prob\left(\mathcal{C}^{\tilde{\gamma}_{0,0}} <
4C_{10} s\lfloor \varepsilon s\rfloor \pi(\lfloor \varepsilon s\rfloor)\; | \;  \gamma_{0,0} = \tilde{\gamma}_{0,0}\right) \ge \delta_{3}(\varepsilon).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $B_{0,0}'=B_{0,0} \setminus A_{0,0}'$.
Let $\hat{Y} = |\{ v \in B_{0,0}': v \lra \partial B_{0,0}\}|$.
Clearly,
\begin{equation}\label{eq:compWithHatY}
 \prob(\mathcal{C}^{\tilde{\gamma}_{0,0}} \ge C_{17}s^2 \pi(s) \; | \; \gamma_{0,0} = \tilde{\gamma}_{0,0})
\ge \prob(\hat{Y} \ge C_{17}s^2 \pi(s)),
\end{equation}
which (for a suitable choice of $C_{17}$)  by Theorem \ref{lem:momentBoundExpConnBoundary}
is at least a positive constant, which we write as $2 \delta_{2}$.
To complete the proof we need to find a $C_{18}>0$ such that
\begin{equation}\label{eq:probLargeClusIsSmall}
 \prob(\mathcal{C}^{\tilde{\gamma}_{0,0}} \ge C_{18}s^2 \pi(s) \; | \; \gamma_{0,0} = \tilde{\gamma}_{0,0}) \le \delta_{2}.
\end{equation}
To do this we look for an upper bound for
$\mathbb{E}[\mathcal{C}^{\tilde{\gamma}_{0,0}} \; | \; \gamma_{0,0} = \tilde{\gamma}_{0,0}]$.
We have
\begin{eqnarray} \label{eq-r-ub}
 \mathbb{E}[\mathcal{C}^{\tilde{\gamma}_{0,0}} \; | \; \gamma_{0,0} =
\tilde{\gamma}_{0,0}] & = & \mathbb{E}[\mathcal{C}^{\tilde{\gamma}_{0,0}} \; |
\; O_{0,0}^{s,\lfloor \varepsilon s\rfloor}; \gamma_{0,0} = \tilde{\gamma}_{0,0}] \\
 & \le & \mathbb{E}[\tilde{\mathcal{C}}(B_{0,0}')]
 + \mathbb{E}[\tilde{\mathcal{C}}(A_{0,0}')\; | \; O_{0,0}^{s,\lfloor \varepsilon s\rfloor};
\gamma_{0,0} = \tilde{\gamma}_{0,0}] \nonumber
\end{eqnarray}
Applying part (iv) of Theorem \ref{lem:PiBounds} to the first expectation in the last line, and the niceness property of
$\tilde{\gamma}_{0,0}$ to the other expectation, shows that
the l.h.s. of \eqref{eq-r-ub}
is at most $C_{19} s^2 \pi(s)$. Finally, Markov's inequality
gives \eqref{eq:probLargeClusIsSmall} with $C_{18} = \frac{C_{19}}{\delta_{2}}$.
This completes the proof of part (i).

Now we prove part (ii). Let $G$ be the event that there is a closed dual circuit in $A_{0,0}^{III}$.
On this event, let $\beta_{0,0}$ denote the innermost of such circuits. Observe that, conditioned on $\be_{0,0}$,
the configuration outside $\beta_{0,0}$
is independent of the configuration inside. Also observe that, on the event $G$, all vertices in the interior
of $\ga_{0,0}$ that are connected to $\ga_{0,0}$ are in $A'_{0,0}$.
By these and related simple observations we have that
\begin{eqnarray*} 
 \lefteqn{\prob\left(\mathcal{C}^{\tilde{\gamma}_{0,0}} < 4C_{10} s\lfloor \varepsilon s\rfloor \pi(\lfloor \varepsilon s\rfloor)\; |
\;  \gamma_{0,0} = \tilde{\gamma}_{0,0}; \beta_{0,0} = \tilde{\beta}\right)}\\
& = &  \prob\left(\mathcal{C}^{\tilde{\gamma}_{0,0}} <
4C_{10} s\lfloor \varepsilon s\rfloor \pi(\lfloor \varepsilon s\rfloor)\; |
\;  O_{0,0}^{s,\lfloor \varepsilon s\rfloor}; \gamma_{0,0} = \tilde{\gamma}_{0,0}; \beta_{0,0} = \tilde{\beta}\right) \\
 & \ge & \prob\left(\tilde{\mathcal{C}}(A_{0,0}') < 4C_{10} s\lfloor \varepsilon s\rfloor \pi(\lfloor \varepsilon s\rfloor)\; |
\;  O_{0,0}^{s,\lfloor \varepsilon s\rfloor}; \gamma_{0,0} = \tilde{\gamma}_{0,0}\right),
\end{eqnarray*}
which, by Markov's inequality and because $\tilde\ga_{0,0}$ is nice, is at least $1/2$.
Hence, the l.h.s. of \eqref{eq-r-leminside-ii} is at least $(1/2) \prob(G)$, which by RSW is larger than
some positive constant which depends only on $\varepsilon$.

\qed
\end{proof}

%where we used the definition of $(s,t)$-nice in the second inequality.
%For the expected number of vertices in $\Lambda_{r}$ connected to the boundary of the box we have
%\begin{eqnarray}
% \mathbb{E}[\tilde{\mathcal{C}}(\Lambda_{r})] & = & \sum_{v \in \Lambda_{r}} \prob(v \lra \partial \Lambda_{r}) \nonumber\\
% & \le & \sum_{v \in \Lambda_{r}} \pi(d(v, \partial \Lambda_{r})) \nonumber\\
% & \le & 8r\sum_{k=0}^{r} \pi(k) \le C_{18}r^{2}\pi(r),\label{eq:expVertConnBound}
%\end{eqnarray}
%where we used Theorem \ref{lem:PiBounds} (ii) in the third inequality. Combining \eqref{eq:upperBoundInGamma} and \eqref{eq:expVertConnBound} gives
%\begin{eqnarray*}
%\lefteqn{\mathbb{E}[\mathcal{C}_{0,0}^{\tilde{\gamma}_{0,0}}(B_{0,0}) \; | \; \gamma_{0,0} = \tilde{\gamma}_{0,0}]} \\
% & \le & C_{18}(s-3t)^2 \pi(s-3t) + C_{19} s^2 \pi(s) \\
% & \le & C_{20}s^2 \pi(s) + C_{19} s^2 \pi(s),
%\end{eqnarray*}
%where we used the inequalities
%\begin{eqnarray*}
% 2C_{12} s\lfloor \varepsilon s\rfloor \pi(\lfloor \varepsilon s\rfloor) & \le & C_{19} s^2 \pi(s); \\
% C_{18}(s-3\lfloor \varepsilon s\rfloor)^2 \pi(s-3\lfloor \varepsilon s\rfloor) & \le & C_{20}s^2 \pi(s),
%\end{eqnarray*}
%which both easily follow from Theorem \ref{lem:PiBounds} (i). Notice that both $C_{19}$ and $C_{20}$ doesn't
%depend on $s$ and $\varepsilon$. Hence, if $C_{17}>\frac{C_{19}+C_{20}}{\delta_{4}}$, \eqref{eq:probLargeClusIsSmall} holds. \qed
%
%\begin{lemma}\label{lem:chooseNxEps} Let $0 < a < b < \infty$. There exists $x>0, \varepsilon \in (0,\frac{1}{12})$ and $N \in \mathbb{N}$
% such that $\frac{1}{x}, \frac{1}{\varepsilon} \in \mathbb{N}$ and for all $n \ge N$,
%\begin{eqnarray}
% C_{16}(\xn)^2 \pi(\xn)\cdot (\frac{1}{x})^{2} & \ge & an^{2}\pi(n); \label{eq:xCondI}\\
% C_{17}(\xn)^{2} \pi(\xn) & \le & \frac{1}{3}(b-a)n^{2}\pi(n); \label{eq:xCondII} \\
% \frac{6C_{14}}{C_{13}} (\frac{1}{x})^{2} \exn\xn\pi(\exn) & \le & \frac{1}{3}(b-a)n^{2}\pi(n).\label{eq:EpsCond}
%\end{eqnarray}
%\end{lemma}
%Note that $4C_{12} \le 4C_{14} \le \frac{6C_{14}}{C_{13}}$.
%\noindent\proof
%Let $0 < a < b$ be given.
%It is easy to see (a weak form of the lower bound in part (i) of Theorem \ref{lem:PiBounds} suffices)
%that if $x$ is sufficiently
%(depending on $a$) small, then
%\eqref{eq:xCondI} holds for all sufficiently large $n$. 
%It also  easily follows (now from the upper bound in the same Lemma) that if $x$ is sufficiently (depending on
%$b-a$) small, \eqref{eq:xCondII} holds for all sufficiently large $n$. 
%Finally, for $x$ fixed, it follows (again from the upper bound in part (i) of Theorem \ref{lem:PiBounds}) that
%if $\ep$ is sufficiently (depending on $b-a$ and $x$) small, then \eqref{eq:EpsCond} holds for all sufficiently large $n$. \\
%{\bf Remark:}
%{\em Note that for this last step it is crucial that the exponent ($1/2$) in the Lemma is strictly smaller than $1$.} \\
%This completes the proof of the Lemma. \qed

%\subsection{If you have a sequence of independent r.v.'s you can get a nice total sum.}
%We will use the following simple general lemma (or `observation'), versions of which have without doubt been
%used in the literature in various contexts.
%\begin{lemma}\label{lem:sumOfRandomVars} Let $0 < \al < \be$,
%and let $k \in \N$ be such that $\al/k < (\be - \al)/2$. 
%Further, let $(X_{i})_{1 \leq i \le k}$ be independent random variables,
%(not necesarrily identically distributed) which satisfy the following:
%\begin{eqnarray*}
% \prob\left(X_{i} \in (\frac{\al}{k}, \frac{\be-\al}{2}\right) & \ge & \eta_{1};\\
% \prob\left(X_{i} \le \frac{\be-\al}{2k} \right) & \ge & \eta_{2}.
%\end{eqnarray*}
%Then
%\begin{equation*}
% \prob\left(\sum_{i=1}^{k} X_{i} \in (\al, \be)\right) \ge (\eta_{1}\eta_{2})^{k}.
%\end{equation*}
%\end{lemma}
%\proof 
%For $1 \leq i \leq k$ we say that `step $i$ is proper' if
%$$
%X_i  \begin{cases} \in (\frac{\al}{k}, \frac{\be - \al}{2}) & \mbox{ if } \sum_{j=1}^{i-1} X_j < \al \\
%\leq \frac{\be - \al}{2 k}, & \mbox{ otherwise } \end{cases}
%$$  
%It is clear that if all steps $i =1, \cdots, k$ are proper, then $\sum_{i=1}^k X_i \in (\al,\be)$.
%It is also easy to see that, for each $i$, the 
%conditional probability that step $i$ is proper, given that all steps $1, \cdots, i-1$ are proper,
%is at least $\min(\eta_1, \eta_2$).
%\qed



\subsection{Completion of the proof of Theorem \ref{thm:maxClusterInInt}} \label{sect-compl}
We are now ready to prove Theorem \ref{thm:maxClusterInInt}. First we still restrict to the case $p = 1/2$.
Let $0 < a < b$ be given.
See the brief outline in Section \ref{sec:proofDefi}.
The lengths of the building blocks $B_{i,j}$ and the widths of the annuli and `corridors'
in the partition of $\La_n$, will be taken proportional to $n$, say (roughly) $x n$ and $\ep x n$ respectively,
with suitably chosen $x$ and $\ep$. For this purpose we will use the following lemma:

\begin{lemma}\label{lem:chooseNxEps}
%Let $0 < a < b < \infty$.
There exist $x>0, \varepsilon \in (0,\frac{1}{12})$ and
$N \in \mathbb{N}$,
with $\frac{1}{x}$ an odd integer, such that for all $n \ge N$ the following inequalities hold:
\begin{eqnarray}
 C_{17}\xn^2 \pi(\xn)\cdot (\frac{1}{x})^{2} & \ge & an^{2}\pi(n); \label{eq:xCondI}\\
 C_{18}\xn^{2} \pi(\xn) & \le & \frac{1}{3}(b-a)n^{2}\pi(n); \label{eq:xCondII} \\
 (4C_{10} \vee C_{15}) (\frac{1}{x})^{2} \exn\xn\pi(\exn) & \le & \frac{1}{3}(b-a)n^{2}\pi(n).\label{eq:EpsCond}
\end{eqnarray}
\end{lemma}
%Note that $4C_{12} \le 4C_{14} \le \frac{6C_{14}}{C_{13}}$. 
\begin{proof}
It is easy to see (a weak form of the lower bound in part (i) of Theorem \ref{lem:PiBounds} suffices)
that if $x$ is sufficiently
(depending on $a$) small, then
\eqref{eq:xCondI} holds for all sufficiently large $n$.
It also  easily follows (now from the upper bound in the same Theorem) that if $x$ is sufficiently (depending on
$b-a$) small, \eqref{eq:xCondII} holds for all sufficiently large $n$.
Finally, for $x$ fixed, it follows (again from the upper bound in part (i) of Theorem \ref{lem:PiBounds}) that
if $\ep$ is sufficiently (depending on $b-a$ and $x$) small, then \eqref{eq:EpsCond} holds for all sufficiently large $n$. \\
{\em Note that for this last step it is essential that the exponent ($1/2$) in part (i) of
Theorem \ref{lem:PiBounds} is strictly smaller than $1$.} \\
This completes the proof of Lemma \ref{lem:chooseNxEps}. \qed
\end{proof}

\medskip
Now let $x$, $\ep$ and $N$ be as in Lemma \ref{lem:chooseNxEps}. Moreover we assume
(which we may, because we can enlarge $N$ if necessary) that
$n - \frac{1}{x} \lfloor x n \rfloor \leq \lfloor \ep \lfloor x n \rfloor \rfloor$ for all $n \ge N$.
%Let $N_{2} \ge \frac{1 + \varepsilon + \frac{1}{x}}{\varepsilon x}$,
%so that for $n \ge N_{2}$: $n - ms \le t$. Then take $N =$ max$(N_{1},N_{2})$.
Denote by $D_{n}$ the event
\begin{equation*}
 D_{n} = \{ \exists v \in \Lambda_{n}: \mathcal{C}_{n}(v) \in (an^2 \pi(n), bn^2 \pi(n)) \}.
\end{equation*}
%We want to prove that there is a $\de >0$ such that $\prob(D_{n})>\delta$ for all $n \ge N$.
Let $n \ge N$ and let $m = \frac{1}{x}, s = \xn$ and $t = \exn$.
By straightforward RSW and FKG arguments, there is a $\delta_{4}(x,\varepsilon)>0$ such that
$\prob(O^{m,s,t}) > \delta_{4}(x,\varepsilon)$.
Hence
\begin{equation}\label{eq:r-end1}
\prob(D_{n}) \ge \delta_{4}(x,\varepsilon)\prob(D_{n} | O^{m,s,t}).
\end{equation}
From Lemma \ref{lem:probGamma} (i) it follows that
\begin{equation} \label{eq:r-end2}
\prob(D_{n} | O^{m,s,t}) \ge (C_{14})^{\frac{1}{x^{2}}}\prob(D_{n} | O^{m,s,t}; \Gamma^{m,s,t}).
\end{equation}
The next step is conditioning on the widest open circuits.
\begin{equation} \label{eq:final-added}
 \prob(D_{n} | O^{m,s,t}; \Gamma^{m,s,t}) = \sum_{(\tilde{\gamma}): (m,s,t)\textrm{-nice}}
\prob(D_{n} | (\gamma) = (\tilde{\gamma}); O^{m,s,t}) \, \prob((\gamma) = (\tilde{\gamma}) | O^{m,s,t}; \Gamma^{m,s,t}).
\end{equation}
For each $(\tilde{\gamma})$ we denote by $\mathcal{C}_{(\tilde{\gamma})}^{in}$ the number of all vertices that are in the
interior of a circuit in the collection $(\tilde{\gamma})$ and connected to that circuit,
and by $\mathcal{C}_{(\tilde{\gamma})}^{out}$ the number of vertices outside these circuits
that are connected to one or more of these circuits, plus the number of vertices on these circuits. We have
\begin{eqnarray} \label{eq:r-end3}
\lefteqn{\prob(D_{n} | (\gamma) = (\tilde{\gamma}); O^{m,s,t})} \\ \nonumber
 & \ge & \prob\left(\frac{\mathcal{C}_{(\tilde{\gamma})}^{in}}{n^{2}\pi(n)} \in (a,b - \frac{1}{3}(b-a)) \; | \; \frac{\mathcal{C}_{(\tilde{\gamma})}^{out}}{n^{2}\pi(n)} \le \frac{1}{3}(b-a); (\gamma) = (\tilde{\gamma}); O^{m,s,t}\right) \\ \nonumber
 & & \cdot \prob\left(\mathcal{C}_{(\tilde{\gamma})}^{out} \le \frac{1}{3}(b-a)n^{2}\pi(n) \; | \; (\gamma) = (\tilde{\gamma});
O^{m,s,t}\right) \\ \nonumber
% & \stackrel{\eqref{eq:EpsCond}}{\ge} &  \prob\left(\frac{\mathcal{C}_{(\tilde{\gamma})}^{in}}{n^{2}\pi(n)}
%\in (a,b - \frac{1}{3}(b-a)) \; | \; \frac{\mathcal{C}_{\tilde{\gamma}}^{out}}{n^{2}\pi(n)} \le \frac{1}{3}(b-a);
%\gamma = \tilde{\gamma}; O^{m,s,t}\right) \\ \nonumber
% & & \cdot \prob\left(\mathcal{C}_{O}(Q) \le \frac{6C_{14}}{C_{13}}m^{2}  st \pi(t) | O^{m,s,t};\gamma =
%\tilde{\gamma}\right) \\ \nonumber
% & \stackrel{\textrm{Lem.}\ref{lem:probGamma}(ii)}{\ge} & \frac{1}{2}
%\prob\left(\frac{\mathcal{C}_{\tilde{\gamma}}^{in}}{n^{2}\pi(n)} \in (a,\frac{1}{3} a + \frac{2}{3} b) \; |
%\; \frac{\mathcal{C}_{\tilde{\gamma}}^{out}}{n^{2}\pi(n)} \le
%\frac{1}{3}(b-a); \gamma = \tilde{\gamma}; O^{m,s,t}\right) \\ \nonumber
 & \geq & \frac{1}{2} \prob\left(\frac{\mathcal{C}_{(\tilde{\gamma})}^{in}}{n^{2}\pi(n)} \in
(a,\frac{1}{3} a + \frac{2}{3} b)\right),
\end{eqnarray}
where the last inequality holds by \eqref{eq:EpsCond} and Lemma \ref{lem:probGamma} (ii), and because the
configurations in the interiors of the $\tilde{\gamma}_{i,j}$'s 
are obviously independent of  the event conditioned on in the expression in the r.h.s. of the first inequality.
%$\{\mathcal{C}_{\tilde{\gamma}}^{out} \le \frac{1}{3}(b-a)n^{2}\pi(n) \}$ and
%$O^{m,s,t}$.
%We write $\mathbb{P}^{\tilde{\gamma}}$ to denote the conditional probability, given $\{\gamma_{i,j} = \tilde{\gamma}_{i,j}\}$.
Note that $\mathcal{C}_{(\tilde{\gamma})}^{in} = \sum_{i,j} \mathcal{C}^{\tilde{\gamma}_{i,j}}$.
The $\mathcal{C}^{\tilde{\gamma}_{i,j}}$'s are independent
and for each $i,j$ we have
\begin{eqnarray*}
 \lefteqn{\prob\left(\mathcal{C}^{\tilde{\gamma}_{i,j}} \in (ax^{2} n^{2}\pi(n), \frac{1}{3}(b-a)n^{2}\pi(n))\right)}\\
 & \stackrel{\eqref{eq:xCondI}, \eqref{eq:xCondII}}{\ge} & \prob\left(\mathcal{C}^{\tilde{\gamma}_{i,j}} \in
\left(C_{17}\xn^{2}\pi(\xn), C_{18}\xn^{2}\pi(\xn)\right)\right) \stackrel{\textrm{Lem.} \ref{lem:sizeIntGam} (i)}{\ge}
\delta_{2},
\end{eqnarray*}
and
\begin{eqnarray*}
 \lefteqn{\prob\left(\mathcal{C}^{\tilde{\gamma}_{i,j}} \le x^{2}\frac{1}{3}(b-a)n^{2}\pi(n)\right)} \\
 & \stackrel{\eqref{eq:EpsCond}}{\ge} & \prob\left(\mathcal{C}^{\tilde{\gamma}_{i,j}}
\le 4C_{10}\exn\xn\pi(\exn)\right) \stackrel{\textrm{Lem.} \ref{lem:sizeIntGam} (ii)}{\ge} \delta_{3}(\varepsilon).
\end{eqnarray*}
%by Lemma \ref{lem:chooseNxEps} and Lemma \ref{lem:sizeIntGam}.
Hence
the conditions of Lemma \ref{lem:sumOfRandomVars} (with $k = (1/x)^2$, $\al = a n^2 \pi(n)$, $\be = (\frac{1}{3} a + 
\frac{2}{3} b) n^2 \pi(n)$) are satisfied. Hence, by that lemma the l.h.s. of \eqref{eq:r-end3} is at least
$\frac{1}{2} (\delta_{2} \wedge \delta_{3}(\ep))^{(1/x)^2}$. Together with \eqref{eq:r-end1} - \eqref{eq:final-added}
this shows that 
\begin{equation}\label{eq:bound-Dn}
\P(D_n) > \de_5,
\end{equation}
with $\de_5$ a positive constant which depends only on $a$ and $b$.
%the probability that there {\em exists} an open cluster with size $\in (a n^2 \pi(n), b n^2 \pi(n))$ 
%is bounded away from $0$.

Now we will show that, by the way we `constructed' the open cluster, a similar result holds for the
{\em maximal} open cluster in $\La_n$.
First note that the `constructed' cluster has the property that it contains an open horizontal and
an open vertical crossing of the box $\La_{m s}$. Also note that there is at most one open cluster
with this property. Given the exact location of the (unique) open cluster with this property, the conditional 
probability that it is the maximal cluster in $\La_n$ is, if its size is larger than $a n^2 \pi(n)$, clearly
larger than or equal to the probability that the remaining part of $\La_n$ contains no open cluster of size
larger than $a n^2 \pi(n)$. By obvious monotonicity this probability is at least $\P(\mathcal{M}_n \leq a n^2 \pi(n))$,
which by Theorem \ref{thm:maxClustCanBeSmall} is at least some positive constant $\de_2$ (which depends only on $a$).
This argument gives
$$\P\left(\mathcal{M}_n \in (a n^2 \pi(n), b n^2 \pi(n))\right) \geq \de_2 \de_5,$$
which completes the proof of Theorem \ref{thm:maxClusterInInt} for $p = 1/2$.

Now let, more generally, $p$ be such that $L(p) \geq n$. It is straightforward to check that (due to the `uniformity'
in $p$ of the results in Section \ref{sect-notation}) each step in the proof
remains essentially valid. For instance, it is easy to see from the arguments used that Lemma \ref{lem:ExpSizeEpsxX} 
(now with $\P$ replaced by $\P_p$) remains valid as long as $L(p) \geq s$. Since we take $s \leq n$ (application of)
this lemma (and, similarly, the other lemma's) can be carried out as before.
This completes the proof of Theorem \ref{thm:maxClusterInInt}. \qed

\medskip\noindent
{\bf \large Acknowledgment.} The first author thanks Antal J\'arai for a useful and pleasant discussion on these an related
problems.

%thus completing the proof of Theorem \ref{thm:maxClusterInInt}:
%completes the proof of Theorem  \ref{thm:existClustInInt} for $p = 1/2$.  

%\medskip
%If $p \neq \frac{1}{2}$ we replace everywhere $\prob, \mathbb{E}$ by $\prob_{p}, \mathbb{E}_{p}$.
%Then constants which come from RSW
%arguments have to be replaced by the corresponding constants depending on $\epsilon$.\\
%ROB:VOLGENDE NOG NAGAAN/POLISHEN:
%In the proof of Lemma \ref{lem:ExpSizeEpsxX}
%replace the $\pi$ by $\pi_{p}$ until \eqref{eq:comAlPrime}. There on the second line
%$\pi_{p}(t) \le C_{5}\pi(t)$ and $\pi_{p}(k) \le C_{5}\pi(k)$, using Theorem \ref{lem:PiBounds} (iii).
%In the proof of Lemma \ref{lem:probGamma} in \eqref{eq:expCR}: $\pi_{p}(k) \le C_{5}\pi(k)$.
%The same for \eqref{eq:expVertConnBound} in the proof of Lemma \ref{lem:sizeIntGam}.
%This completes the proof for all $p$.\qed

%Recall that $E$ denotes the set of edges in the square lattice. Denote by $E_{n}$ the
%edges in $\La_n \cap E$. Denote for any connected set $F \subset E_{n}$, by $Cl(F)$ the event that $F$ is an open cluster. 
%(So $Cl(F)$ is the event that all edges in $F$ are open and all edges not in $F$ with at least one endvertex in $F$
%are closed). By the constructions we used, we have in fact already proved that
%there exist $x \in (0,1), \delta_{1}>0$ and $N \in \mathbb{N}$ such that, for all
%$p \in (0,1), n \in \{N, \cdots, L(p)\}$:
%\begin{equation}\label{eq:clustWithCrosInInt}
% \prob_{p}\left(\exists v \in \Lambda_{n}:
%v \lra l,r,t,b(\Lambda_{\frac{1}{x}\xn}), \frac{\mathcal{C}_{n}(v)}{n^{2}\pi(n)} \in (a, b)\right) > \delta_{1},
%\end{equation}
%where $v \lra l,r,t,b(\Lambda_{m})$ means that $v$ is connected to the lefthand side, righthand side, top
%and bottom of the box $\Lambda_{m}$ inside the box $\Lambda_{m}$.
%Notice that, for $m \le n$, there can be at most one open cluster in $\Lambda_{n}$ such that
%$v \lra l,r,t,b(\Lambda_{m})$ for all $v$ in this cluster.
%For any connected subset $F \subset E_{n}$ with $|F| \in (an^{2}\pi(n),bn^{2}\pi(n))$, where $|F|$ denotes the
%number of vertices in the cluster $F$, we have
%\begin{eqnarray}
% \prob_{p}\left( \frac{\mathcal{M}_{n}}{n^{2}\pi(n)} \in (a,b) \; | \; Cl(F) \right)
%& \geq & \prob_{p}\left( \frac{\mathcal{M}(E_{n} \setminus \bar{F})}{n^{2}\pi(n)} < a\right)\nonumber\\
% & \ge & \prob_{p}\left( \frac{\mathcal{M}_{n}}{n^{2}\pi(n)} < a \right) \ge \delta_{2},\label{eq:maxClustCondOnF}
%\end{eqnarray}
%where the existence of $\delta_{2} > 0$ follows from Theorem \ref{thm:maxClustCanBeSmall},
%$\mathcal{M}(F)$ denotes the size of the maximal open cluster in the subgraph $F$ and $\bar{F}$ denotes the union of the edges
%in $F$ with the edges which have one endvertex in common with an edge in $F$.
%
%So conditioning on the cluster $F$ which contains vertex $v \in F \cap \Lambda_{\frac{1}{x}\xn}$ with
%$v \lra l,r,t,b(\Lambda_{\frac{1}{x}\xn})$ and combining \eqref{eq:clustWithCrosInInt} and \eqref{eq:maxClustCondOnF}
%gives the result.\qed

%\bibliographystyle{amsplain}
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\bibitem[BCKS99]{BCKS99} C. Borgs, J.T. Chayes, H. Kesten and J. Spencer,
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\bibitem[BCKS01]{BCKS01} C. Borgs, J.T. Chayes, H. Kesten and J. Spencer,
The birth of the infinite cluster: finite-size scaling in percolation, 
{\it Comm. Math. Phys.} {\bf 224}, 153--204 (2001).

\bibitem[BK85]{BK85} J. van den Berg and H. Kesten, Inequalities with applications
to percolation and reliability, {\it J. Appl. Probab.} {\bf 22}, 556--569 (1985).

\bibitem[BLN12]{BLN12} J. van den Berg, B. de Lima and P. Nolin,
A percolation process on the square lattice where large finite clusters
are frozen, {\it Random Structures \& Algorithms} {\bf 40}, 220--226 (2012).

\bibitem[G99]{G99} G. Grimmett, Percolation. 2nd edition. {\it Springer} (1999).

\bibitem[J03]{J03} A. J{\'a}rai, Incipient infinite percolation clusters in 2$D$, 
{\it Ann. Probab.} {\bf 31}, 444--485 (2003).

\bibitem[K86]{K86} H. Kesten, The incipient infinite cluster in two-dimensional percolation,
{\it Probab. Theory Rel. Fields} {\bf 73} (1986).

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\end{thebibliography}