text:
\begin{document}

\maketitle

\begin{abstract}
  We prove a long-standing conjecture on random-cluster models, namely
  that the critical point for such models with parameter $q\geq1$ on the
  square lattice is equal to the self-dual point $p_{sd}(q) = \sqrt{q} /
  (1+\sqrt{q})$. This gives a proof that the critical
  temperature of the $q$-state Potts model is equal to $\log (1+\sqrt
  q)$ for all $q\geq 2$. We further prove that the transition is sharp,
  meaning that there is exponential decay of correlations in the
  sub-critical phase. The techniques of this paper are rigorous and
  valid for all $q\geq 1$, in contrast to earlier methods valid only for
  certain given $q$. The proof extends to the triangular and the
  hexagonal lattices as well.
\end{abstract}

\section*{Introduction}

Since random-cluster models were introduced by Fortuin and Kasteleyn in
1969~\cite{FortuinKasteleyn}, they have become an important tool in the
study of phase transitions. The spin correlations of Potts models are
rephrased as cluster connectivity properties of their random-cluster
representations. This allows the use of geometric techniques, thus
leading to several important applications. Nevertheless, only few
aspects of the random-cluster models are understood in full generality.

The \emph{random-cluster model} on a finite connected graph is a model
on the edges of this graph, each one being either closed or open. The
probability of a configuration is proportional to $$p^{\# \;\text{open
edges}} (1-p)^{\# \; \text{closed edges}} q^{\# \; \text{clusters}},$$
where the \emph{edge-weight} $p\in[0,1]$ and the \emph{cluster-weight}
$q\in(0,\infty)$ are the parameters of the model. For $q\geq 1$, this
model can be extended to infinite-volume lattices where it exhibits a
phase transition at some critical parameter $p_c(q)$ (depending on the
lattice). There are no general conjectures for the value of the critical
point.

However, in the case of planar graphs, there is a connection (related to
the Kramers-Wannier duality for the Ising model \cite{KramersWannier})
between random-cluster models on a graph and on its dual with the same
cluster-weight $q$ and appropriately related edge-weights $p$ and
$p^\star=p^\star(p)$. This relation leads in the particular case of
$\mathbb Z^2$ (which is isomorphic to its dual) to a natural conjecture:
the critical point is the same as the so-called \emph{self-dual point}
satisfying $p_{sd}=p^\star(p_{sd})$, which has a known value
$$p_{sd}(q) = \frac{\sqrt{q}}{1+\sqrt{q}}.$$

\bigskip

In the present article, we prove this conjecture for all $q\geq 1$:
\begin{theorem}\label{critical_value}
  Let $q\geq 1$. The critical point $p_c=p_c(q)$ for the random-cluster
  model with cluster-weight $q$ on the square lattice satisfies $$p_c =
  \frac{\sqrt{q}}{1+\sqrt{q}}.$$
\end{theorem}

A rigorous derivation of the critical point was previously known in
three cases. For $q=1$, the model is simply bond percolation, proved by
Kesten in 1980~\cite{Kesten} to be critical at $p_c(1)=1/2$. For $q=2$,
the self-dual value corresponds to the critical temperature of the Ising
model, as first derived by Onsager in 1944~\cite{Onsager}; one can
actually couple realizations of the Ising and FK models to relate their
critical points, see \cite{Grimmett} and references therein for details.
For modern proofs in that case, see \cite{AizenmanBarskyFernandez} or
the short proof of \cite{BeffaraDuminilSmirnov}. Finally, for
sufficiently large $q$, a proof is known based on the fact that the
random-cluster model exhibits a first order phase transition
(see~\cite{LaanaitMessagerMiracleSoleRuizShlosman, LaanaitMessagerRuiz},
the proofs are valid for $q$ larger than $25.72$). We mention that
physicists derived the critical temperature for the Potts models with
$q\geq4$ in 1978, using non-geometric arguments based on analytic
properties of the Hamiltonian~\cite{HintermannKunzWu}.

In the sub-critical phase, we prove that the probability for two points
$x$ and $y$ to be connected by a path decays exponentially fast with
respect to the distance between $x$ and $y$. In the super-critical
phase, the same behaviour holds in the dual model. This phenomenon is
known as a \emph{sharp phase transition}:
\begin{theorem}\label{exponential_decay}
  Let $q\geq1$. For any $p<p_c(q)$, there exist $0<C(p,q),c(p,q)<\infty$
  such that for any $x,y\in \Z^2$, \begin{equation}
    \ph(x\leftrightarrow y)\leq C(p,q)\e^{-c(p,q)|x-y|},
  \end{equation}
  where $|\cdot|$ denotes the Euclidean norm.
\end{theorem}

\bigskip

The proof involves two main ingredients. The first one is an estimate on
crossing probabilities at the self-dual point $p=\sqrt{q}/(1+\sqrt{q})$:
the probability of crossing a rectangle with aspect ratio $(1,\alpha)$
--- meaning that the ratio between the height and the width is of order
$\alpha$ --- in the vertical direction is bounded away from $0$ and $1$
uniformly in the size of the box.  This result is the main new
contribution of this paper. It is a generalization of the celebrated
Russo-Seymour-Welsh theorem for percolation.

The second ingredient is a collection of sharp threshold theorems, which
were originally introduced for product measures. They have been used in
many contexts, and are a powerful tool for the study of phase
transitions, see Bollob\'as and Riordan
\cite{BollobasRiordan,BollobasRiordanperco}. These theorems were later
extended to positively associated measures by Graham and Grimmett
\cite{GrahamGrimmett,Grimmett}. In our case, they may be used to show
that the probability of crossings goes to $1$ when
$p>\sqrt{q}/(1+\sqrt{q})$.

Actually, the situation is slightly more complicated than usual: the
dependence inherent in the model makes boundary conditions difficult to
handle, so that new arguments are needed. More precisely, one can use a
classic sharp threshold argument for symmetric increasing events in
order to deduce that the crossing probabilities of larger and larger
domains, under \emph{wired boundary condition}, converge to $1$ whenever
$p>\sqrt{q}/(1+\sqrt{q})$.  Moreover, the theorem provides us with
bounds on the speed of convergence for rectangles with \emph{wired
boundary condition}. A new way of
combining long paths allows us to create an infinite cluster. We
emphasize that the classic construction, used by Kesten \cite{Kesten}
for instance, does not work in our case.

The approach allows the determination of the critical value, but it
provides us with a rather weak estimate on the speed of convergence for
crossing probabilities. Nevertheless, combining the fact that the
crossing probabilities go to $0$ when $p<p_{sd}$ with a very general
threshold theorem, we deduce that the cluster-size at the origin has
finite moments of any order. It is then an easy step to derive the
exponential decay of the two-point function.

\bigskip

Theorem~\ref{critical_value} has several notable consequences. First, it
extends up to the critical point results that are known for the
subcritical random-cluster models under the exponential decay condition
(for instance, Ornstein-Zernike estimates \cite{CampaninoIoffeVelenik}
or strong mixing properties). Second, it identifies the critical value
of the Potts models via the classical coupling between random-cluster
models with cluster-weight $q\in \mathbb{N}$ and the $q$-state Potts
models:
\begin{theorem}
  Let $q\geq2$ be an integer; consider the $q$-state Potts model on
  $\Z^2$, defined by the Hamiltonian $$H_q(c) := - \sum_{(xy)\in E}
  \delta_{c_x,c_y}$$ (where $c_x\in\{1,\ldots,q\}$ is the color at site
  $x$ and $E$ the set of edges of the lattice). The model exhibits a
  phase transition at the critical inverse temperature $$\beta_c(q)=\log
  (1+\sqrt q).$$
\end{theorem}

\bigskip

The methods of this paper harness symmetries of the graph, together with
the self-dual property of the square lattice. In the case of the
hexagonal and triangular lattice, the symmetries of the graphs, the
duality property between the hexagonal and the triangular lattices and
the star-triangle relation allow to extend the crossing estimate proved
in Section~\ref{sec:crossings} subject to certain additional technical
difficulties. The rest of the proof can be carried over to the
triangular and the hexagonal lattices as well, yielding the following
result:
\begin{theorem}\label{triangular}
  The critical value $p_c=p_c(q)$ for the random-cluster model with
  cluster-weight $q\geq 1$ satisfies
  $$\begin{array}{ll}
    y_c^3+3y_c^2-q=0  & \text{on the triangular lattice and}\\
    y_c^3-3qy_c-q^2=0 & \text{on the hexagonal lattice,}
  \end{array}$$
  where $y_c=p_c/(1-p_c)$. Moreover, there is exponential decay in the
  sub-critical phase.
\end{theorem}

\bigskip

There are many unanswered questions related to
Theorem~\ref{critical_value}. First, the behaviour at criticality is not
well understood in the general case, and it seems that new techniques
are needed. It is conjectured that the random-cluster model undergoes a
second-order phase transition for $q\in(0,4)$ (it is further believed
that the scaling limit is then conformally invariant), and a first-order
phase transition when $q\in(4,\infty)$. Proving the above for every $q$
remains a major open problem.
 We mention that the random-cluster models with parameter $q=2$
 \cite{Smirnov} or $q$ very large
 \cite{LaanaitMessagerMiracleSoleRuizShlosman, LaanaitMessagerRuiz} are
 much better understood.  Second, the technology developed in the
 present article relies heavily on the positive association property of
 the random-cluster measures with $q\geq 1$. Our strategy does not
 extend to random-cluster models with $q<1$. Understanding these models
 is a challenging open question.

\bigskip

The paper is organized as follows. In Section~\ref{sec:basic} we review
some basic features of random-cluster models.
Section~\ref{sec:crossings} is devoted to the statement and the proof of
the crossing estimates. In Section~\ref{sec:sharp_threshold}, we briefly
present the theory of sharp threshold that we will employ in the next
section.  Section~\ref{sec:proofs} contains the proofs of
Theorems~\ref{critical_value} and~\ref{exponential_decay}.
Section~\ref{sec:triangular} is devoted to extensions to other lattice
and contains the proof of Theorem~\ref{triangular}.

\section{Basic features of the model}
\label{sec:basic}

We start with an introduction to the basic features of random-cluster
models; more detail and proofs can be found in Grimmett's
book~\cite{Grimmett}.

\paragraph{Definition of the random-cluster model.}

The random-cluster measure can be defined on any graph. However, we will
restrict ourselves to the square lattice (of mesh size $1$), or more
precisely a rotated version by an angle $\pi/4$, see
Figure~\ref{fig:lattice_dual}. We denote this lattice by $\mathbb{L} =
(\mathbb{V},\mathbb{E})$, with $\mathbb{V}$ denoting the set of
\emph{sites} and $\mathbb{E}$ the set of \emph{edges}. In this paper,
$G$ will always denote a connected subgraph of $\mathbb{L}$, \ie\  a
subset of vertices of $\mathbb{V}$ together with all the edges between
them. We denote by $\partial G$ the boundary of $G$, \ie\  the set of
sites of $G$ linked by an edge of $\mathbb{E}$ to a site of
$\mathbb{V}\setminus G$.

A \emph{configuration} $\omega$ on $G$ is a subgraph of $G$, composed of
the same sites and a subset of its edges. We will call the edges
belonging to $\omega$ \emph{open}, the others \emph{closed}. Two sites
$a$ and $b$ are said to be \emph{connected} if there is an \emph{open
path}, \ie\ a path composed of open edges only, connecting them (this
event will be denoted by $a\leftrightarrow b$). Two sets $A$ and $B$ are
\emph{connected} if there exists an open path connecting them (denoted
$A\leftrightarrow B$). The maximal connected components will be called
\emph{clusters}. We will often simply use the term \emph{path} for
\emph{open path} when there is no possible ambiguity.

A \emph{boundary condition} $\xi$ is a partition of $\partial G$. We
denote by $\omega \cup \xi$ the graph obtained from the configuration
$\omega$ by identifying (or \emph{wiring}) the edges in $\xi$ that
belong to the same component of $\xi$. Boundary conditions should be
understood as encoding how sites are connected outside $G$. Let
$o(\omega)$ (resp. $c(\omega)$) denote the number of open (resp.\
closed) edges of $\omega$ and $k(\omega,\xi)$ the number of connected
components of $\omega\cup\xi$.  The probability measure
$\phi^{\xi}_{p,q,G}$ of the random-cluster model on $G$ with parameters
$p$ and $q$ and boundary condition $\xi$ is defined by
\begin{equation}
  \label{probconf}
  \phi_{p,q,G}^{\xi} (\left\{\omega\right\}) :=
  \frac {p^{o(\omega)}(1-p)^{c(\omega)}q^{k(\omega,\xi)}}
  {Z_{p,q,G}^{\xi}}
\end{equation}
for every configuration $\omega$ on $G$, where $Z_{p,q,G}^{\xi}$ is a
normalizing constant referred to as the \emph{partition function}.

Let $t<x$ and $y<z$; we will identify the rectangle $[t,x)\times[y,z)$
with the set of vertices in $\mathbb{V}$ that lie within it. The graph
with vertex set $[t,x)\times[y,z)$ together with the induced subset of
$\mathbb{E}$ is called a \emph{rectangle of \,$\mathbb{L}$}.

\begin{figure}[ht!]
    \begin{center}
     \includegraphics[width=\hsize]{lattice+dual}
    \end{center}
    \caption{\textbf{Left:} Example of a configuration on the rotated
    lattice.  \textbf{Right:} A configuration together with its dual
    configuration.}
    \label{fig:lattice_dual}
  \end{figure}

\paragraph{The finite energy property.}

This is a very simple property of random-cluster models. Let
$\epsilon\in(0,1/2)$. The conditional probability for an edge to be
open, knowing the states of all the other edges, is bounded away from
$0$ and $1$ uniformly in $p\in(\epsilon,1-\epsilon)$ and in the
configuration away from this edge. This property extends to any finite
family of edges.

\paragraph{The domain Markov property.}

One can encode, using appropriate boundary condition $\xi$, the
influence of the configuration outside a sub-graph on the measure within
it.  Consider a graph $G=(V,E)$ and a random-cluster measure
$\phi^{\psi}_{p,q,G}$ on it.  For $F\subset E$, consider $G'$ with $F$
as the set of edges and the endpoints of it as the set of sites. Then,
the restriction to $G'$ of \smash{$\phi^{\psi}_{p,q,G}$} conditioned to
match some configuration $\omega$ outside $G'$ is exactly
\smash{$\phi_{p,q,G'}^{\xi}$}, where $\xi$ describes the connections
inherited from $\omega\cup \psi$ (two sites are wired if they are
connected by a path in $\omega\cup\psi$ outside the box).

\paragraph{The FKG inequality and comparison between boundary
conditions.}

An event is called \emph{increasing} if it is preserved by addition of
open edges, see~\cite{Grimmett}. Random-cluster models with parameter
$q\geq 1$ are \emph{positively correlated}. This property has two
important consequences,  the first one being the
\emph{Fortuin-Kasteleyn-Ginibre inequality}:
\begin{equation}\label{FKG}
  \phi_{p,q,G}^{\xi}(A\cap B)\geq
  \phi_{p,q,G}^{\xi}(A)\phi_{p,q,G}^{\xi}(B),
\end{equation}
which holds for every pair of increasing events $A$ and $B$ and any
boundary conditions $\xi$. This correlation inequality is extremely
important in the study of random-cluster models.

The second property is a \emph{comparison between boundary conditions}:
for any boundary conditions $\psi\leq\xi$, meaning that sites wired in
$\psi$ are wired in $\xi$, we have
\begin{equation}\label{comparison_between_boundary_conditions}
  \phi^{\psi}_{p,q,G}(A)\leq \phi^{\xi}_{p,q,G}(A)
\end{equation}
for any increasing event $A$. We say that $\phi_{p,q,G}^\xi$
\emph{stochastically dominates} $\phi_{p,q,G}^\psi$. Combined with the
domain Markov property, the comparison between boundary conditions
allows to estimate conditional probabilities.

\paragraph{Examples of boundary conditions: free, wired and periodic.}

Two boundary conditions play a special role in the study of the
random-cluster model. The \emph{wired} boundary condition, denoted by
$\phi_{p,q,G}^1$, is specified by the fact that all the vertices on the
boundary are pairwise wired (only one set in the partition). The
\emph{free} boundary condition, denoted by $\phi_{p,q,G}^0$, is
specified by no wiring between sites. These boundary conditions are
extremal for the stochastic ordering, since any boundary condition has
less (resp.\ more) wired sites than in the wired (resp.\ free) boundary
condition.

We will also consider periodic boundary conditions: for $n\geq1$, the
torus of size $n$ can be seen as the box $[0,n)^2$ with the boundary
condition $(i,0)$ is wired to $(i,n)$ for every $i\in[0,n]$ and $(0,j)$
is connected to $(n,j)$ for every $j\in[0,n]$.  We will denote the
random-cluster measure on the torus of size $n$ by
$\phi_{p,q,[0,n]^2}^{\rm p}$ or more concisely $\phi_{p,q,n}^{\rm p}$.
Note that the torus is naturally embedded into the plane.

\paragraph{Dual graph and planar duality.}

In two dimensions, one can associate with any random-cluster model a
dual model. Let $G$ be a finite graph embedded in a surface.  Define the
dual graph $G^{\star}=(V^{\star},E^{\star})$ as follows: place a dual
site at the centres of the faces of $G$ (the external face, when
considering a graph on the plane, must be counted as a face of the
graph), and for every bond $e\in E$, place a dual bond between the two
sites corresponding to faces bordering $e$. Given a subgraph
configuration $\omega$, construct a bond model on $G^\star$ by declaring
any bond of the dual graph to be open (resp.\ closed) if the
corresponding bond of the primal lattice is closed (resp.\ open) for the
initial configuration. The new configuration is called the \emph{dual
configuration} of $\omega$.

When defining the dual model properly, one must be careful about
boundary conditions (it will be crucial in this article). We first
present an easy case: consider the random-cluster measure with
parameters $(p,q)$ on the torus of size $n$ (in other words, with
periodic boundary condition). The dual model on the dual graph (which is
also a torus of size $n$) given by the dual configurations is then a
random-cluster measure with the same parameter $q$ and a dual parameter
$p^\star = p^\star(p,q)$ satisfying $$p^\star(p,q) :=
\frac{(1-p)q}{(1-p)q+p}, \; \text{or equivalently} \; \frac{p^\star
p}{(1-p^\star)(1-p)}=q.$$ In other words, the dual measure
$(\phi_{p,q,n}^{\rm p})^{\star}$ of $\phi_{p,q,n}^{\rm p}$ is
$\phi_{p^{\star},q,n}^{\rm p}$.  This relation is an instance of
\emph{planar duality}. It is then natural to define the self-dual point
$p_{sd}=p_{sd}(q)$ by solving the equation $p^\star(p_{sd},q)=p_{sd}$,
thus obtaining
\begin{equation}
  p_{sd}(q)=\frac{\sqrt{q}}{1+\sqrt{q}}.
\end{equation}

\paragraph{Infinite-volume measures and the definition of the critical
point.}

The domain Markov property and comparison between boundary conditions
allow us to define infinite-volume measures. Indeed, one can consider a
sequence of measures on boxes of increasing size with free boundary
condition. This sequence is increasing in the sense of stochastic
domination, which implies that it converges weakly to a limiting
measure, called the random-cluster measure on $\mathbb{L}$ with free
boundary condition (denoted by $\ph^0$). This classic construction can
be performed with many other sequences of measures, defining several
\emph{a priori} different infinite-volume measures on $\mathbb{L}$. For
instance, one can define the random-cluster measure $\ph^1$ with wired
boundary condition, by considering the decreasing sequence of
random-cluster measures on finite boxes with wired boundary condition.

For given $q\geq 1$, it is shown that uniqueness can fail only for $p$
in a countable set $\mathcal{D}_q$, see Theorem (4.60) of
\cite{Grimmett}.  Therefore, there exists a \emph{critical point} $p_c$
such that for any infinite-volume measure with $p<p_c$ (resp.\ $p>p_c$),
there is almost surely no infinite component of connected sites (resp.\
at least one infinite component).

\begin{remark}\label{conjecture_uniqueness}
  It is natural to conjecture that the critical point satisfies
  $p_c=p_{sd}$. Indeed, if one assumes $p_c\neq p_{sd}$, there would be
  two phase transitions, one at $p_c$, due to the change of behaviour in
  the primal model, and one at $p_c^\star$, due to the change of
  behaviour in the dual model.
\end{remark}

\paragraph{The inequality $p_c\geq p_{sd}$.}

As in the case of percolation, a lower bound for the critical value can
be derived using the uniqueness of the infinite cluster above the
critical point.  Indeed, if one assumes that $p_c<p_{sd}$, the
configuration at $p_{sd}$ must contain one infinite open cluster and one
infinite dual open cluster (since the random-cluster model in the dual
is then super-critical as well). Intuition indicates that such
coexistence would imply that there is more than one infinite open
cluster; an elegant argument (due to Zhang in the case of percolation)
formalizes this idea.  We refer to the exposition in Theorem (6.17)
of~\cite{Grimmett} for full detail, but still give a sketch of the
argument.

The proof goes as follows, see Figure~\ref{fig:Zhang}. Assume that
$p_c<p_{sd}$ and consider the random-cluster model with $p=p_{sd}$.
There is an infinite open cluster, and therefore, we can choose a large
box such that the infinite open cluster and the dual infinite open
cluster touch the boundary with probability greater than $1-\ep$.  The
FKG inequality (through the so-called ``square-root trick'') implies
that the infinite open cluster actually touches the top side of the box,
using only edges outside the box, with probability greater than
$1-\ep^{1/4}$. We deduce that with probability at least $1-2\ep^{1/4}$,
the infinite open cluster touches both the top and bottom sides, using
only edges outside of the box.

A similar argument implies that the infinite dual open cluster touches
both the left- and right-hand sides of the box with probability at least
$1-2\ep^{1/4}$.  Therefore, with probability at least $1-4\ep^{1/4}$,
the complement of the box contains an infinite open path touching the
top of the box, one touching the bottom, and infinite dual open paths
touching each of the vertical edges.  Enforcing edges in the box to be
closed, which brings only a positive multiplicative factor due to the
finite energy property of the model, and choosing $\ep$ sufficiently
small, we deduce that there are two infinite open clusters with positive
probability.  Since the infinite open cluster must be unique, this is a
contradiction, which implies that $p_c\geq p_{sd}$.

\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=1.0\hsize]{zhang} \end{center}
    \caption{Steps of the (sketchy) argument $p_c\geq p_{sd}$.}
    \label{fig:Zhang}
  \end{figure}
\bigskip

When $p<p_{sd}\leq p_c$, there is no infinite cluster for any
infinite-volume measure. General arguments imply uniqueness of the
infinite-volume measure whenever $p\neq p_{sd}$ and $q\geq 1$ (see
Theorem~(6.17) of \cite{Grimmett}). This fact will be useful in the
sequel since, \emph{except at criticality}, we do not have to specify
which infinite-volume measure is under consideration. We will denote the
unique infinite-volume measure by $\phi_{p,q}$ when $p\neq p_{sd}$.

\section{Crossing probabilities for rectangles at the self-dual point}
\label{sec:crossings}

In this section, we prove crossing estimates for rectangles of
prescribed aspect ratio. This is an extension of the Russo-Seymour-Welsh
theory for percolation. We will work with $p=p_{sd}(q)$ and the measures
$\phi_{p_{sd},q}^1$ and $\phi_{p_{sd},q,n}^{\rm p}$; we present the
proof in the periodic case. The case of the (bulk) wired boundary
condition can be derived from this case (see Corollary~\ref{RSW_bulk}).

For a rectangle $R$, let $\C_v(R)$ denote the event that there exists a
path between the top and the bottom sides which stays inside the
rectangle. Such a path is called a \emph{vertical (open) crossing} of
the rectangle. Similarly, we define $\C_h$ to be the event that there
exists an \emph{horizontal open crossing} between the left and the right
sides.  Finally, $\C_v^\star(R^\star)$ denotes the event that there
exists a dual-open crossing from top to bottom in the dual.

The following theorem states that, at the self-dual point, the
probability of crossing a rectangle horizontally is bounded away from
$0$ uniformly in the sizes of both the rectangle and the torus provided
that the aspect ratio of the rectangles remains constant. The size of
the ambient torus is denoted by $m$. Note that $p=p^\star$ when
$p=p_{sd}$, and hence the random-cluster measure on the torus is
self-dual.

\begin{theorem}\label{RSW_torus}
  Let $\al>1$ and $q\geq 1$. There exists $c(\al)>0$ such that for every
  $m>\alpha n>0$,
  \begin{equation}
    \phi_{p_{sd},q,m}^{\rm p}\big(\C_h([0,\al n)\times[0,n))\big)\geq
    c(\al).
  \end{equation}
\end{theorem}

We begin the proof with a lemma, which corresponds to the existence of
$c(1)$ and is based on the self-duality of random-cluster measures on
the torus.  This lemma is classic and is the natural starting point for
any attempt to prove RSW-like estimates.

\begin{lemma}\label{square_crossings}
  For every $m>n\geq 1$, $\displaystyle \phi_{p_{sd},q,m}^{\rm
  p}\big(\C_h([0,n)^2)\big)=\frac{1}{2}$.
\end{lemma}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=\hsize]{square_symmetry}
  \end{center}
  \caption{\textbf{Left:} The square $[0,n)^2$ and its dual have the
  same graph structure.  \textbf{Right:} The events $\C_h([0,n)^2)$ and
  $\C_v^\star([0,n)^2)$.}
  \label{fig:square_symmetry}
\end{figure}

\begin{proof}
  Note that the dual of $[0,n)^2$ is $[0,n)^2$ (meaning the sites of the
  dual torus inside $[0,n)^2$), see Figure~\ref{fig:square_symmetry}. If
  there is no open crossing from left to right in $[0,n)^2$, there
  exists necessarily a dual-open crossing from top to bottom in the dual
  configuration. Hence, the complement of $\C_h([0,n)^2)$ is
  $\C_v^\star([0,n)^2)$, thus yielding  \begin{equation*}
    \phi_{p_{sd},q,m}^{\rm
    p}\big(\C_h([0,n)^2)\big)+\phi_{p_{sd},q,m}^{\rm
    p}\big(\C_v^\star([0,n)^2)\big) = 1.
  \end{equation*}
  By self-duality and symmetry of the lattice, the probability
  $\phi_{p_{sd},q,m}^{\rm p}(\C_v^\star([0,n)^2))$ is equal to the
  probability of $\phi_{p_{sd},q,m}^{\rm p}(\C_h([0,n)^2))$, giving
  \begin{equation*}
    1=2\phi_{p_{sd},q,m}^{\rm p}\big(\C_h([0,n)^2)\big),
  \end{equation*}
  which concludes the proof.
\end{proof}

The only major difficulty is to prove that rectangles of aspect ratio
$\al$ are crossed in the horizontal direction --- with probability
uniformly bounded away from $0$ --- for \emph{some} $\al>1$. There are
many ways to prove this in the case of percolation. Nevertheless, they
always involve independence in a crucial way; in our case, independence
fails, so we need a new argument. The main idea is to invoke
self-duality in order to enforce the existence of crossings, even in the
case where boundary conditions could look disadvantageous. In order to
do that, we introduce the following family of domains, which are in some
sense nice symmetric domains.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\hsize]{symmetric_domain}
  \end{center}
  \caption{Two paths $\g_1$ and $\g_2$ satisfying Hypothesis $(\star)$
  and the graph $G(\g_1,\g_2)$.}
  \label{fig:symmetric_domain}
\end{figure}

Define the line $d:=-\sqrt{2}/4+{\rm i}\R$. The orthogonal symmetry
$\sigma_d$ with respect to this line maps $\mathbb{L}$ to
$\mathbb{L}^\star$. Let $\g_1$ and $\g_2$ be two paths satisfying the
following Hypothesis $(\star)$, see Figure~\ref{fig:symmetric_domain}:
\begin{itemize}
  \item $\g_1$ remains on the left of $d$ and $\g_2$ remains on the
    right;
  \item $\g_2$ begins at $0$ and $\g_1$ begins on a site of
    $\mathbb{L}\cap (-\sqrt 2/ 2+{\rm i}\mathbb{R}_+)$;
  \item $\g_1$ and $\sigma_d(\g_2)$ do not intersect (as curves in the
    plane);
  \item $\g_1$ and $\sigma_d(\g_2)$ end at two sites (one primal and one
    dual) which are at distance $\sqrt2/2$ from each other.
\end{itemize}
The definition extends trivially via translation, so we will say that
the pair $(\g_1,\g_2)$ satisfies Hypothesis $(\star)$ if one of its
translates does.

When following the paths in counter-clockwise order, we can create a
circuit by linking the end points of $\g_1$ and $\sigma_d(\g_2)$ by a
straight line, the start points of $\sigma_d(\g_2)$ and $\g_2$, the end
points of $\g_2$ and $\sigma_d(\g_1)$, and the start points of
$\sigma_d(\g_1)$ and $\g_1$. The circuit $(\g_1,\sigma_d(\g_2), \g_2,
\sigma_d(\g_1))$ surrounds a set of vertices of $\mathbb{L}$. Define the
graph $G(\g_1,\g_2)$ composed of sites of $\mathbb{L}$ that are
surrounded by the circuit $(\g_1,\sigma_d(\g_2),\g_2,\sigma_d(\g_1))$,
and of edges of $\mathbb{L}$ that remain entirely inside the circuit
(boundary included).

The \emph{mixed boundary condition} on this graph is wired on $\g_1$
(all the edges are pairwise connected), wired on $\g_2$, and free
elsewhere.  We denote the measure on $G(\g_1,\g_2)$ with parameters
$(p_{sd},q)$ and mixed boundary condition by $\phi_{p_{sd},q,\g_1,\g_2}$
or more simply $\phi_{\g_1,\g_2}$.

\begin{lemma}\label{self-duality}
  For any pair $(\g_1,\g_2)$ satisfying Hypothesis $(\star)$, the
  following estimate holds: $$\phi_{\g_1,\g_2}(\g_1\leftrightarrow
  \g_2)\geq\frac{1}{1+q^2}.$$
\end{lemma}

\begin{proof}
  On the one hand, if $\g_1$ and $\g_2$ are not connected,
  $\sigma_d(\g_1)$ and $\sigma_d(\g_2)$ must be connected by a dual path
  in the dual model (event corresponding to
  $\sigma_d(\g_1)\leftrightarrow\sigma_d(\g_2)$ in the dual model).
  Hence,
  \begin{equation}\label{self-duality1}
    1= \phi_{\g_1,\g_2} (\g_1\leftrightarrow\g_2) +
    \phi^\star_{\g_1,\g_2} (\sigma_d(\g_1)\leftrightarrow\sigma_d(\g_2))
    = \phi_{\g_1,\g_2} (\g_1\leftrightarrow\g_2) +
    \sigma_d*\phi^\star_{\g_1,\g_2} (\g_1\leftrightarrow\g_2),
  \end{equation}
  where $\sigma_d*(\phi_{\g_1,\g_2}^\star)$ denotes the image under
  $\sigma_d$ of the dual measure of $\phi_{\g_1,\g_2}$. This
  measure lies on $G(\g_1,\g_2)$ as well and has parameters
  $(p_{sd},q)$.

  When looking at the dual measure of a random-cluster model, the
  boundary condition is transposed into a new boundary condition for the
  dual measure. In the case of the periodic boundary condition, we
  obtained the same boundary condition for the dual measure. Here, the
  boundary condition becomes wired on $\g_1\cup \g_2$ and free elsewhere
  (this is easy to check using Euler's formula).

  It is very important to notice that the boundary condition is
  \emph{not} exactly the mixed one, since $\g_1$ and $\g_2$ are wired
  \emph{together}.  Nevertheless, the Radon-Nikodym derivative of
  $\sigma_d*\phi_{\g_1,\g_2}^\star$ with respect to $\phi_{\g_1,\g_2}$
  is easy to bound. Indeed, for any configuration $\omega$, the number
  of cluster can differ only by $1$ when counted in
  $\sigma_d*\phi_{\g_1,\g_2}^\star$ or $\phi_{\g_1,\g_2}$ so that the
  ratio of partition functions belongs to $[1/q,q]$. Therefore, the
  ratio of probabilities of the configuration $\omega$ remains between
  $1/q^2$ and $q^2$. This estimate extends to events by summing over all
  configurations. Therefore, $$\sigma_d*\phi_{\g_1,\g_2}^\star
  (\g_1\leftrightarrow\g_2) \leq q^2 \phi_{\g_1,\g_2}
  (\g_1\leftrightarrow\g_2).$$ When plugging this inequality into
  \eqref{self-duality1}, we obtain $$\phi_{\g_1,\g_2}
  (\g_1\leftrightarrow\g_2) + q^2 \phi_{\g_1,\g_2}
  (\g_1\leftrightarrow\g_2) \geq 1$$ which implies the claim.
\end{proof}

We are now in a position to prove the key result of this section.

\begin{proposition}\label{box_crossing}
  For all $m>3n/2>0$, the following holds: $$\phi_{p_{sd},q,m}^{\rm p}
  \big[ \C^v\big([0,n)\times[0,3/2n)\big) \big] \geq
  \frac{1}{16(1+q^2)}.$$
\end{proposition}

Before proving this proposition, we show how it implies
Theorem~\ref{RSW_torus}. The strategy is straightforward and classic: we
combine crossings together, using only the FKG inequality.

\begin{proof}[Proof of Theorem~\ref{RSW_torus}]
  If $\al<3/2$, Proposition~\ref{box_crossing} implies the claim so we
  can assume $\al>3/2$. Define the following rectangles, see
  Figure~\ref{fig:a}: $$R^h_j = [j n/2,jn/2+3n/2) \times [0,n) \quad
  \text{and} \quad R^v_j=[jn/2,j n/2+n)\times[0,n)$$ for
  $j\in[0,\lfloor2\alpha\rfloor-1]$, where $\lfloor x\rfloor$ denotes
  the integer part of $x$. If every rectangle $R^h_j$ is crossed
  horizontally, and every rectangle $R^v_j$ is crossed vertically, then
  $[0,\alpha n)\times[0,n)$ is crossed horizontally. We denote this
  event by $B$.  The rectangle $R^h_j$ is crossed horizontally with
  probability greater than $[16(1+q^2)]^{-1}$
  (Proposition~\ref{box_crossing}), the rectangle $R^v_j$ is crossed
  vertically with probability $1/2$ (Lemma~\ref{square_crossings}) and
  so, using the FKG inequality, \begin{equation*}
    \phi_{p_{sd},q,m}^{\rm p}\big(\C_h([0,\al n)\times[0,n))\big)\geq
    \phi_{p_{sd},q,m}^{\rm p}(B)\geq
    \left(\frac{1}{32(1+q^2)}\right)^{\lfloor2\alpha\rfloor}.
  \end{equation*}
The claim follows with $c(\alpha) := [32(1+q^2)]
  ^{-\lfloor2\alpha\rfloor}$.
\end{proof}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.5\hsize]{combo}\quad \quad \quad
    \includegraphics[width=0.3\hsize]{event_A}
  \end{center}
  \caption{\textbf{Left:} A combination of crossings in smaller
  rectangles creating a horizontal crossing of a very long rectangle.
  \textbf{Right:} The rectangles $R$, $R_1$ and $R_2$ and the event
  $A$.}
  \label{fig:a}
\end{figure}

\begin{proof}[Proof of Proposition~\ref{box_crossing}]
  The proof goes as follows: we start by creating two paths crossing
  square boxes, and we then prove that they are connected with good
  probability.

  \paragraph{Setting of the proof.}

  Consider the rectangle $R=[0,3n/2)\times[0,n)$ which is the union of
  the rectangles $R_1=[0,n)\times[0,n)$ and $R_2=[n/2,3n/2)\times[0,n)$,
  see Figure~\ref{fig:a}. Let $A$ be the event defined by the following
  conditions:
  \begin{itemize}
    \item $R_1$ and $R_2$ are both crossed horizontally (these events
      have probability at least $1/2$ to occur, using
      Lemma~\ref{square_crossings});
    \item $[n/2,n)\times\{0\}$ is connected inside $R_2$ to the top side
      of $R_2$ (which has probability greater than 1/4 to occur using
      symmetry and Lemma~\ref{square_crossings}).
  \end{itemize}
  Employing the FKG inequality, we deduce that:
  \begin{equation}\label{abc}
    \phi_{p_{sd},q,m}^{\rm p}(A)\geq \frac{1}{16}.
  \end{equation}
  When $A$ occurs, define $\Ga_1$ to be the top-most horizontal crossing
  of $R_1$, and $\Ga_2$ the right-most vertical crossing of $R_2$ from
  $[n/2,n)\times\{0\}$ to the top side. Note that this path is
  automatically connected to the right-hand side of $R_2$ --- which is
  the same as the right-most side of $R$. If $\Ga_1$ and $\Ga_2$ are
  connected, then there exists a horizontal crossing of $R$. In the
  following, we show that $\Ga_1$ and $\Ga_2$ are connected with good
  probability.

  \paragraph{Exploration of the paths $\Ga_1$ and $\Ga_2$.}

  There is a standard way of exploring $R$ in order to discover $\Ga_1$
  and $\Ga_2$. Start an exploration from the top-left corner of $R$ that
  leaves open edges on its right, closed edges on its left and remains
  in $R_1$. If $A$ occurs, this exploration will touch the right-hand
  side of $R_1$ before its bottom side; stop it the first time it does.
  Note that the exploration process ``slides'' between open edges of the
  primal lattice and dual open edges of the dual (formally, this
  exploration process is defined on the medial lattice, see \emph{e.g.}
  \cite{BeffaraDuminilSmirnov}). The open edges that are adjacent to the
  exploration form the top-most horizontal crossing of $R_1$, \ie\
  $\Ga_1$. At the end of the exploration, the process has \emph{a
  priori} discovered a set of edges which lies above $\Ga_1$, so that
  the remaining part of $R_1$ is undiscovered.

  By starting an exploration at point $(n,0)$, leaving open edges on its
  left and closed edges on its right, we can explore the rectangle
  $R_2$. If $A$ holds, the exploration ends on the top side of $R_2$.
  The open edges adjacent to the exploration constitute the path $\Ga_2$
  and the set of edges already discovered lies ``to the right'' of
  $\Gamma_2$.

  \begin{figure}[ht!]
    \begin{center}
      \includegraphics[width=0.8\hsize]{rectangle_R}
    \end{center}
    \caption{The light grey area is the part of $R$ that is \emph{a
    priori} discovered by the exploration processes (note that this area
    can be much smaller). The dark grey is the domain
    $G_0(\tilde{\g}_1,\tilde{\g}_2)$. We have depicted all the paths
    involved in the construction. Note that dashed curves are ``virtual
    paths''  of the dual lattice obtained by the reflection $\sigma_d$:
    they are not necessarily dual open.}
    \label{fig:rectangle_R}
  \end{figure}

  \paragraph{The reflection argument.}

  Assume first that we know $\Ga_1=\g_1$ and $\Ga_2=\g_2$ and that they
  do not intersect. Let $x$ be the end-point of $\g_1$, \ie\ its unique
  point on the right-hand side of $R_1$. We want to define a set
  $G_0(\g_1,\g_2)$ similar to those considered in
  Lemma~\ref{self-duality}. Apply the following ``surgical procedure,''
  see Figure~\ref{fig:rectangle_R}:
  \begin{itemize}
    \item First, define the symmetric paths $\sigma_d(\g_1)$ and
      $\sigma_d(\g_2)$ of $\g_1$ and $\g_2$ with respect to the line
      $d:=(n-\sqrt{2}/4)+{\rm i}\R$;
    \item Then, parametrize the path $\sigma_d(\g_1)$ by the distance
      (along the path) to its starting point $\sigma_d(x)$ and define
      $\tilde{\g}_1\subset \g_1$ so that $\sigma_d(\tilde{\g}_1)$ is the
      part of $\sigma_d(\g_1)$ between the start of the path and the
      first time it intersects $\g_2$. As before, the paths are
      considered as curves of the plane; we denote $z$ the intersection
      point of the two curves. Note that $\g_1$ and $\g_2$ are not
      intersecting, which enforces $\sigma_d(\g_1)$ and $\g_2$ to be;
    \item From this, parametrize the path $\g_2$ by the distance to its
      starting point $(n,0)$ and set $y$ to be the last visited site in
      $\mathbb{L}$ before the intersection $z$. Define $\tilde{\g}_2$ to
      be the part of $\g_2$ between the last point intersecting $n+{\rm
      i}\R$ before $y$ and $y$ itself;
    \item Paths $\tilde{\g}_1$ and $\tilde{\g}_2$ satisfy Hypothesis
      $(\star)$ so that the graph $G(\tilde{\g}_1,\tilde{\g}_2)$ can be
      defined;
    \item We construct a sub-graph $G_0(\g_1,\g_2)$ of
      $G(\tilde{\g}_1,\tilde{\g}_2)$ as follows: the edges are given by
      the edges of $\mathbb{L}$ included in the connected component of
      $G(\tilde{\g}_1,\tilde{\g}_2)\setminus [\g_2\cup\g_1]$ containing
      $d$ (it is the connected component which contains $x-\varepsilon
      {\rm i}$), and the sites are given by their endpoints.
  \end{itemize}

  The graph $G_0(\g_1,\g_2)$ has a very useful property: none of its
  edges has been discovered by the previous exploration paths.  Indeed,
  $\sigma_d(\g_1)$ and $\sigma_d(x)$ are included in the unexplored
  connected component of $R\setminus R_1$, and so does
  $G_0(\g_1,\g_2)\cap (R\setminus R_1)$. Edges of $G_0(\g_1,\g_2)$ in
  $R_1$ are in the same connected component of $R\setminus (\g_1\cup
  \g_2)$ as $x-\epsilon {\rm i}$, and thus lie `below' $\g_1$.

  \paragraph{Conditional probability estimate.}

  Still assuming that $\g_1$ and $\g_2$ do not intersect, we would like
  to estimate the probability of $\g_1$ and $\g_2$ being connected by a
  path knowing that $\Ga_1=\g_1$ and $\Ga_2=\g_2$.  Following the
  exploration procedure described above, we can discover $\g_1$ and
  $\g_2$ without touching any edge in the interior of $G_0(\g_1,\g_2)$.
  Therefore, the process in the domain is a random-cluster model with
  specific boundary condition.

  The boundary of $G_0(\g_1,\g_2)$ can be split into several sub-arcs of
  various types (see Figure~\ref{fig:rectangle_R}): some are sub-arcs of
  $\g_1$ or $\g_2$, while the others are (adjacent to) sub-arcs of their
  symmetric images $\sigma_d(\g_1)$ and $\sigma_d(\g_2)$. The
  conditioning on $\Ga_1=\g_1$ and $\Ga_2=\g_2$ ensures that the edges
  along the sub-arcs of the first type are open; the connections along
  the others depend on the exact explored configuration in a much more
  intricate way, but in any case the boundary condition imposed on the
  configuration inside $G(\tilde\g_1, \tilde\g_2)$ dominates the mixed
  boundary condition --- notice that \emph{any} boundary condition
  dominates the free one and that $\tilde{\g}_1$ and $\tilde{\g}_2$ are
  two sub-arcs of the first type (they are then wired).

  From these observations, we deduce that for any increasing event $B$
  \emph{depending only on edges in $G_0(\g_1,\g_2)$},
  \begin{equation}
    \phi^{\rm p}_{p_{sd},q,m}(B|\Ga_1=\g_1,\Ga_2=\g_2) \geq
    \phi_{\tilde{\g}_1,\tilde{\g}_2}(B).
  \end{equation}
  In particular, we can apply this inequality to $\{\g_1\leftrightarrow
  \g_2\text{ in }G_0(\g_1,\g_2)\}$. Note that if $\tilde{\g}_1$ and
  $\tilde{\g}_2$ are connected in $G(\tilde{\g}_1,\tilde{\g}_2)$, $\g_1$
  and $\g_2$ are connected in $G_0(\tilde{\g_1},\tilde{\g}_2)$. The
  first event is of $\phi_{\tilde{\g}_1,\tilde{\g}_2}$-probability
  $1/(1+q^2)$, implying
  \begin{align}
    \phi^{\rm p}_{p_{sd},q,m}
    (\g_1\leftrightarrow\g_2|\Ga_1=\g_1,\Ga_2=\g_2) &\geq
    \phi_{\tilde{\g}_1,\tilde{\g}_2}
    (\g_1\leftrightarrow\g_2\text{~in~}G_0(\g_1,\g_2)) \nonumber \\
    & \geq  \phi_{\tilde{\g}_1,\tilde{\g}_2}(\tilde{\g}_1\leftrightarrow
    \tilde{\g}_2)\geq\frac1{1+q^2}.\label{estimate_duality}
  \end{align}

  \paragraph{Conclusion of the proof.}

  Note the following obvious fact: if $\g_1$ and $\g_2$ intersect, the
  conditional probability that $\Ga_1$ and $\Ga_2$ intersect, knowing
  $\Ga_1=\g_1$ and $\Ga_2=\g_2$ is equal to $1$ --- in particular, it is
  greater than $1/(1+q^2)$. Now,
  \begin{align*}
    \phi^{\rm p}_{p_{sd},q,m}(\C_h(R))&\geq  \phi^{\rm
    p}_{p_{sd},q,m}(\C_h(R)\cap A)\\
    &\geq\phi^{\rm p}_{p_{sd},q,m}(\{\Ga_1\leftrightarrow\Ga_2\}\cap
    A)\\
    &=\phi^{\rm p}_{p_{sd},q,m}\big( \phi^{\rm
    p}_{p_{sd},q,m}(\Ga_1\leftrightarrow\Ga_2|\Ga_1,\Ga_2)\mathbbm{1}_{A}\big)\\
    &\geq \frac{1}{1+q^2}\phi(A)\geq \frac{1}{16(1+q)^2}
  \end{align*}
  where the first two inequalities are due to inclusion of events, the
  third one to the definition of conditional expectation, and the fourth
  and fifth ones, to \eqref{estimate_duality} and \eqref{abc}.
\end{proof}

An equivalent of Theorem~\ref{RSW_torus} holds in the case of the
infinite-volume random-cluster measure with wired boundary condition.

\begin{corollary}\label{RSW_bulk}
  Let $\al>1$ and $q\geq 1$; there exists $c(\al)>0$ such that for every
  $n\geq1$,
  \begin{equation}
    \phi_{p_{sd},q}^1\big[\C_h\big([0,\al n)\times[0,n)\big)\big]\geq
    c(\al).
  \end{equation}
\end{corollary}

\begin{proof}
  Let $\alpha>1$ and $m>2\alpha n>0$. Using the invariance under
  translations of $\phi^{\rm p}_{p_{sd},q,m}$ and comparison between
  boundary conditions, we have $$\phi^{1}_{p_{sd},q,[-\frac m2,\frac
  m2)^2}\big[\C_h\big([0,\al n)\times[0,n)\big)\big]\geq \phi^{\rm
  p}_{p_{sd},q,m}\big[\C_h\big([0,\al n)\times[0,n)\big)\big]\geq
  c(\alpha).$$ When $m$ goes to infinity, the left hand side converges
  to the probability in infinite volume, so that
  $$\phi^{1}_{p_{sd},q}\big[\C_h\big([0,\al n)\times[0,n)\big)\big]\geq
  c(\alpha).$$
\end{proof}

\begin{remark}
  The only place where we use the periodic and the (bulk) wired boundary
  conditions is in the estimate of Lemma~\ref{box_crossing}. For
  instance, if one could prove that the probability for a square box to
  be crossed from top to bottom with free boundary conditions stays
  bounded away from $0$ when $n$ goes to infinity, then an equivalent of
  Theorem~\ref{RSW_torus} would follow with free boundary conditions.

  Uniform estimates with respect to boundary conditions should be true
  for $q\in[1,4)$; we expect the random-cluster model to be conformally
  invariant in the scaling limit. It should be false for $q\geq4$.
  Indeed, for $q>4$, the phase transition is (conjecturally) of first
  order in the sense that there should not be uniqueness of the
  infinite-volume measure. At $q=4$, the random-cluster model should be
  conformally invariant, but the probability of a crossing with free
  boundary condition should converge to $0$. Nevertheless, the
  probability that there is an open circuit surrounding the box of size
  $n$ in the box of size $2n$ with free boundary condition should stay
  bounded away from $0$.

  Proving an equivalent of Theorem~\ref{RSW_torus} with uniform
  estimates with respect to boundary conditions is an important
  question, since it allows to study the critical phase. The special
  case $q=2$ has been derived recently in \cite{DuminilHonglerNolin}.
\end{remark}

\section{A sharp threshold theorem for crossing probabilities}
\label{sec:sharp_threshold}

The aim of this section is to understand the behaviour of the function
$p\mapsto \phi_{p,q,n}^\xi(A)$ for an increasing event $A$. This
increasing function is equal to $0$ at $p=0$ and to $1$ at $p=1$, and we
are interested in the range of $p$ for which its value is between $\ep$
and $1-\ep$ for some positive $\ep$ (this range is usually referred to
as a \emph{window}). Under mild conditions on $A$, the window will be
narrow for large graphs, and its width can be bounded above in terms of
the size of the underlying graph, which is known as a \emph{sharp
threshold} behaviour.

Historically, the general theory of sharp thresholds was first developed
by Kahn, Kalai and Linial~\cite{KahnKalaiLinial} (see also
\cite{Friedgut,FriedgutKalai,KalaiSafra}) in the case of product
measures. In lattice models such as percolation, these results are used
via a differential equality known as Russo's formula,
see~\cite{GrimmettPerco,Russo}. Both sharp threshold theory and Russo's
formula were later extended to random-cluster measures with $q\geq1$,
see references below. These arguments being not totally standard, we
remind the readers of the classic results we will employ and refer them
to \cite{Grimmett} for general results. Except for
Theorem~\ref{maximum_influence}, the proofs are quite short so that it
is natural to include them. The proofs are directly extracted from the
Grimmett's monograph \cite{Grimmett}.

\bigskip

In the whole section, $G$ will denote a finite graph; if $e$ is an edge
of $G$, let $J_e$ be the random variable equal to $1$ if the edge $e$ is
open, and $0$ otherwise. We start with an example of differential
inequality, which will be useful in the proof of
Theorem~\ref{exponential_decay}.

\begin{proposition}[see \cite{Grimmett,GrimmettPiza}]
  \label{russo_hamming_proposition}
  Let $q\geq 1$; for any random-cluster measure $\phi_{p,q,G}^\xi$ with
  $p\in(0,1)$ and any increasing event $A$, $$\frac{d}{dp}
  \phi_{p,q,G}^\xi(A)\geq 4\phi_{p,q,G}^\xi(A)\phi_{p,q,G}^\xi(H_A),$$
  where $H_A(\omega)$ is the Hamming distance between $\omega$ and $A$.
\end{proposition}

\begin{proof}
  Let $A$ be an increasing event. The key step is the following
  inequality, see \cite{BezuidenhoutGrimmettKesten,Grimmett}, which can
  be obtained by differentiating with respect to $p$ (for details of the
  computation, see Theorem (2.43) of \cite{Grimmett}):
  \begin{equation}\label{russo}
    \frac{d}{dp}\phi_{p,q,G}^\xi(A)=\frac{1}{p(1-p)}\sum_{e\in E} \left[
    \phi_{p,q,G}^\xi (\mathbbm{1}_AJ_e) - \phi_{p,q,G}^\xi (J_e)
    \phi_{p,q,G}^\xi(A) \right].
  \end{equation}
  A similar differential formula is actually true for any random
  variable $X$, but we will not use this fact in the proof. Define
  $|\eta|$ to be the number of open edges in the configuration, it is
  simply the sum of the random variables $J_e$, $e\in E$. With this
  notation, one can rewrite \eqref{russo} as
  \begin{align*}
    \frac{d}{dp} \phi_{p,q,G}^\xi(A) &= \frac{1}{p(1-p)} \left[
    \phi_{p,q,G}^\xi (|\eta|\mathbbm{1}_A) - \phi_{p,q,G}^\xi (|\eta|)
    \phi_{p,q,G}^\xi(A) \right] \\
    &= \frac{1}{p(1-p)} \left[ \phi_{p,q,G}^\xi \big((|\eta|+H_A)
    \mathbbm{1}_A\big) - \phi_{p,q,G}^\xi \big(|\eta|+H_A\big)
    \phi_{p,q,G}^\xi(A) \right. \\
    &\hspace{6em} \left. {} - \phi_{p,q,G}^\xi(H_A\mathbbm{1}_A) +
    \phi_{p,q,G}^\xi (H_A) \phi_{p,q,G}^\xi(A) \right]\\
    &\geq \frac{1}{p(1-p)}\phi_{p,q,G}^\xi(H_A)\phi_{p,q,G}^\xi(A).
  \end{align*}
  To obtain the second line, we simply add and subtract the same
  quantity. In order to go from the second line to the third, we remark
  two things: in the second line, the third term equals $0$ (when $A$
  occurs, the Hamming distance to $A$ is $0$), and the sum of the first
  two terms is positive thanks to the FKG inequality (indeed, it is easy
  to check that $|\eta|+H_A$ is increasing). The claim follows since
  $p(1-p)\leq 1/4$.
\end{proof}

This proposition has an interesting reformulation: integrating the
formula between $p_1$ and $p_2>p_1$, we obtain
\begin{equation}\label{russo_hamming}
  \phi_{p_1,q,G}^\xi (A) \leq \phi_{p_2,q,G}^\xi (A) \; {\rm
  e}^{-4(p_2-p_1)\phi_{p_2,q,G}^\xi(H_A)}
\end{equation}
(note that $H_A$ is a decreasing random variable).  If one can prove
that the typical value of $H_A$ is sufficiently large, for instance
because $A$ occurs with small probability, then one can obtain bounds
for the probability of $A$. This kind of differential formula is very
useful in order to prove the existence of a sharp threshold.  The next
example presents a sharper estimate of the derivative.

Intuitively, the derivative of $\phi_{p,q,G}^\xi(A)$ with respect to $p$
is governed by the influence of one single edge, switching from closed
to open (roughly speaking, considering the increasing coupling between
$p$ and $p+{\rm d}p$, it is unlikely that two edges switch their state).
The following definition is therefore natural in this setting. The
\emph{(conditional) influence} on $A$ of the edge $e\in E$, denoted by
$I_A(e)$, is defined as \[ I_A(e) := \phi_{p,q,G}^\xi (A|J_e=1) -
\phi_{p,q,G}^\xi (A|J_e=0). \]

\begin{proposition}\label{russo_influence}
  Let $q\geq 1$ and $\ep>0$; there exists $c=c(q,\ep)$ such that for any
  random-cluster measure $\phi_{p,q,G}^\xi$ with $p\in[\ep,1-\ep]$ and
  any increasing event $A$, \begin{equation*}
    \frac{d}{dp}\phi_{p,q,G}^\xi(A)\geq c\sum_{e\in E}I_A(e).
  \end{equation*}
\end{proposition}

\begin{proof}
  Note that, by definition of $I_A(e)$, \[ \phi_{p,q,G}^\xi
  (\mathbbm{1}_A\cap J_e) - \phi_{p,q,G}^\xi (A) \phi_{p,q,G}^\xi (J_e)
  = I_A(e) \phi_{p,q,G}^\xi (J_e) \big( 1 - \phi_{p,q,G}^\xi (J_e) \big)
  \] so that \eqref{russo} becomes
  \begin{align*}
    \frac{d}{dp} \phi_{p,q,G}^\xi (A) &= \frac{1}{p(1-p)} \sum_{e\in E}
    \phi_{p,q,G}^\xi (J_e) \big( 1 - \phi_{p,q,G}^\xi(J_e) \big)
    I_A(e)\\
    &= \frac {\phi_{p,q,G}^\xi(J_e)\big(1-\phi_{p,q,G}^\xi(J_e)\big)}
    {p(1-p)} \sum_{e\in E}I_A(e)
  \end{align*}
  from which the claim follows since the term \[ \frac
  {\phi_{p,q,G}^\xi(J_e)\big(1-\phi_{p,q,G}^\xi(J_e)\big)} {p(1-p)} \]
  is bounded away from $0$ uniformly in $p\in[\ep,1-\ep]$ when $q$ is
  fixed.
\end{proof}

\bigskip

There has been an extensive study of the largest influence in the case
of product measures. It was initiated in \cite{KahnKalaiLinial} and
recently lead to important consequences in statistical models, see
\emph{e.g.}~\cite{BollobasRiordan, BollobasRiordanperco}. The following
theorem is a special case of the generalization to positively-correlated
measures.

\begin{theorem}[see \cite{GrahamGrimmett}]\label{maximum_influence}
  Let $q\geq 1$ and $\ep>0$; there exists a constant $c = c(q,\ep) \in
  (0,\infty)$ such that the following holds. Consider a random-cluster
  model on a graph $G$ with $|E|$ denoting the number of edges of $G$.
  For every $p\in[\ep,1-\ep]$ and every
  increasing event $A$, there exists $e\in E$ such that \[ I_A(e)\geq c
  \,\phi_{p,q,G}^\xi(A)\big(1-\phi_{p,q,G}^\xi(A)\big)\frac{\log
  |E|}{|E|}.  \]
\end{theorem}

There is a particularly efficient way of using
Proposition~\ref{russo_influence} together with
Theorem~\ref{maximum_influence}. In the case of a translation-invariant
event on a torus of size $n$, horizontal (resp.\ vertical) edges play a
symmetric role, so that the influence is the same for all of them. In
particular, Proposition~\ref{russo_influence} together with
Theorem~\ref{maximum_influence} provide us with the following
differential inequality:
\begin{theorem}\label{symmetric_sharp_threshold}
  Let $q\geq1$ and $\ep>0$. There exists a constant $c=c(q,\ep)
  \in(0,\infty)$ such that the following holds. Let $n\geq 1$ and let
  $A$ be a translation-invariant event on the torus of size $n$: for any
  $p\in[\ep,1-\ep]$, \[ \frac{d}{dp}\phi_{p,q,n}^{\rm p}(A)\geq c
  \big(\phi_{p,q,n}^{\rm p}(A)(1-\phi_{p,q,n}^{\rm p}(A)\big)\log n.  \]
\end{theorem}

In particular, for a non-empty increasing event $A$, we can integrate
the previous inequality between two parameters $p_1<p_2$ to obtain \[
\frac{\phi_{p_1,q,n}^{\rm p}(A)}{1-\phi_{p_1,q,n}^{\rm p}(A)}\geq
\frac{1-\phi_{p_2,q,n}^{\rm p}(A)}{\phi_{p_2,q,n}^{\rm
p}(A)}n^{c(p_2-p_1)}.\] If we further assume that
$\phi_{p_1,q,n}^\xi(A)$ stays bounded away from $0$ uniformly in $n\geq
1$, we can find $c'>0$ such that
\begin{equation}\label{important}
  \phi_{p_2,q,n}^{\rm p}(A)\geq 1-c'n^{-c(p_2-p_1)}.
\end{equation}
This inequality will be instrumental in the next section.

\section{The proofs of Theorems~\ref{critical_value}
and~\ref{exponential_decay}}
\label{sec:proofs}

The previous two sections combine in order to provide estimates on
crossing probabilities (see \cite{BollobasRiordan, BollobasRiordanperco}
for applications in the case of percolation). Indeed, one can consider
the event that \emph{some} long rectangle is crossed in a torus. At
$p=p_{sd}$, we know that the probability of this event is bounded away
from $0$ uniformly in the size of the torus (thanks to
Theorem~\ref{RSW_torus}).  Therefore, we can apply
Theorem~\ref{symmetric_sharp_threshold} to conclude that the probability
goes to $1$ when $p>p_{sd}$ (we also have an explicit estimate on the
probability). It is then an easy step to deduce a lower bound for the
probability of crossing a \emph{particular} rectangle.

Theorem~\ref{critical_value} is proved by constructing a path from $0$
to infinity when $p>p_{sd}$, which is usually done by combining
crossings of rectangles. There is a major difficulty in doing such a
construction: one needs to transform estimates in the torus into
estimates in the whole plane. One solution is to replace the periodic
boundary condition by wired boundary condition. The path construction is
a little tricky since it must propagate wired boundary conditions
through the construction (see Proposition~\ref{path_circuits}); it does
not follow the standard lines.

Theorem~\ref{exponential_decay} follows from a refinement of the
previous construction in order to estimate the Hamming distance of a
typical configuration to the event $\{0\leftrightarrow
\mathbb{L}\setminus [-n,n)^2\}$. It allows the use of
Proposition~\ref{russo_hamming_proposition}, which improves bounds on
the probability that the origin is connected to distance $n$. With these
estimates, we show that the cluster size at the origin has finite
moments of any order, whenever $p<p_{sd}$. Then, it is a standard step
to obtain exponential decay in the sub-critical phase.

\bigskip

The following two lemmas will be useful in the proofs of both theorems.
We start by proving that crossings of long rectangles exist with very
high probability when $p>p_{sd}$.

\begin{lemma}\label{long_path}
  Let $\alpha>1$, $q\geq 1$ and $p>p_{sd}$; there exists
  $\ep_0=\ep_0(p,q,\al)>0$ and $c_0=c_0(p,q,\al)>0$ such that
  \begin{equation}
    \phi_{p,q,\alpha^2 n}^{\rm p}\left(\C^v\big([0,n)\times[0,\alpha
    n)\big)\right)\geq 1-c_0n^{-\ep_0}
  \end{equation}
  for every $n\geq1$.
\end{lemma}

\begin{proof}
  The proof will make it clear that it is sufficient to treat the case
  of integer $\al$, we therefore assume that $\al$ is a positive integer
  (not equal to $1$).  Let $B$ be the event that there exists a vertical
  crossing of a rectangle with dimensions $(n/2,\al^2 n)$ in the torus
  of size $\al^2n$. This event is invariant under translations and
  satisfies
  \begin{equation*}
    \phi_{p_{sd},q,\al^2 n}^{\rm p}(B)\geq \phi_{p_{sd},q,\alpha^2
    n}^{\rm p}\big(\C^v([0,n/2)\times[0,\al^2 n))\big)\geq c(2\al^2)
  \end{equation*}
  uniformly in $n$.

  Let $p>p_{sd}$. Since $B$ is increasing, we can apply
  Theorem~\ref{symmetric_sharp_threshold} (more precisely
  \eqref{important}) to deduce that there exist $\ep=\ep(p,q,\al)$ and
  $c=c(p,q,\al)$ such that
  \begin{equation}\label{aa}
    \phi_{p,q,\alpha^2 n}^{\rm p}(B)\geq 1-cn^{-\ep}.
  \end{equation}
  If $B$ holds, one of the $2\alpha^3$ rectangles
  \begin{equation*}
    [in/2,in/2+n)\times[j\alpha n,(j+1)\alpha n),
    \quad(i,j)\in\{0,\cdots,2\al^2-1\}\times\{0,\cdots,\al-1\}
  \end{equation*}
  must be crossed from top to bottom. We denote these events by $A_{ij}$
  --- they are translates of $\C^v([0,n)\times[0,\alpha n))$.  Using the
  FKG inequality in the second line (this is an instance of the
  ``square-root trick'' mentioned earlier), we find
  \begin{align*}
    \phi_{p,q,\alpha^2 n}^{\rm p}(B)&\leq 1-\phi_{p,q,\alpha^2 n}^{\rm
    p}(B^c)\leq1-\phi_{p,q,\alpha^2 n}^{\rm p}(\cap_{i,j}A_{ij}^c)\\
    &\leq 1-\prod_{i,j}\phi_{p,q,\alpha^2 n}^{\rm
    p}(A_{ij}^c)=1-\left[1-\phi_{p,q,\alpha^2 n}^{\rm
    p}\big(\C^v([0,n)\times[0,\alpha n)\big)\right]^{2\alpha^3}.
  \end{align*}
  Plugging \eqref{aa} into the previous inequality, we deduce
  \begin{equation*}
    \phi_{p,q,\alpha^2 n}^{\rm p}\left(\C^v\big([0,n)\times[0,\alpha
    n)\big)\right)\geq 1-(cn^{-\ep})^{1/(2\alpha^3)}.
  \end{equation*}
  The claim follows by setting $c_0:=c^{1/(2\alpha)^3}$ and
  $\ep_0:=\ep/(2\alpha^3)$.
\end{proof}

\bigskip

Let $\al>1$ and $n\geq 1$; we define the annulus \[ A_n^\al :=
[-\al^{n+1},\al^{n+1}]^2\setminus [-\al^n,\al^n]^2.  \] An \emph{open
circuit} in an annulus is an open path which surrounds the origin.
Denote by $\A^\al_n$ the event that there exists an open circuit
surrounding the origin and contained in $A_n^\al$, together with an open
path from this circuit to the boundary of $[-\al^{n+2},\al^{n+2}]^2$,
see Figure~\ref{fig:event_A_n}. The following lemma shows that the
probability of $\A_n^\al$ goes to $1$, provided that $p>p_{sd}$ and that
we fixed wired boundary conditions on $[-\al^{n+2},\al^{n+2}]^2$.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\hsize]{event_A_n}
  \end{center}
  \caption{\textbf{Left:} The event $\A_n^\alpha$.  \textbf{Right:} The
  combination of events $\A_n^\alpha$: we see that it indeed constructs
  a path from the origin to infinity.}
  \label{fig:event_A_n}
\end{figure}

\begin{lemma}\label{circuit}
  Let $\al>1$, $q\geq 1$ and $p>p_{sd}$; there exists $c_1=c_1(p,q,\al)$
  and $\ep_1=\ep_1(p,q,\al)$ such that for every $n\geq 1$,
  \begin{equation*}
    \phi_{p,q,\al^{n+2}}^{1}(\A^{\al}_n)\geq 1-c_1e^{-\ep_1n}.
  \end{equation*}
\end{lemma}

\begin{proof}
  First, observe that $\A^{\alpha}_n$ occurs whenever the following
  events occur simultaneously:
  \begin{itemize}
    \item The following rectangles are crossed vertically:
      \begin{align*}
        R_1 &:=
        [\al^n,\alpha^{n+1}]\times[-\alpha^{n+1},\alpha^{n+1}],\\
        R_2 &:=
        [-\alpha^{n+1},-\al^n[\times[-\alpha^{n+1},\alpha^{n+1}];
      \end{align*}
    \item The following rectangles are crossed horizontally:
      \begin{align*}
        R_3 &:=
        [-\alpha^{n+1},\alpha^{n+1}]\times[\al^n,\alpha^{n+1}],\\
        R_4 &:=
        [-\alpha^{n+1},\alpha^{n+1}]\times[-\alpha^{n+1},-\al^n],\\
        R_5 &:=
        [-\alpha^{n+2},\alpha^{n+2}]\times[-\alpha^{n+1},\alpha^{n+1}].
      \end{align*}
  \end{itemize}
  For the measure in the torus, these events have probability greater
  than $1-c(\al^n)^{-\ep}$ with $c=c_0(p,q,2\al/(\al-1))$ and
  $\ep=\ep_0(p,q,2\al/(\al-1))$. Using the FKG inequality, we obtain
  $$\phi_{p,q,\alpha^{n+2}}^{\rm p}(\A^\alpha_n)\geq
  (1-c(\al^n)^{-\ep})^5$$ from which we deduce the following estimate,
  harnessing the comparison between boundary conditions,
  $$\phi_{p,q,\alpha^{n+2}}^{1} (\A^\alpha_n) \geq
  (1-c(\al^n)^{-\ep})^5.$$ The claim follows by setting $c_1:=5c$ and
  $\ep_1:=\ep\log \al$.
\end{proof}

\bigskip

The following proposition readily implies Theorem~\ref{critical_value};
It will also be useful in the proof of Theorem~\ref{exponential_decay}.
We want to prove that the probability of the intersection of events
$\A_n^\al$ is of positive probability when $p>p_{sd}$. So far, we know
that there is an open circuit with very high probability when we
consider the random-cluster measure with wired boundary condition in a
slightly larger box. In order to prove the result, we assume the
existence of a large circuit. Then, we iteratively condition on events
$\A_{n-k}^\al$, $k\geq 0$. When conditioning `from the outside to the
inside', we guarantee that at step $k$, there exists an open circuit in
$A_{n-k+1}^\al$ that surrounds $A_{n-k}^\al$. Using comparison between
boundary conditions, we can assure that the measure in $A_{n-k}^\al$
stochastically dominates the measure in $A_{n-k+1}^\al$ with wired
boundary condition. In other words, we keep track of advantageous
boundary conditions. Note that the reasoning, while reminiscent of
Kesten's construction of an infinite path for percolation, is not
standard.

\begin{proposition}\label{path_circuits}
  Let $\alpha>1$, $q\geq 1$ and $p>p_{sd}$; there exists $c,c_1,\varepsilon_1$ (depending on $p,q$ and $\alpha$) so that for every $N\geq 1$,

  $$\phi_{p,q}\left(\bigcap_{n\geq N}\A_N^{\alpha}\right)\geq c\prod_{k=N}^{\infty}(1-c_1{\rm e}^{-\ep_1k}). $$
\end{proposition}

\begin{proof}
  Let $\al>1$, $q\geq1$, $p>p_{sd}$, $N\geq 1$ and recall that there is a unique
  infinite-volume measure $\ph$. For every $n\geq N$, we know that
  \begin{equation}\label{b}
    \ph\left(\bigcap_{k=N}^n\A_n^\al\right)=
    \ph(\A_n^\al)\prod_{k=N}^{n-1}\ph(\A^\al_k|\A^\al_j,k+1\leq j\leq
    n).
  \end{equation}

  On the one hand, let $k\in[N,n-1]$. Conditionally to $\A^\al_j$,
  $k+1\leq j\leq n$, we know that there exists a circuit in the annulus
  $A_{k+1}^\al$.  Exploring from the outside, we shall consider the most
  exterior such circuit, denoted by $\Ga$. Conditionally to $\Ga=\g$,
  the unexplored part of the box $[-\al^{k+2},\al^{k+2}]^2$ follows the
  law of a random-cluster configuration with wired boundary condition.
  In particular, the conditional probability that there exists a circuit
  in $A^\al_k$ connected to $\g$ is greater than the probability that
  there exists a circuit in $A^\al_k$ connected to the boundary of
  $[-\al^{k+2},\al^{k+2}]^2$ with wired boundary condition. Therefore,
  we obtain that almost surely
  \begin{align*} \ph(\A^\al_{k}|\A^\al_{j},k+1\leq j\leq
    n)&=\ph\big(\ph(\A^\al_k|\Ga=\g)\big)\\
    &\geq\ph\big(\phi_{p,q,\al^{k+2}}^{1}(\A^\al_k)\big)\\
    &\geq 1-c_1{\rm e}^{-\ep_1k}
  \end{align*}
  where we have harnessed Lemma~\ref{circuit} in the last inequality.

  On the other hand, for $p=p_{sd}$, consider the event $\A_n^{\al}$ in
  the bulk. Thanks to Corollary~\ref{RSW_bulk}, its probability is
  bounded away from $0$ uniformly in $n$. Since the event is increasing,
  we obtain that there exists $c=c(\al)>0$ such that
  \begin{equation*}
    \ph(\A_n^\al)=\ph^1(\A_n^\al)\geq c
  \end{equation*}
  for any $n\geq N$ and $p> p_{sd}$. Plugging the two estimates into
  \eqref{b}, we obtain \[ \ph\left(\bigcap_{k=N}^n\A_n^\al\right)\geq
  c\prod_{k=N}^{n-1}(1-c_1{\rm e}^{-\ep_1k})\geq
  c\prod_{k=N}^{\infty}(1-c_1{\rm e}^{-\ep_1k}).  \] Letting $n$ go to
  infinity concludes the proof.
\end{proof}

\bigskip

\begin{proof}[Proof of Theorem~\ref{critical_value}]
  The bound $p_c\geq p_{sd}$ is provided by Zhang's argument, as
  explained in Section~\ref{sec:basic}. For $p>p_{sd}$, fix
  $\alpha>1$. Applying Proposition \ref{path_circuits} to $N=1$, we find \[ \ph(0\leftrightarrow\infty)\geq
  \ph\left(\bigcap_{n\geq1}\A_n^\al\right)>0 \] so that $p$ is
  super-critical. Since $p$ is super-critical for every $p>p_{sd}$, we
  deduce $p_c\leq p_{sd}$.
\end{proof}

\bigskip

\begin{proof}[Proof of Theorem~\ref{exponential_decay}]
  Let $x$ be a site of $\Z^2$, and let $\C_x$ be the cluster of $x$,
  \ie\  the maximal connected component containing the site $x$. We
  denote by $|\C_x|$ its cardinality. We first prove that $|\C_x|$ has
  finite moments of any order. Then we deduce that the probability of
  $\{|\C_x|\geq n\}$ decays exponentially fast in $n$.  The proof of the
  Step 2 is extracted from \cite{Grimmett}.

  \paragraph{Step 1: finite moments for $|\C_x|$.}

  Let $d>0$ and $p<p_{sd}$; we want to prove that
  \begin{equation}\label{moment_estimate}
    \ph(|\C_x|^d)<\infty.
  \end{equation}
  In order to do so, let $p_1 := (p+p_{sd})/2$ and define $D_n :=
  \{x\leftrightarrow \Z^2\setminus (x+[-n,n)^2)\}$; denote by $H_n$ the
  Hamming distance to $D_n$. Note that $H_n$ is the minimal number of
  closed edges that one must cross in order to go from $x$ to the
  boundary of the box of size $n$ centered at $x$. Let $$\al := \exp
  \left[ \frac{p_1-p}{2d+3} \right]>1.$$ We know from
  Proposition~\ref{path_circuits}, applied to the (super-critical) dual
  model, that the probability of $\bigcap_{n>N}(\A_n^\al)^\star$ is
  larger than $c\prod_N^\infty (1-c_1e^{-\varepsilon_1n})$ ($(\A_n^\al)^\star$ is the occurrence of $\A_n^\al$ in
  the dual model). Hence, there exists $N=N(p_1,q,\al)$ sufficiently
  large such that \[ \phi_{p_1,q}\left(\bigcap_{k\geq N}^\infty
  (\A_n^\al)^\star \right)\geq \frac12. \]On this event, $H_{n}$ is
  greater than $(\log n/\log \al)-N$ since there is at least one closed
  circuit in each annulus $A_k^\al$ with $k\geq N$ (thus increasing the
  Hamming distance by 1). We obtain \[ \phi_{p_1,q}(H_{n})\geq
  \left(\frac{\log n}{\log
  \al}-N\right)\phi_{p_1,q}\left(\bigcap_{k\geq N}^\infty (\A_n^\al)^\star
  \right)\geq\frac{\log n}{4\log \al} \] for $n$ sufficiently large. We
  can use \eqref{russo_hamming} to find \[ \phi_{p,q}(D_n)\leq
  \phi_{p_1,q}(D_n)\exp \big[-4(p_1-p)\phi_{p_1,q}(H_n)\big]\leq
  n^{-(2d+3)} \] for $n$ sufficiently large, from which
  \eqref{moment_estimate} follows readily.

  \paragraph{Step 2: an expression for $H_n$.}

  Consider a (non necessarily open) self-avoiding path $\g$ going from
  the origin to the boundary of the box of size $n$. We can bound from
  below the number $T(\g)$ of closed edges along this path by the
  following quantity: \[ \frac{T(\g)}{n}\geq\frac1{|\g|}T(\g) \geq
  \frac1{|\g|}\sum_{x\in \g} \frac{1}{|\C_x|}\geq
  \left(\frac1{|\g|}\sum_{x\in \g}|\C_x|\right)^{-1}.  \] Indeed, the
  number of closed edges in $\g$ is larger than the number of distinct
  clusters intersecting $\g$.  Moreover, if $\mathcal{C}$ denotes such a
  cluster, we have that $1\geq \sum_{x\in\g}|\C|^{-1}\mathbbm{1}_{x\in
  \C}$. The last inequality is due to Jensen's inequality. Hence,
  \begin{equation}\label{expression_H_n}
    \frac{H_n}n\geq \inf_{\g:0\leftrightarrow
    \Z^2\setminus\B_n}\left(\frac{1}{|\g|}\sum_{x\in
    \g}|\C_x|\right)^{-1}.
  \end{equation}
  Let $(\tilde{\C}_x)_{x\in\B_n}$ be a family of independent subsets of
  $\Z^2$ distributed as $\C_x$. We claim that $(|{\C}_x|)_{x\in \B_n}$
  is stochastically dominated by the family $(M_x)_{x\in \B_n}$ defined
  as $$M_x:=\sup_{y\in \Z^2:x\in\tilde{\C}_y}|\tilde{\C}_y|.$$

  Let $v_1, v_2, \ldots$ be a deterministic ordering of $\mathbb{Z}^2$.
  Given the random family $(\tilde{\C}_x)_{x\in\B_n}$, we shall
  construct a family $(D_x)_{x\in \B_n}$ having the same joint law as
  $(\C_x)_{x\in \B_n}$ and satisfying the following condition: for each
  $x$, there exists $y$ such that $D_x\subset\tilde{\C}_y$. First, set
  $D_{v_1}=\tilde{C}_{v_1}$.  Given $D_{v_1}$, $D_{v_2}$, \ldots,
  $D_{v_n}$, define $E=\bigcup_{i=1}^n D_{v_1}$. If $v_{n+1}\in E$, set
  $D_{v_{n+1}}=D_{v_j}$ for some $j$ such that $v_{n+1}\in D_{v_j}$. If
  $v_{n+1}\notin E$, we proceed as follows. Let $\Delta_eE$ be the set
  of edges of $\Z^2$ having exactly one end-vertex in $E$. We may find a
  (random) subset $F$ of $\tilde{C}_{v_{n+1}}$ such that $F$ has the
  conditional law of $C_{n+1}$ given that all edges in $\Delta_eE$ are
  closed; we now set $D_{v_{n+1}}=F$. We used the domain Markov property
  and the positive association. Indeed, we use that the law of
  $C_{v_{n+1}}$ depends only on $\Delta_eE$, and stochastically
  dominates the law of the cluster in the bulk without any conditioning.
  We obtain the required stochastic domination accordingly. In
  particular, $|\C_x|\leq M_x$ and $M_x$ has finite moments.

  \paragraph{Step 3: conclusion of the proof.}

  Note that, from \eqref{russo_hamming}, it is sufficient to prove that
  with probability bounded away from 0, $H_n$ is of order $n$. From
  \eqref{expression_H_n} and the previous stochastic domination, we get
  \begin{equation*}
    \liminf_{n\rightarrow \infty}\frac{H_n}n\geq
    \liminf_{n\rightarrow\infty}\inf_{\g:0\leftrightarrow
    \Z^2\setminus\B_n}\left(\frac{1}{|\g|}\sum_{x\in
    \g}|\C_x|\right)^{-1}\geq
    \left(\limsup_{n\rightarrow\infty}\sup_{\g:0\leftrightarrow
    \Z^2\setminus\B_n}\frac{1}{|\g|}\sum_{x\in \g}M_x\right)^{-1}.
  \end{equation*}
  We can harness Lemma~2 of \cite{FontesNewman} to show that
  \begin{equation*}
    \liminf_{n\rightarrow \infty}\frac{H_n}n\geq
    \left(2\limsup_{n\rightarrow\infty}\sup_{|\Ga|\geq n}
    \frac{1}{|\Ga|}\sum_{x\in \g}|\tilde{\C}_x|^2\right)^{-1}
  \end{equation*}
  where the supremum is over all lattice animals $\Gamma$ of cardinality
  larger than $m$ that contain the origin. We need a uniformly positive
  lower bound on the right hand side. This can be thought of as a
  two-dimensional law of large numbers. Since the $|\tilde{\C}_x|^2$
  have finite moments of any order, it is actually one of the results of
  \cite{CoxGandolfiGriffinKesten}, to which we refer the interested
  reader. Therefore, with positive probability, $\liminf H_n/n$ is
  greater than a given constant, which concludes the proof.
\end{proof}

\section{The critical point for the triangular and hexagonal lattices}
\label{sec:triangular}

Let $\mathbb T$ be the triangular lattice of mesh size $1$, embedded in
the plane in such a way that the origin is a vertex and the edges of
$\mathbb{T}$ are parallel to the lines of equations $y=0$,
$y=\sqrt{3}x/2$ and $y=-\sqrt{3}x/2$. The dual graph of this lattice is
a hexagonal lattice, denoted by $\mathbb{H}$, see
Figure~\ref{fig:reseau_triangular}. Via planar duality, it is sufficient
to handle the case of the triangular lattice in order to prove
Theorem~\ref{triangular}. Define $p_{\mathbb{T}}$ as being the unique
$p\in (0,1)$ such that $y^3+3y^2-q=0$, where $y := p_{\mathbb{T}} /
(1-p_{\mathbb{T}})$. The goal is to prove that
$p_c(\mathbb{T})=p_{\mathbb{T}}$.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.8\hsize]{reseau_triangular}
  \end{center}
  \caption{\textbf{Left:} The triangular lattice $\mathbb{T}$ with its
  dual lattice $\mathbb{H}$.    \textbf{Right:} The exchange of the two
  patterns does not alter the random-cluster connective properties of
  the black vertices.}
  \label{fig:reseau_triangular}
\end{figure}

\bigskip

The general strategy is the same as in the square lattice case. We prove
that at $p=p_\mathbb{T}$, a crossing estimate similar to
Theorem~\ref{RSW_torus} holds. Sharp threshold arguments and proofs of
Section~\ref{sec:proofs} can be adapted \emph{mutatis mutandis},
replacing square-shaped annuli by hexagonal-shaped annuli. The crossing
estimate must be slightly modified, and we present the few changes. It
harnesses the planar-duality between the triangular and the hexagonal
lattices, and the so-called \emph{star-triangle transformation} (see
\emph{e.g.} Section 6.6 of \cite{Grimmett} and
Figure~\ref{fig:reseau_triangular}). We assume that the reader already
knows the star-triangle transformation.

Let $e_1=\sqrt{3}/2+{\rm i}/2$ and $e_2={\rm i}$; whenever we write
coordinates, they are understood as referring to the basis $(e_1,e_2)$.
A `rectangle' $[a,b)\times[c,d)$ is the set of points in $z\in
\mathbb{T}$ such that $z=\lambda e_1+\mu e_2$ with $\lambda\in[a,b)$ and
$\mu\in[c,d)$ (it has a lozenge shape, see \emph{e.g.}
Figure~\ref{fig:event_A_triangular}). By analogy with the case of the
square lattice, $\C_v(D)$ denotes the event that there exists a path
between the top and the bottom sides of $D$ which stays inside $D$. Such
a path is called a \emph{vertical open crossing} of the rectangle. Other
quantities are defined similarly. We present the crossing estimate in
the case of the torus $\mathbb{T}_m$ of size $m$ (deriving the bulk
estimate follows the same lines as in the square lattice case);
$\phi_{p_{sd},q,m}^{\rm p}$ denotes the random-cluster measure on
$\mathbb{T}_m$.

\begin{theorem}\label{RSW_torus_triangular}
  Let $\al>1$ and $q\geq 1$. There exists $c(\al)>0$ such that for every
  $m>\alpha n>0$,
  \begin{equation}
    \phi_{p_\mathbb{T},q,m}^{\rm p}\big(\C_h([0,n)\times[0,\al
    n))\big)\geq c(\al).
  \end{equation}
\end{theorem}

The main difficulty is the adaptation of Lemma~\ref{self-duality}.
Define the line $d:=-\sqrt 3/3+{\rm i}\R$. The orthogonal symmetry
$\sigma_d$ with respect to $d$ maps $\mathbb{T}$ to another triangular
lattice. Note that this lattice is a sub-lattice of $\mathbb{H}$ (in the
sense that its vertices are also vertices of $\mathbb H$).  Let $\g_1$
and $\g_2$ be two paths satisfying the following Hypothesis $(\star)$,
see Figure~\ref{fig:triangular_symmetry}:
\begin{itemize}
  \item $\g_1$ remains on the left of $d$ and $\g_2$ remains on the
    right,
  \item $\g_2$ begins at $0$ and $\g_1$ begins on a site of
    $\mathbb{L}\cap (-\sqrt 3/ 2+{\rm i}\mathbb{R}_+)$,

  \item $\g_1$ and $\sigma_d(\g_2)$ do not intersect (as curves in the
    plane),
  \item $\g_1$ and $\sigma_d(\g_2)$ end at two sites (one primal and one
    dual) which are at distance $\sqrt3/3$ from one another.
\end{itemize}
When following the paths in counter-clockwise order, we can create a
circuit by linking the end points of $\g_1$ and $\sigma_d(\g_2)$ by a
straight line, the start points of $\sigma_d(\g_2)$ and $\g_2$, the end
points of $\g_2$ and $\sigma_d(\g_1)$, and the start points of
$\sigma_d(\g_1)$ and $\g_1$. The circuit $(\g_1,\sigma_d(\g_2), \g_2,
\sigma_d(\g_1))$ surrounds a set of vertices of $\mathbb{T}$. Define the
graph $G(\g_1,\g_2)$ with sites being site of $\mathbb{T}$ that are
surrounded by the circuit $(\g_1,\sigma_d(\g_2),\g_2,\sigma_d(\g_1))$,
and with edges of $\mathbb{T}$ that remain entirely inside the circuit
(boundary included).

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.4\hsize]{triangular_symmetry}
  \end{center}
  \caption{The graph $G(\g_1,\g_2)$ with the two solid arcs $\g_1$ and
  $\g_2$ and the dashed arcs $\sigma_d(\g_1)$ and $\sigma_d(\g_2)$. The
  dual arcs $\g_1^\star$ and $\g_2^\star$ are
  dotted.}\label{fig:triangular_symmetry}
\end{figure}

We will need an additional technical condition, that we present now.
Note that for any edge of $\sigma_d(\mathbb{T})$ there is one vertex of
$\mathbb{T}$ and one vertex of $\mathbb{H}$ at distance $\sqrt 3/6$ from
its midpoint. We assume that for any edge of $\sigma_d(\g_1)$ and
$\sigma_d(\g_2)$, the associated vertex of $\mathbb{T}$ is in the
interior of the domain $G(\g_1,\g_2)$ (therefore, the associated vertex
of $\mathbb{H}$ is outside the domain, see white vertices in
Fig~\ref{fig:triangular_symmetry}). This conditions is denoted
Hypothesis $(\star\star)$.

The \emph{mixed boundary condition} on this graph is wired on $\g_1$
(all the edges are pairwise connected), wired on $\g_2$, and free
elsewhere.  We denote the measure on $G(\g_1,\g_2)$ with parameters
$(p_\mathbb{T},q)$ and mixed boundary condition by
$\phi_{p_\mathbb{T},q,\g_1,\g_2}$ or more simply $\phi_{\g_1,\g_2}$.
With these definitions, we have an equivalent of
Lemma~\ref{self-duality}:

\begin{lemma}\label{self-duality_triangular}
  For any $\g_1,\g_2$ satisfying Hypotheses $(\star)$ and
  $(\star\star)$, we have $$\phi_{\g_1,\g_2}(\g_1\leftrightarrow
  \g_2)\geq \frac 1{1+q^2}.$$
\end{lemma}

\begin{proof}
  As previously, if $\g_1$ and $\g_2$ are not connected, $\g_1^\star$
  and $\g_2^\star$ are connected in the dual model, where
  $\g_1^\star,\g_2^\star\subset\mathbb{H}$ are the dual arcs bordering
  $G(\g_1,\g_2)$ close to $\sigma_d(\g_1)$ and $\sigma_d(\g_2)$. Thanks
  to Hypothesis $(\star\star)$ and the mixed boundary condition, this
  event is equivalent to the event that $\sigma_d(\g_1)$ and
  $\sigma_d(\g_2)$ are dual connected. Using Hypothesis $(\star)$ and
  the symmetry, we deduce $$\phi_{\g_1,\g_2} \big(\g_1\leftrightarrow
  \g_2\big) + \sigma_d*\phi_{\g_1,\g_2}^\star
  \big(\g_1\leftrightarrow\g_2\big)=1,$$ where as before $\sigma_d *
  \phi_{\g_1,\g_2}^\star$ denotes the push-forward under the symmetry
  $\sigma_d$ of the dual measure of $\phi_{\g_1,\g_2}$ --- in
  particular, it lies on $\sigma_d(\mathbb{H})$ and the edge-weight is
  $p_\mathbb{T}^\star$.  This lattice contains the sites of $\mathbb{T}$
  and those of another copy of the triangular lattice which we will
  denote by $\mathbb{T}'$.  Since $\g_1$ and $\g_2$ are two paths of
  $\mathbb{T}$, one can use the star-triangle transformation for any
  triangle of $\mathbb{T}$ included in $G(\g_1,\g_2)$ that contains a
  vertex of $\mathbb{T}'$: one obtains that
  $\sigma_d*\phi_{\g_1,\g_2}^\star\big(\g_1\leftrightarrow\g_2\big)$ is
  equal to the probability of $\g_1$ and $\g_2$ being connected, in a
  model on $\mathbb{T}$ with edge-weight $p_\mathbb{T}$. Here, we need
  Hypothesis $(\star\star)$ again in order to ensure that all the
  triangles containing a vertex of $\mathbb{T}'$ have no edges on the
  boundary (which would have forbidden the use of the star-triangle
  transformation). The same observation as in the case of the square
  lattice shows that the boundary conditions are the same as for
  $\phi_{\g_1,\g_2}$, except that arcs $\g_1$ and $\g_2$ are wired
  together. The same reasoning as in Lemma~\ref{self-duality} implies
  that $$\sigma_d*\phi_{\g_1,\g_2}^\star \big( \g_1\leftrightarrow\g_2
  \big) \leq q^2 \phi_{\g_1,\g_2}\big(\g_1\leftrightarrow \g_2\big),$$
  and the claim follows readily.
\end{proof}

The existence of $c(1)$ is obtained in the same way as in the case of
the square lattice, with only the obvious modifications needed; we leave
the details as an ``exercise for the reader''.
Theorem~\ref{RSW_torus_triangular} is derived exactly as in
Section~\ref{sec:crossings}, as soon as an equivalent of
Proposition~\ref{box_crossing} holds:
\begin{proposition}\label{box_crossing_triangular}
  There exists a constant $c(3/2)>0$ such that, for all $m>3n/2>0$,
  $$\phi_{p_{\mathbb{T}},q,m}^{\rm p} \big( \C^v([0,3n/2)\times[0,n))
  \big) \geq c(3/2).$$
\end{proposition}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.8\hsize]{event_A_triangular}
  \end{center}
  \caption{\textbf{Left:} The set $[0,3n/2)\times[0,n)$ and the event
  $A$. \textbf{Right:} One can obtain the path $\Ga_1'$ from $\Ga_1$ by
  replacing any bad edge with two edges. Since $\Ga_1$ is the top-most
  crossing, it contains no double edges and this construction can be
  done.}
  \label{fig:event_A_triangular}
\end{figure}

\begin{proof}[Proof of Proposition~\ref{box_crossing_triangular}]
  The general framework of the proof is the same as before, but some
  technicalities occur because the underlying lattice is not self-dual.
  Consider the rectangle $D=[0,3n/2)\times[0,n)$, which is the union of
  rectangles $D_1=[0,n)\times[0,n)$ and $D_2=[n/2,3n/2)\times[0,n)$, see
  Figure~\ref{fig:event_A_triangular}. Let $A$ be the event that:
  \begin{itemize}
    \item $D_1$ and $D_2$ are both crossed horizontally (each crossing
      has probability at least $c(1)$ to occur);
    \item $[n/2,n)\times\{0\}$ (resp. $[n,3n/2)\times\{n\}$) is
      connected inside $D_2$ to the top side (resp.\ to the bottom).
      Using the FKG inequality and symmetries of the lattice, this event
      occurs with probability larger than $c(1)^2/4$.
  \end{itemize}
  Therefore, $A$ has probability larger than $c(1)^4/4$.

  When $A$ occurs, define $\Ga_1$ to be the top-most crossing of the
  rectangle $D_1$, and $\Ga_2$ the right-most crossing in $D_2$ between
  $[n/2,n)\times\{0\}$ and the top side of $D_2$. Note that $\Ga_2$ is
  automatically connecting $[n/2,n)\times\{0\}$ to the right edge and to
  $[n,3n/2)\times\{n\}$. In order to conclude, it is sufficient to prove
  that $\Ga_1$ and $\Ga_2$ are connected with probability larger than
  some positive constant.

  Consider the lowest path $\Ga'_1$ above $\Ga_1$ which satisfies the
  following property: for any edge $e$ in $\Ga'_1$, the associated site
  of $\sigma_d(\mathbb{H})$ (see the definition of Hypothesis
  $(\star\star)$) is in the connected component of $D_1\setminus \Ga'_1$
  \emph{above} $\Ga_1'$. Such a path can be obtained from $\Ga_1$ by
  replacing every `bad' edge with the other two edges of a triangle, as
  shown in Figure~\ref{fig:event_A_triangular}. Since $\Ga_1$ is the
  top-most crossing, it cannot have double edges and the path $\Ga'_1$
  can be constructed. In particular it ends at the same point as
  $\Ga_1$, and it goes from left to right. Note that it is not
  necessarily open. We define $\Ga_2'$ similarly in the obvious way (the
  left-most path on the right of $\Ga_2$ such that for any edge of
  $\Ga_2'$, the associate site of $\sigma_d(\mathbb{H})$ is on the right
  of $\Ga'_2$).

  We now sketch the end of the proof. Apply a construction similar to
  the proof of Proposition~\ref{box_crossing} in order to create a
  domain $G(\Ga'_1,\Ga_2')$. With mixed boundary conditions, the
  probability of connecting $\Ga'_1$ to $\Ga'_2$ in $G(\Ga'_1,\Ga'_2)$
  is larger than $1/(1+q^2)$ ($\Ga_1'$ and $\Ga_2'$ have been
  constructed in such a way that Hypothesis $(\star\star)$ can be
  fulfilled). But $\Ga_1$ disconnects $\Ga'_1$ from $\Ga'_2$, and
  $\Ga_2$ disconnects $\Ga'_2$ from $\Ga_1$. Using boundary conditions
  inherited from the fact that $\Ga_1$ and $\Ga_2$ are crossings, one
  can prove that $\Ga_1$ is connected to $\Ga_2$ in $G(\Ga'_1,\Ga_2')$
  with probability larger than $1/(1+q^2)$. The end of the proof follows
  exactly the same lines as in the case of the square lattice case.
\end{proof}

\paragraph{Acknowledgements.}

The authors were supported by the ANR grant BLAN06-3-134462, the EU
Marie-Curie RTN CODY, the ERC AG CONFRA, as well as by the Swiss {NSF}.
The authors would like to thank Geoffrey Grimmett for many fruitful
discussions and useful comments on the manuscript. This works was mostly
done during the stay of the authors at IMPA: they would like to thank
Vladas Sidoravicius for his hospitality. The second author would like to
thank Stanislav Smirnov for his constant support.


TITLE: Ambiguity in the definition of unmixed ideal
QUESTION [7 upvotes]: Compare the definitions:
Page 136, Matsumura, Commutative Ring Theory:

A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal.

This means that      $\forall P\in {\rm Ass}(I) ,\quad \operatorname{ht} I = \operatorname{ht} P\quad (1) $
Page 59, Bruns&Herzog, Cohen-Macaulay Rings, 1998:

One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors, or in modern language if the associated prime ideals of $R/I$ are the minimal prime ideals of $I$.

This means that ${\rm Min}({\rm Ass}(I))={\rm Ass}(I)\quad \quad (2)$
Clearly $(1)$ implies $(2)$ but the reverse isn't true. How these definitions are equivalent?  I mean while (2) doesn't imply (1) then how can we say the two definitions are the same?

REPLY [4 votes]: You are correct in observing this.
In the old literature, unmixedness means that all associated primes are of the same height. However, in modern literature, the definition of unmixedness is equidimensional and no embedded prime, that is, all its associated primes are minimal above it and are of the same height.