T_{cr}$. The Potts model has one Gibbs state in that regime, which is called the chaotic state. Therefore the typical configuration of the system in the box $\Lambda$ under the boundary condition $\eta$ is the following: near the boundary it is dictated by the boundary condition $\eta$, whereas somewhere close to the boundary $\partial\Lambda$ there is a long contour $\Gamma$, separating the boundary layer from the rest of the box, where the system behaves chaotically. So the partition function can be written as a sum over such contours, $$Z^P_{\Lambda,\eta,T}=\sum_{\Gamma} Z^P_{\Lambda,\eta,T}(\Gamma).$$ Within the precision we need, we can rewrite it as $$Z^P_{\Lambda,\eta,T}\approx {\sum_{\Gamma}}^{(')}Z^P_{\Lambda,\eta,T}(\Gamma) ,\Eq(appr)$$ where the summation is restricted to those $\Gamma$ which are close to the boundary $\partial\Lambda$. The partition function $Z^P_{\Lambda\setminus A,\eta,T}$ can be written in the same way. However, the boundary of the box $\Lambda\setminus A$ is not connected, so the analogue of \equ(appr) is the following: $$Z^P_{\Lambda \setminus A,\eta,T}\approx {\sum_{\Gamma, \bar\Gamma}}^{(')}Z^P_{\Lambda\setminus A,\eta,T}(\Gamma,\bar\Gamma),\Eq(appr1)$$ where again the summation is restricted to $\Gamma$ lying close to the boundary $\partial\Lambda$ and to $\bar\Gamma$ lying close to the boundary $\partial A$. We have, therefore, approximate equalities: $$Z^P_{\Lambda,\eta^i,T}\approx {\sum_{\Gamma_i}}^{(')}Z^P_{\Lambda,\eta^i,T}(\Gamma_i),\Eq(appr2)$$ $$Z^P_{\Lambda \setminus A,\eta^i,T}\approx {\sum_{\Gamma_i, \bar\Gamma}}^{(')}Z^P_{\Lambda\setminus A,\eta^i,T}(\Gamma_i,\bar\Gamma).\Eq(appr3)$$ So it is enough to estimate the ratio $$\frac{Z^P_{\Lambda\setminus A,\eta^1,T}(\Gamma_1,\bar\Gamma) Z^P_{\Lambda,\eta^2,T}(\Gamma_2)} {Z^P_{\Lambda,\eta^1,T}(\Gamma_1) Z^P_{\Lambda\setminus A,\eta^2,T}(\Gamma_2,\bar\Gamma)} $$ for every triple $(\Gamma_1,\Gamma_2,\bar\Gamma)$ of contours, which are close to the corresponding parts of the boundaries of our subsets. To see the desired cancellation we observe that the logarithm of the partition function $Z^P_{\Lambda\setminus A,\eta,T}(\Gamma,\bar\Gamma)$ can be represented as a volume term plus a boundary term, and if the two boundaries $\partial\Lambda$, $\partial A$ are well separated, this boundary term is nearly a sum of two terms corresponding to the contours $\Gamma,\bar\Gamma$ (again with the same precision). This is the strategy we are going to follow. There are different options to study these partition functions. One way is to use the variant of the Pirogov-Sinai theory for the Potts model, developed in [Mar1]. Technically however it is easier to pass first to the FK representation, introduced above, and then use the Pirogov-Sinai theory for it, developed in [LMMRS]. To implement this program we rewrite \equ(otn1) with the help of the identity \equ(tozh) as $$\frac{Z^P_{\Lambda\setminus A,\eta^1,T}Z^P_{\Lambda,\eta^2,T}} {Z^P_{\Lambda,\eta^1,T}Z^P_{\Lambda\setminus A,\eta^2,T} }= \frac{Z^{FK}_{\Lambda\setminus A,\eta^1,T}Z^{FK}_{\Lambda,\eta^2,T}} {Z^{FK}_{\Lambda,\eta^1,T}Z^{FK}_{\Lambda\setminus A,\eta^2,T} }. \Eq(otn2) $$ We want to express the above partition functions in terms of more familiar FK partition functions with free and wired boundary conditions. We then want to use the corresponding contour models to treat the latter. In order to proceed we need some more notation, notions and results which we borrow from [LMMRS]. \subheading{3.2. Pirogov-Sinai theory of the FK model and cluster expansions} We call a {\it plaquette\/} $p$ any four-tuple of bonds in $\Bbb B$, which form an elementary cell. We call two bonds adjacent, if they share a vertex, and we call them coadjacent if they belong to the same plaquette. These definitions lead to natural notions of connectedness and coconnectedness of a subset of $\Bbb B$. Let $X\subset\Bbb B$ be a subgraph with no isolated sites. We denote by $v(X)\subset \Bbb Z^2$ the set of its vertices, and by $|X|$ the number of its bonds. The subset $v_I(X)\subset v(X)$ of {\it inner\/} vertices consists of all vertices which belong to four bonds of $X$. The bond $b\in X$ belongs to the boundary $\partial X\subset X$ iff $b\in p$, where $p$ is a plaquette such that $p\not\subset X$. The bond $b\notin X$ belongs to the coboundary $\delta X\subset X^c$ iff $v(b)\cap v(X) \ne\emptyset$. We denote by $C(X)$ the number of connected components of the graph $X$. Let now $V\subset \vi$ be a finite subgraph without isolated vertices. We introduce the partition functions with free and wired boundary conditions by $$Z^f(V)=\sum_{X\subset V, X\cap \delta V^c=\emptyset} (e^{\beta}-1)^{|X|}q^{C(X)+|v_I(V)\setminus v(X)|},$$ $$Z^w(V)=\sum_{X\subset V, \pa V\subset X} (e^{\beta}-1)^{|X|}q^{C(X)-C(V)+|v_I(V)\setminus v(X)|}.$$ The following limits exist and are equal: $$\lim_{V\to \vi}(1/|V|)\ln Z^f(V)= \lim_{V\to \vi}(1/|V|)\ln Z^w(V)=f(\beta).$$ A coconnected subset $\gamma\subset\vi$ is called a {\it contour\/}, if it is a coboundary of some $X\subset\vi$. If $\gamma$ is finite, then either $X$ or $X^c$ is finite. The unique infinite component of $\vi \setminus \gamma$ is called the exterior of $\gamma$ and is denoted by $\text{Ext}(\gamma)$. We also introduce $V(\gamma)=\vi\setminus\text{Ext}(\gamma)$, and $\text{Int}(\gamma)=V(\gamma)\setminus \gamma$. For $b\in\delta X$ we introduce $d(b)$ as the number of endpoints of $b$, which belong to $X$, and we define the length of the contour $\gamma$ by $$||\gamma||=\sum_{b\in \gamma}d(b).$$ If $X$ is finite, then $\gamma$ is called a {\it contour of the free class\/}, and if $X^c$ is finite, then $\gamma$ is called a {\it contour of the wired class\/}. Note, that some of the contours belong to both classes. For each of the classes one introduces in the standard way the notions of compatible contours and external contours. For a family $\theta=\{\gamma_1, \dots,\gamma_n\}$ of mutually compatible external contours in $V$ we introduce $V(\theta)=\cup_iV(\gamma_i)$, $\text{Int}(\theta )=V(\theta)\setminus \theta$, $\text{Ext}(\theta )= \vi \setminus V(\theta)$, $\text{Ext}_V(\theta )= V \setminus V(\theta)$. With these definitions we obtain the following relations between the partition functions:$$ Z^f(V)=\sum_{\theta _f\subset V} q^{|v_I(V\setminus \text{Int}(\theta _f))|} Z^w(\text{Int}(\theta _f)),\Eq(statf)$$ where the sum is over the families $\theta _f$ of mutually compatible external $f$-contours in $V$, and $$Z^w(V)=\sum_{\theta _w\subset V} (e^{\beta}-1)^{|\text{Ext}_V(\theta _w)|}Z^f(V(\theta _w)),\Eq(statw)$$ where the sum runs over the families $\theta _w$ of mutually compatible external $w$-contours in $V$, which do not intersect with the boundary $\pa V$. \medskip {\it A contour model\/} is specified by assigning weights $\varphi (\gamma)$ to contours. The corresponding partition function is defined by $$\Cal Z(V|\fii)=\sum_{\pa\subset V}\prod_{\gamma\in\pa} \fii(\gamma),\Eq(statk)$$ where the sum is over admissible families $\pa$ of contours in $V$. We are going to consider contour models both for $f$- and $w$-contours; in the first case admissibility means that contours $\ga_f$ are compatible and are in $V$, while in the second case it means that contours $\ga_w$ are compatible, are in $V$ and moreover $\ga_w\cap \pa V=\emptyset$. For every family $\pa$ of admissible contours we introduce the subset $\theta(\pa)\subset \pa$ as the collection of all external contours in $\pa$. Evidently, $$\Cal Z(V|\fii)=\sum_{\theta\subset V}\prod_{\gamma\in\theta} \fii(\gamma)\Cal Z(\text{Int}(\ga)|\fii),\Eq(statk1)$$ where the summation is over all families $\theta$ of external contours. We are going to consider the probability distribution $\nu_{V,\fii}$ on the ensemble of the admissible contours in $V$, corresponding to the contour functional $\fii$. Namely, we define the probability to observe the family $\pa$ by $$\nu_{V,\fii}(\pa)=\frac{\prod_{\gamma\in\pa} \fii(\gamma)}{\Cal Z(V|\fii)}.\Eq(raspk1)$$ By applying the Peierls transformation one gets immediately from this definition, that the probability of a given contour $\ga$ to appear in $V$ satisfies the Peierls estimate: $$\nu_{V,\fii}\{\pa:\ga\in\pa\}\le\fii(\ga).\Eq(peierls)$$ {\it The contour model with a parameter\/} $a\ge 0$ is defined by the following partition function: $$\Cal Z(V|\fii,a)=\sum_{\theta\subset V}\prod_{\gamma\in\theta} e^{a|V(\ga)|}\fii(\gamma)\Cal Z(\text{Int}(\ga)|\fii),\Eq(statkp)$$ where the sum runs over all families $\theta$ of external contours. We introduce also the probability distribution $\nu_{V,\fii,a}(\pa)$ for the contours of the contour model with parameter by modifying the definition \equ(raspk1) in an obvious way. The important difference is that once $a>0$, then the estimate \equ(peierls) is no longer valid in general. \addition{Paragraph added} A contour model with parameter is in fact associated to an ``unstable phase'' or ``wrong'' boundary condition. The presence of a parameter $a>0$ favors the formation of a ``large'' contour representing a flip into a ``stable phase'', taking place very close to the boundary (Lemma 1 below). The advantage of contour models lies in the fact that they can be treated by means of the cluster expansion technique. However, that is possible only for those contour models, whose contour functional satisfies the estimate \addition{Absolute value removed from the following formula. Functionals are assumed to be real and positive in this paper; the proof of Lemma 1 crucially depends on this fact. Disagreed. Later we have positivity. But here it can be general.} $$|\fii(\ga)|\le e^{-\tau ||\ga||},$$ with $\tau$ reasonably big. In that case the functional is called a $\tau$-functional, following [PS1], [PS2], [Sin]. This ensures the existence of the free energy per bond:$$f(\fii)= \lim_{V\to \vi}(1/|V|)\ln \Cal Z(V|\fii).$$ Actually, it implies much more. Namely, one has the following formula for the partition function: $$\ln \Cal Z(V|\fii)=\sum_{B\subset V}\phi(B),$$ where the sum runs over all connected subsets of $V$, and $\phi$ is a $\fii$-dependent function, which satisfies the bound $$\phi(B)\le e^{-\frac{\tau}{2} d(B)},$$ where $d(B)$ is the number of bonds in the smallest connected set which contains all boundary bonds of $B$. In particular, one has the following formula for the logarithm of the partition function: $$\ln \Cal Z(V|\fii)=|V|f(\fii)+\sum_{b\in\pa V}g_{\fii}(b,V),\Eq(grch)$$ where the function $g_{\fii}(b,V)$ is defined for every pair consisting of a bond $b$ and a box $V$, such that $b\in\pa V$, and has the following regularity properties: $$|g_{\fii}(b,V)|\le Ce^{-\frac{\tau}{2}},\Eq(grch1)$$ $$|g_{\fii}(b,V_1)-g_{\fii}(b,V_2)|\le Ce^{-\frac{\tau}{2}\dist(b,V_1\triangle V_2)}\Eq(grch2)$$ for $b\in \pa V_1\cap \pa V_2$, where $V_1\triangle V_2$ stands for the symmetric difference. (The above statements are standard from the point of view of the theory of cluster expansions and can be found, for example, in [DKS], sect. 3.11.) In [LMMRS] the contour functionals, which describe the FK model (in a sense which will be explained later) were constructed. We will need the following result, which is part of the main result of [LMMRS]: \proclaim{Theorem A} Consider the two-dimensional FK model for the $q$-state Potts model, $q$ being large enough, in the regime when $\be<\be_{cr}(q)$. Then there exist $\tau$-functionals $\fii_f, \fii_w$ and a real parameter $a=a(\be)>0$ such that $$Z^f(V)=q^{|v_I(V)|}\Cal Z(V|\fii_f),\Eq(f)$$ $$Z^w(V)=(e^{\beta}-1)^{|V|}\Cal Z(V|\fii_w,a).\Eq(w)$$ The following relations hold: $$a+\ln (e^{\be}-1)+f(\fii_w)=\frac 12 \ln q+f(\fii_f)=f(\be),\Eq(sootn)$$ $$\fii_f(\ga_f){\Cal Z}(\text{\rm Int}(\ga_f)|\fii_f)= q^{-|v(\text{\rm Int}(\ga_f))|}Z^w(\text{\rm Int}(\ga_f)), \Eq(indf)$$ $$\fii_w(\ga_w){\Cal Z}(\text{\rm Int}(\ga_w)|\fii_w)= e^{-a|V(\ga_w)|}(e^{\be}-1)^{-|V(\ga_w)|}Z^f(V(\ga_w)). \Eq(indw)$$ The parameter $\tau$ can be chosen arbitrarily large, provided $q$ is sufficiently large. \endproclaim The relation between the contour models and the initial FK model comes from comparing the formulas \equ(statf), \equ(statw) with \equ(statk), \equ(statkp), \equ(f) and \equ(w): the distribution of the external contours of the FK model in the box $V$ with free b.c. coincides with the distribution of the external contours in $V$ defined by the contour model with contour functional $\fii_f$, while that of the FK model with wired b.c. coincides with the distribution of the contour model with the functional $\fii_w$ and parameter $a$. Indeed, in both cases the partition function is written as a sum of products of terms, corresponding to compatible external contours. Since the formulas \equ(statf), \equ(statw), \equ(statk), \equ(statkp), \equ(f) and \equ(w) are valid for all volumes, it implies that the factors corresponding to external contours are actually the same. \subheading{3.3. The boundary clusters} We are ready now to rewrite the ratio \equ(otn2) with the help of the partition functions introduced above. We will consider first the case when $\la$ is the square box $\la(l)$. Let $n\in\Cal B_{\Lambda(l),\eta}$, and consider all open clusters $K$ of $n$, which have sites in $\la(l)^c$. Such clusters will be called {\it boundary clusters\/}. By $\Cal K=\Cal K(n)$ we denote the collection of all boundary clusters $K$ of $n$. The set of all possible collections of boundary clusters $\Cal K$ of configurations in $\Cal B_{\Lambda(l),\eta}$ will be denoted by $\Cal S_\eta$. Denote by $O=O(\Cal K)$ the complement $$O=\vi_{\Lambda(l)}\setminus \cup _{K\in \Cal K}K.$$ It is immediate to see that $$Z^{FK}_{\Lambda(l),\eta,T}=\sum_{\Cal K\in \Cal S_\eta} Z^f(O(\Cal K)) (e^\beta-1)^{\sum_{ K\in \Cal K}|K|}.\Eq(split)$$ Let us introduce the shorthand notation $\la(l,l^p)$ for the annulus $\la(l)\setminus \la(l-l^p)$. Then for every configuration $n\in\Cal B_{\Lambda(l,l^p),\eta}$ we can introduce the set of its boundary clusters in the same manner as it was done above. This set splits into two families: the family $\Cal K$ of boundary clusters which are attached to the exterior boundary of the annulus $\la(l,l^p)$ and the family $\bar\Cal K$ of boundary clusters which are attached to the interior boundary of $\la(l,l^p)$ and are disjoint from the exterior one. The set of all such pairs $(\Cal K, \overline{\Cal K})$ will be denoted by $\widetilde{\Cal S}_\eta$. In the obvious notation one has the following analogue of the formula \equ(split): $$Z^{FK}_{\Lambda(l,l^p),\eta,T}=\sum_{(\Cal K,\overline{\Cal K}) \in \widetilde{\Cal S}_\eta} Z^f(O(\Cal K\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K\cup\overline{\Cal K}}|K|}. \Eq(split1)$$ \addition{The paragraph until formula (3.27) has been substantially rewritten. I prefer to avoid the word ``distance'' because, rigorously speaking, the ``distance'' from a boundary cluster to the boundary is zero. I replace it with ``height''} Let us introduce the subset $\Cal S_\eta '\subset \Cal S_\eta$ formed by all families $\Cal K$, such that every $K\in \Cal K$ has a {\it height}\/ $$\aligned\thick(K, \partial \Lambda(l)) &\bydef \max\Bigl\{\dist\bigl(u,\partial\Lambda(l)\bigr) \,:\, u\in K\Bigr\}\\ &\le l^p/3\;. \endaligned$$ In the same way we define the subset $\widetilde{\Cal S}_\eta ' \subset \widetilde{\Cal S}_\eta$ as the collection of all pairs $(\Cal K,\overline{\Cal K})$ with heights $\thick(K, \partial \Lambda(l))\le l^p/3$ and $\thick(\overline{K}, \partial \Lambda(l-l^p))\le l^p/3$. If we denote by $\overline{\Cal S}_\eta '$ the set of all families $\overline{\Cal K}$ of boundary clusters $\overline{K}$ satisfying the last restriction, then clearly $$\widetilde{\Cal S}_\eta '=\Cal S_\eta '\times \overline{\Cal S}_\eta '.\Eq(s)$$ Suppose now for a moment that we are able to show that \addition{In next three formulas $\tau/10$ has been replaced by $\widetilde\tau$. The ``10'' looked a bit arbitrary} $$Z^{FK}_{\Lambda(l),\eta,T}=\biggl[\sum_{\Cal K\in \Cal S_\eta '} Z^f(O(\Cal K)) (e^\beta-1)^{\sum_{ K\in \Cal K}|K|}\biggr] (1+Ce^{-\widetilde\tau l^p}),\Eq(split')$$ and that $$Z^{FK}_{\Lambda(l,l^p),\eta,T}= \biggl[\sum_{(\Cal K,\overline{\Cal K})\in \widetilde{\Cal S}_\eta '} Z^f(O(\Cal K\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K\cup\overline{\Cal K}}|K|}\biggr] (1+C'e^{-\widetilde\tau l^p}), \Eq(split1')$$ where the constants $C=C(l,p,\eta),C'=C'(l,p,\eta)$ are uniformly bounded in $l$ and $\eta$, and $\widetilde\tau=\widetilde\tau(\tau)>0$ is independent of $l$ and $\eta$. We claim that in such a case the relation \equ(otn0) follows from the expansion \equ(grch) and the relation \equ(grch2). Indeed, let us insert the expansions \equ(split') and \equ(split1') into \equ(otn2), with $\la=\la(l), A=\la(l^p)$. Using \equ(s), we have \addition{The summation range of the following formula has been modified} $$\aligned &\frac{Z^{FK}_{\Lambda(l,l^p),\eta^1,T}Z^{FK}_{\Lambda(l),\eta^2,T}} {Z^{FK}_{\Lambda(l),\eta^1,T}Z^{FK}_{\Lambda(l,l^p),\eta^2,T} }=\\ % &\ \frac{\dsize\sum_{\scriptstyle \Cal K_1\in \Cal S_{\eta_1}' , \Cal K_2\in \Cal S_{\eta_2} '\atop \scriptstyle \overline{\Cal K}\in \overline{\Cal S}_{\eta_1'} (=\overline{\Cal S}_{\eta_2'})} %{\Cal K_1,\Cal K_2\in \Cal S_\eta ',\overline{\Cal K} %\in \overline{\Cal S}_\eta '} Z^f(O(\Cal K_1\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K_1\cup\overline{\Cal K}}|K |}Z^f(O(\Cal K_2)) (e^\beta-1)^{\sum_{ K\in \Cal K_2}|K|}} {\dsize\sum_{\scriptstyle \Cal K_1\in \Cal S_{\eta_1}' , \Cal K_2\in \Cal S_{\eta_2} '\atop \scriptstyle \overline{\Cal K}\in \overline{\Cal S}_{\eta_2'} (=\overline{\Cal S}_{\eta_1'})} %{\Cal K_1,\Cal K_2\in \Cal S_\eta ',\overline{\Cal K} %\in \overline{\Cal S}_\eta '} Z^f(O(\Cal K_1)) (e^\beta-1)^{\sum_{ K\in \Cal K_1}|K|}Z^f(O(\Cal K_2\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K_2\cup\overline{\Cal K}}|K |}}\\ % &\ \ \quad {}\times (1+C''e^{-\widetilde\tau l^p}).\endaligned $$ Consider the ratio of the corresponding terms: $$\frac{Z^f(O(\Cal K_1\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K_1\cup\overline{\Cal K}}|K |}Z^f(O(\Cal K_2)) (e^\beta-1)^{\sum_{ K\in \Cal K_2}|K|}}{Z^f(O(\Cal K_1)) (e^\beta-1)^{\sum_{ K\in \Cal K_1}|K|}Z^f(O(\Cal K_2\cup\overline{\Cal K})) (e^\beta-1)^{\sum_{K\in\Cal K_2\cup\overline{\Cal K}}|K |}}.$$ Note that the total sets of the boundary clusters $K$, appearing in the numerator or in the denominator, are the same, and each is equal to $\Cal K_1\cup\Cal K_2\cup\overline{\Cal K}$. Hence all the factors $(e^\beta-1)^{\sum_{*}}$ cancel out. Now, the sets $O(\Cal K_*\cup\overline{\Cal K}), O(\Cal K_*)$ are in general not connected, so the corresponding partition functions split into products, and the factors which appear both in the numerator and in the denominator also cancel. A moment's thought leads to the conclusion that what is left equals the ratio $$\frac{Z^f(\widetilde O(\Cal K_1\cup\overline{\Cal K})) Z^f(\widetilde O(\Cal K_2)) }{Z^f(\widetilde O(\Cal K_1))Z^f(\widetilde O(\Cal K_2\cup\overline{\Cal K}))},$$ where $\widetilde O(\Cal K_*\cup\overline{\Cal K})$, $\widetilde O(\Cal K_*)$ are those connected components of the sets $O(\Cal K_*\cup\overline{\Cal K})$, $O(\Cal K_*)$, which contain the whole ``middle level'', i.e. the set $\pa \la(l-\frac 12 l^p).$ The application of the expansion \equ(grch) and the relation \equ(grch2) implies immediately, that the last ratio is equal to $1+Ce^{-\widetilde\tau l^p}$ with $C=C(\Cal K_1,\Cal K_2,\overline{\Cal K},l,p)$ uniformly bounded in $\Cal K_1,\Cal K_2,\overline{\Cal K},l$, which proves our statement \equ(otn0). The above argument shows, that the only things that remain to be proven are the relations \equ(split'), \equ(split1'). We will do this in the next subsection. The reason why our project is bound to succeed is that above the critical temperature $\be_{cr}^{-1}(q)$ the FK model (as well as the Potts model) has a unique state --- the chaotic one --- which is characterized by the appearance of a large amount of small connected clusters. So the boundary conditions, fixed around some box $V$, are unable to influence the behavior of the system in the bulk. More precisely, no matter which boundary conditions we choose, there will be a contour in the vicinity of the boundary $\pa V$, separating the boundary influenced behavior outside it from the chaotic one inside. We start the rigorous proof of this picture by considering the wired b.c. In that case the formula \equ(w) tells us that the corresponding distribution of the external contours coincides with the one for the contour model with parameter. In light of that the appearance of the following statement is natural: \proclaim{Lemma 1. Estimate on the volume of the unstable phase} \noindent Let $\theta_w=\{\gamma_1, \dots,\gamma_n\}$ be a family of mutually compatible external $w$-contours in $V$. Consider the event that the contours $\{\gamma_1, \dots,\gamma_n\}$ are the only external contours in the ensemble defined by the contour $\tau$-functional $\fii_w$ with parameter $a$. That is, we consider the probability distribution $$\nu_{V,\fii_w, a}(\theta_w)= \nu_{V,\fii_w, a}(\gamma_1, \dots,\gamma_n)= \frac{\prod_1^n e^{a|V(\ga_i)|}\fii_w(\gamma_i)\Cal Z(\text{\rm Int}(\ga_i)|\fii_w)} {\Cal Z(V|\fii_w,a)}.\Eq(raspk)$$ Introduce the random variable $u_V=u_V(\theta_w) =|\text{\rm Ext}_V(\theta_w )|$. Then $$\nu_{V,\fii_w, a}(u_V\ge N)\le\exp \{ -aN+C|\pa V|\},\Eq(N)$$ where $C=C(\tau,\be).$ \endproclaim \demo{Note} It is worth noting that our statement {\it does not\/} hold for an arbitrary contour model with parameter, even for large $\tau$. The reason is that when one discusses general contour models, one asks for the upper bound $|\fii(\ga)|\le e^{-\tau ||\ga||}$ only, and so one does not rule out the possibility that $\fii(\ga)$ is actually much smaller and even vanishes for some contours. But in such a case the number of sites in the box $V$ which stay outside all external contours is of the order of $|V|$, and the estimate \equ(N) breaks down. However for the situation at hand we have also the lower bound \addition{Absolute value removed from next formula} $$\fii(\ga)\ge e^{-\bar\tau ||\ga||}\Eq(1)$$ for some real $\bar\tau$, and this is enough to prove the estimate \equ(N). \enddemo \demo{Proof of Lemma 1} The idea of the proof of the upper bound is to replace the partition function in the denominator of \equ(raspk) by a lower bound which has the form of one of the factors of the numerator of \equ(raspk). To do this we consider the collection $\Theta_w(V) =\{\Gamma_1, \dots,\Gamma_k, k=k(V)\}$ of mutually compatible external $w$-contours in $V$ which minimizes the variable $u_V$. It is clear that $u_V(\Theta_w(V))=C|\pa V|$ for some $C$. Then $$\aligned &\nu_{V,\fii_w, a}(u_V\ge N) =\sum_{\theta _w\subset V:u_V(\theta_w)\ge N}\nu_{V,\fii_w, a}(\theta_w)=\\ &\frac{\sum_{\theta _w\subset V:u_V(\theta_w)\ge N}\prod_{\ga\in\theta _w} e^{a|V(\ga)|}\fii_w(\gamma)\Cal Z(\text{Int}(\ga)|\fii_w)} {\Cal Z( V|\fii_w,a)}\le\\ &\frac{\sum_{\theta _w\subset V:u_V(\theta_w)\ge N}\prod_{\ga\in\theta _w} e^{a|V(\ga)|}\fii_w(\gamma)\Cal Z(\text{Int}(\ga)|\fii_w)} {e^{a(|V|-u_V(\Theta_w(V)))} \prod_{\Gamma\in\Theta _w(V)}\fii_w(\Gamma)\Cal Z(\text{Int}(\Gamma)|\fii_w)}\le\\& \le e^{a(u_V(\Theta_w(V))-N)}\frac{1}{\prod_{\Gamma\in\Theta _w(V)}\fii_w(\Gamma)} \frac{\Cal Z(V|\fii_w)}{\prod_{\Gamma\in\Theta _w(V)}\Cal Z(\text{Int}(\Gamma)|\fii_w)}. \endaligned$$ We now claim that each of the last two factors admits an upper bound of the order of $\exp \{ C|\pa V|\}$ for some $C$. For the last factor this follows from the expansion \equ(grch), since the complement $V\setminus \cup_{\Gamma\in\Theta _w(V)} \text{Int}(\Gamma)$ is contained in the neighborhood of $\pa V$ of radius 2. For the first one we use \equ(indw) and \equ(f) to express the contour functional $\fii_w$ via partition functions $\Cal Z(*|\fii_w), \Cal Z(*|\fii_f)$ of contour models (with no parameters). We obtain that $$\fii_w(\ga_w)= e^{-a|V(\ga_w)|}(e^{\be}-1)^{-|V(\ga_w)|}q^{|v_I(V(\ga_w))|}\frac{\Cal Z(V(\ga_w)|\fii_f)} {\Cal Z(\text{Int}(\ga_w)|\fii_w)}. $$ We then use the expansion \equ(grch) to write each partition function as an exponent of the volume term and the boundary term and the relation \equ(sootn) to observe that all volume terms cancel out. (The fact that we are dealing not with just an abstract contour model, but with a specific one which admits the lower bound \equ(1) on the contour functional is made explicit by our use of the relation \equ(indw), which implies in particular the strict positivity of the contour functional.) $\blacksquare$ \enddemo \subheading{3.4. Fingers of the boundary clusters and their surgeries} In what follows we are proving the relation \equ(split1') for the case of the square box $\la(l)$. The relation \equ(split') is easier and can be proven by the same argument with simpler notation. In the following statement we estimate the probability of the event that the boundary cluster goes deep inside the box. \addition{The beginning sentence of the theorem has been slightly rewritten and (3.34) corrected to incorporate the factor $c$ (now $C$)} % \proclaim{Lemma 2. Estimate of the probability of a long finger} \noindent Let $q$ be such that Theorem A above holds. Fix a real number $0

0$ and $b>0$ is an absolute constant (e.g.~$1/6$).
\endproclaim
\demo{Proof of Lemma 2} The idea of the proof is to study ``fingers'', which are
protruding parts of
the boundary clusters. The finger can be either attached to the exterior
boundary of $\la(l,l^p)$ or it joins the exterior and the interior boundaries of $\la(l,l^p)$.
If the finger is ``thin'' somewhere --- which means that its length is
of higher order that its thickness --- then one can cut across it,
obtaining an exterior contour of the length of the order $l^p$, which implies
the estimate
needed. If the finger is ``fat'' everywhere, that implies that the number of open bonds
inside it
is much larger than the perimeter, so one can hope to control the situation by using the
estimate \equ(N).
To implement this program we start by defining fingers and their parameters.
\addition{The definition of finger has been rewritten. I think is better not
to choose the bond $b$ beforehand. Agreed. Senya. Also, I take the bases to be sets of BONDS
and the finger to be a set of SITES (it is a cluster).
Disagreed. It has to be a set of bonds. Senya.}
For a boundary cluster $K$ and fixed numbers $0