instructions: paper with 3 figures. The figures come as part 2, 3 ,4/4. The figures are .ps files. They start with %!PS-Adobe-2.0 EPSF-1.2 and end with %%Trailer end. BODY \magnification 1200 \input amstex \documentstyle{amsppt} \TagsOnRight \NoRunningHeads \define\e{\eta} \define\s{\sigma} \define\reald{\Bbb R^d} \define\Zetad{\Bbb Z^d} \define\real2{\Bbb R^2} \define\Zeta2{\Bbb Z^2} \topmatter \title Agreement percolation and phase coexistence \\ in some Gibbs systems \endtitle \leftheadtext\nofrills {G. Giacomin, J.L. Lebowitz, C. Maes} \rightheadtext\nofrills {Percolation in Gibbs states} % % \author G. Giacomin, J.L. Lebowitz and C. Maes \endauthor \affil Department of Mathematics, Rutgers University and Instituut voor Theoretische Fysica, KU Leuven \endaffil \address Giambattista Giacomin \hfill\newline Department of Mathematics \hfill\newline Hill Center, Rutgers University \hfill\newline New Brunswick, N.J. 08903, U.S.A. \endaddress \email giacomin\@boltzmann.rutgers.edu \endemail \address Joel L. Lebowitz \hfill\newline Departments of Mathematics and Physics \hfill\newline Hill Center, Rutgers University \hfill\newline New Brunswick, N.J. 08903, U.S.A. \endaddress \email lebowitz\@math.rutgers.edu \endemail \address Christian Maes \hfill\newline Instituut voor Theoretische Fysica, KU Leuven \hfill\newline Celestijnenlaan 200D \hfill\newline B--3001 Leuven, Belgium \endaddress \email FGBDA35\@BLEKUL11.bitnet \endemail \keywords percolation, Gibbs measures, nonuniqueness, antiferromagnets, hard-core models, Widom--Rowlinson continuum model \endkeywords \abstract We extend some relations between percolation and the dependence of Gibbs states on boundary conditions known for Ising ferromagnets to other systems and investigate their general validity: percolation is defined in terms of the agreement of a configuration with one of the ground states of the system. This extension is studied via examples and counterexamples, including the antiferromagnetic Ising and hard core models on bipartite lattices, Potts, many layered Ising and continuum Widom--Rowlinson models. In particular our results on the hard square lattice model make rigorous observations made in Hu and Mak (1989) and (1990) on the basis of computer simulations. Moreover we observe that the (naturally defined) clusters of the Widom--Rowlinson model play (for the WR--model itself) the same role that the clusters of the Fortuin--Kasteleyn measure play for the ferromagnetic Potts models. The phase transition and percolation in this system can be mapped into the corresponding liquid--vapor transition of a one component fluid. \endabstract \endtopmatter \document \head 1. Introduction \endhead \baselineskip=25pt The low temperature phases of matter can generally be thought of as small thermal perturbations of corresponding ground states. This is particularly simple for the case of a classical lattice system whose configuration is specified by $\{\s(x)\}$ with $x$ on some regular lattice ${\Cal L}$, and $\s(x)$ a spin variable taking one of a finite number of values at each site. Given a local interaction having a finite number of periodic ground state configurations (PGSC), we can then take these PGSC as boundary conditions for the Gibbs measures in $\Lambda$, i.e. consider the Gibbs distribution at inverse temperature $\beta$ with boundary conditions given by a PGSC on $\Lambda^c$, the complement of $\Lambda$, of some box $\Lambda$ (see Section 2 for precise definitions). For fixed $\Lambda$ and $\beta > 0$ the boundary condition has a definite influence on the probability distribution of the spins in the bulk of $\Lambda$. In particular, in the limit $\beta \nearrow \infty$, the Gibbs measure in $\Lambda$ becomes concentrated on the extension inside $\Lambda$ of the PGSC imposed outside $\Lambda$. We are interested in knowing whether this influence persists for $\beta$ large but finite when we take the volume $\Lambda$ to be macroscopic. In other words, is there an ordered state in the infinite volume ($\Lambda\nearrow \Cal L$) system dependent upon the PGSC imposed as boundary conditions on the finite box $\Lambda$? \newline If there is such a {\it memory} of the state with respect to boundary conditions {\it at infinity}, one sometimes calls the corresponding ground state thermally stable. The well known Peierls argument provides such a result for the ferromagnetic Ising model and the Pirogov--Sinai theory, Pirogov and Sinai (1976), and its extensions study the genericity of this scenario. Under certain conditions, e.g. when there are a finite number of thermally stable periodic ground states, it allows one to construct the low temperature phase diagram of the system (for a review see Sinai (1982), (1986) Zahradn\'{\i}k (1984), Bricmont and Slawny (1985), Slawny (1986)). In this note, we investigate the geometric or percolation picture of this memory effect, considered by many authors (Russo (1979), Fortuin and Kasteleyn (1972), Georgii (1988)): to make precise the intuition that the influence of the boundary conditions must propagate via ``interacting sites'' from infinity to the center of the system if the corresponding non--zero temperature state is to be stable. We are thus led to the question: in what sense is the finite temperature state corresponding to a ground state accompanied by the presence of an infinite connected cluster on which the PGSC is realized. This question is fully answered at sufficiently low temperatures where the proof of the existence of different phases, determined by different GSC boundary conditions, via the Peierls argument or Pirogov--Sinai theory actually proves the existence in each such phase of just such a cluster and no other. Our interest here is therefore primarily the extension of well known low temperature results to higher temperatures where the Peierls and Pirogov-Sinai arguments fail. We are particularly interested in the question of whether, for two dimensional systems with particular symmetries, the existence of such an ordered state is equivalent to the percolation of that and only that ground state configuration. This is known for example in the case of the standard ferromagnetic Ising model without external magnetic field. As we show below, this picture extends to other models such as the antiferromagnetic Ising model, the hard square model and the Widom--Rowlinson model. A weaker statement is proven for the Potts model. However, we will also give examples (see Section 3(c) below) in which the existence of an ordered state does not imply percolation of the corresponding ground state. We emphasize that our setup is different from that in the Fortuin--Kasteleyn representation. Here the percolation clusters are defined directly in terms of the Gibbs state configuration. Therefore in general we do not expect to have direct relations between e.g. correlation functions in the Gibbs state and corresponding percolation probabilities. Still we will see that for the model in Section 4 such a relation can in fact be established. In the next Section we present the general framework. This is implemented by the examples of Section 3 in the case of lattice systems. Section 4 is devoted to the continuum Widom--Rowlinson model. \head 2. General framework \endhead We present the notation here in case of lattice systems. The continuum model is contained in Section 4. \definition{The lattice} We restrict our attention to the $d-$dimensional lattice $\Bbb Z^d$, $d\ge 2$. This restriction is made for notational convenience and possible generalizations will be noted later on. By $x\sim y$ we mean that $x$ and $y \in \Bbb Z^d$ are nearest neighbors. Given $\Lambda \subset \Bbb Z^d$, $\partial \Lambda$ will denote the outer boundary of $\Lambda $, i.e. $\partial \Lambda =\{x\in \Bbb Z ^d \setminus \Lambda : y \sim x \text{ for some } y\in \Lambda\}$. In the sequel $\Lambda$ is always a finite subset of $\Bbb Z ^d$. \enddefinition \definition{The configuration space} The single site state space is denoted by $S$ and it contains a finite number of elements ($\vert S \vert =q$). An infinite volume configuration $\s = \{\s(x)\}_{ x\in \Bbb Z^d } $ is an element of $\Omega=S^{\Bbb Z ^d}$. A configuration $\sigma $ is periodic if there is a $(k_1,\ldots , k_d) \in \Bbb Z ^d$ such that if $y=x+(n_1 k_1 ,\ldots ,n_d k_d)$ for some $(n_1, \ldots , n_d )\in \Bbb Z^d$, then $\sigma (x)=\sigma (y)$. \enddefinition \definition{The Hamiltonian and its ground states} The translation invariant Hamiltonian $H$ consists (for definiteness) of the nearest neighbor interaction $U:S\times S \rightarrow {\Bbb R}$ and the self--energy $V: S \rightarrow{\Bbb R}$ $$ H(\s)=H(\s ;J,h) = -J \sum_{x\sim y } U (\s(x),\s(y)) - h \sum_x V (\s(x)) \tag2.1 $$ in which the infinite sums are formal. The real parameters $J\ge 0$ and $h$ in (2.1) play the roles of a coupling coefficient and magnetic field (or chemical potential). Consider $\s ,\e \in \Omega$ such that $\sigma (x) =\eta (x)$ for all $x \in {\Bbb Z ^d}\setminus \Lambda$, for some finite $\Lambda\subset \Bbb Z ^d$. Then $\s$ and $\e$ are said to be {\it equal at infinity} or one is an {\it excitation} of the other. Their finite relative Hamiltonian $H(\s|\e)$ is then $$ \aligned H(\s|\e) &\equiv H(\s) - H(\e) \\ &\equiv -J \sum_{x\sim y } [U (\s(x),\s(y)) - U (\e(x),\e(y)) ] - h \sum_x [V(\s(x))-V(\e(x))] \endaligned \tag 2.2 $$ in which the sum is now only over a finite number of terms. A configuration $\e\in \Omega$ is a {\it ground state} if $H(\s|\e)\geq0$ for any excitation $\s$ of $\e$. The set of periodic ground state configurations (PGSC) is denoted by $g(H)$. \enddefinition \definition{The Gibbs measure} Take $\e \in g(H)$. The finite volume ($\Lambda$) Gibbs state with $\e$ boundary conditions is the probability measure on the product space $S^\Lambda$ defined by $$ \mu_{\beta ,\Lambda}^{\e }(\s_\Lambda) = \frac1{Z_\Lambda (\beta,\e)} \exp[-\beta H(\s_\Lambda^\e|\e)] \tag2.3 $$ for $\sigma _\Lambda\in S^\Lambda$. In (2.3) $\beta >0$ plays the role of the inverse temperature, $Z_\Lambda (\beta,\e)$ is the normalization constant and $$ \s_\Lambda^\e (x) = \cases \s _\Lambda (x)& \text{ if } x\in \Lambda\\ \e(x)& \text{ if } x\in \Lambda^c=\Bbb Z ^d \setminus \Lambda \endcases \tag2.4 $$ Observe that $\sigma _\Lambda^\e$ is an excitation of $\e$. The infinite volume Gibbs measure $\mu _\beta ^\eta$ is obtained taking the limit as $\Lambda \nearrow {\Bbb Z ^d}$ along suitable subsequences: $\mu ^\e _\beta$ is a measure on $\Omega$ endowed with the product sigma algebra (for details see Georgii (1988), Simon (1993)). By phase coexistence at $\beta$ we mean that $\mu ^\eta _\beta \not= \mu ^\sigma _\beta $ for some $\eta, \sigma \in g (H)$. \enddefinition \definition{Agreement with the PGSC} To understand the geometrical structure of such phases fix $\eta \in g(H)$ and the map $s_{\e }:\Omega \rightarrow \{0,1\}^{\Bbb Z ^d}$ expressing agreement or disagreement with the ground state $\eta$ $$ (s _{\eta } (\sigma))(x) = \cases 1 &\text{ if } \s (x) = \e(x) \\ 0 &\text{ if } \s(x) \neq \e(x) \\ \endcases \tag2.5 $$ We denote by $\nu _{\beta, \Lambda} ^\eta =\mu ^\e _{\beta, \Lambda} s^{-1}_{\eta}$ the measure on $\{0,1\}^\Lambda$ induced by the map $s_{\e }$ (with a slight abuse of notation). As before we can take the limit $\Lambda \nearrow \Bbb Z ^d$; write it as $\nu _\beta ^\eta$. \enddefinition \definition{Percolation of the ground states} Percolation is defined in terms of regular site percolation for the measure $\nu ^\e _\beta$: for $s \in \{0,1\}^{\Bbb Z ^d}$ define the clusters $$ C (s ,\alpha)=\{x \in {\Bbb Z ^d}: s (x)=\alpha \} \tag2.6 $$ in which $\alpha=0,1$. Call $\overline{C}(\s ,\alpha)$ the maximal connected component containing the origin of ${\Bbb Z ^d}$ ($\overline{C}(s ,\alpha)=\emptyset$ if $s (0)\neq \alpha$). The probability of $\eta-$percolation (respectively non $\eta-$percolation) in the phase with boundary condition $\eta$ is defined as $$ \theta ^\eta _\beta (\alpha)= \nu^\e _\beta (\{s: \vert \overline{C}(s ,\alpha)\vert =\infty\}) \tag2.7 $$ with $\alpha =1$ (respectively $\alpha =0$). We have $\eta-$percolation, or agreement percolation, (respectively non $\eta-$percolation) if $\theta ^\eta _\beta (1) >0$ (respectively $\theta ^\eta _\beta (0)>0$). Observe that not having $\eta-$percolation in general does not imply having non $\eta-$percolation. \enddefinition \definition{Attractive measures} If $S$ is ordered ($\ge$), $\Omega$ is ordered ($\succ$) by defining $\sigma ^\prime \succ \sigma $ if $\sigma ^\prime (x) \ge \sigma(x)$ for all $x \in \Bbb Z ^d$. The order chosen for $S$ may depend on the site $x$. A measurable event $A$ is said to be increasing if $\sigma \in A$ and $\sigma ^\prime \succ \sigma $ implies $\sigma ^\prime\in A$. A measure $\mu$ on $\Omega$ is attractive (FKG) if $\mu (A \cap B )\ge \mu (A) \mu (B)$ for all $A$, $B$ increasing events, see Fortuin {\it et al.} (1971). \enddefinition We are interested in the verification of three main statements: \newline $a)$ we are asking whether phase coexistence implies percolation, i.e. given $\eta , \eta ^\prime \in g(H)$ is it true that $$ \mu_\beta ^{\eta} \not= \mu_\beta ^{\eta^\prime } \Longrightarrow \ \ \theta ^\eta _\beta (1)>0 \tag2.8a $$ \newline $b)$ whether the reverse implication is true when $d=2$ $$ \theta ^\eta _\beta (1)>0 \ \ \Longleftrightarrow \ \ \mu_\beta ^{\eta } \not=\mu_\beta ^{\eta^\prime } \tag2.8b $$ \newline $c)$ still for $d=2$, can we rule out the possibility of disagreement percolation $$ \theta _\beta ^\eta (0)=0 \tag2.8c $$ For the models we will be dealing with, if $d>2$ the statement $(2.8b)$ fails or it is expected to fail (see Aizenman {\it et al.} (1987) and references therein). \head 3. Lattice models \endhead As already mentioned in the introduction, the case in which $S=\{-1,+1\}$, $U(a,b)=ab$ and $h=0$ (ferromagnetic Ising model) is well understood. In this case $g(H)=\{\e_+ ,\e_-\}$ ($\e_+(x)=+1, x\in \Bbb Z ^d$ and $\e_-=-\e _+$) and for $\beta >\beta _c$ ($1/\beta _c$ is the critical temperature) there are two distinct translation invariant extremal states obtained by taking $\eta _+$ and $\eta _-$ as boundary conditions and the proof of the statements $(2.8a)$, $(2.8b)$ and $(2.8c)$ can be found in Coniglio {\it et al} (1976). \subhead a) Ising antiferromagnet \endsubhead $\Bbb Z ^d$ is bipartite, that is it can be split into two sublattices (in this case the points with even sum of coordinates $\Bbb Z ^d_e$ and the ones with odd sum $\Bbb Z ^d _o$) such that if $x,y\in\Bbb Z ^d_e$ (or $x,y\in\Bbb Z ^d _o$) then $x \not\sim y$. Fix $S=\{-1,+1\}$ and take $$ U(a,b)=-ab \ \ \ \ \ \ \ \ \ V(a)=a \tag3.1 $$ If $|h|<2dJ$, then $g(H)=\{ \eta _e , \eta _o\}$ in which $\eta _e (x)=+1$ if $x\in \Bbb Z ^d_e$, $\eta _e (x)=0$ otherwise and $\eta _o =-\eta _e$ (see for example Dobrushin and Shlosman (1985)). The phase transition in this model has been studied in Dobrushin (1968) and Heilmann (1974). Because of the bipartite structure, flipping the spins on the even (or odd) sites makes the model into a ferromagnetic Ising model with a staggered magnetic field. In particular for magnetic field $h=0$, there is the usual Curie point $T_c$ (the critical temperature for the ferromagnetic model above). \proclaim{Proposition 3.1:} With the choices of (3.1) in (2.1), for any $h$ and any $J \ge 0$, statements (2.8a), (2.8b) and (2.8c) hold. \endproclaim \demo{Proof} Take $\e =\e_e$. For any $\Lambda $ containing the origin, we have that if $\mu _\beta ^{\eta _e}\not=\mu _\beta ^{\eta _o}$ $$ \nu_\beta ^{\eta _e} (s(0)=1 \vert s(x)=0 \text{ for all } x\in {\partial \Lambda} )= $$ $$ \mu _\beta ^{\eta _e} (\sigma (0)=+1 \vert \sigma (x)=\eta _o(x), \text{ for all } x \in \partial \Lambda)= $$ $$ \mu _\beta ^{\eta _o} (\sigma (0)=+1 \vert \sigma (x)=\eta_o(x), \text{ for all } x \in \partial \Lambda) \le \tag3.2 $$ $$ \mu_\beta ^{\eta _o} (\sigma (0)=+1) < \nu ^{\eta _e} _\beta (s (0)=+1) $$ The first equality is a change of notation and the second one is the Markov property. The first inequality follows from the attractivity of the measure $\mu_\beta ^{\eta _o}$ with respect to the order relation $\sigma \succ \sigma ^\prime$ iff $\sigma (x)\eta _e(x) \ge \sigma ^\prime (x) \eta _e (x)$ for all $x \in \Bbb Z ^d$. By Theorem 2 of Bricmont {\it et al.} (1987) we have $(2.8a)$. \newline If $\mu _\beta ^{\eta _e}=\mu _\beta ^{\eta _o}$ then, by the fact that $\mu _\beta ^{\eta _e} T_i^{-1}=\mu _\beta ^{\eta _o}$ ($T_i \sigma (x)=\sigma (x-e_i)$, $e_i$ is the unit vector in the direction $i\in \{1 ,\ldots ,d\}$) we have that $\nu _\beta ^{\eta _e}=\nu _\beta ^{\eta _o}\equiv \nu _\beta$. This implies that $\nu _\beta $, besides being attractive (in the usual order), is reflection invariant and ergodic under translations. Hence if $d=2$ the main Theorem of Gandolfi {\it et al.} (1988) applies and $(2.8b)$ is proven. If $\mu _\beta ^{\eta _e}\not=\mu _\beta ^{\eta _o}$, $\mu _\beta ^{\eta }$'s are not $T_i-$invariant, but only $T_i^2-$invariant. Hence we have to use an extension (Klein and Yang (1993)) of the result in Gandolfi {\it et al.} (1988) to get $(2.8c)$. \qed \enddemo The results of Proposition 3.1 can be extended to more general bipartite lattices. \subhead b) Hard core lattices \endsubhead The hard core (or hard squares) lattice model with activity $\lambda$ is defined by the infinite volume limit ($\Lambda \nearrow \Bbb Z^d$) of the measure $\mu _{\lambda , \Lambda}$ on the product space $\{0,1\}^\Lambda $ $$ \mu_{\lambda , \Lambda}^\gamma (\sigma_\Lambda)= {\chi (\sigma_\Lambda^{\eta_\gamma}) \lambda ^{N_\Lambda} \over Z_\gamma (\lambda , \Lambda)} \tag3.3 $$ In (3.3) $\gamma \in \{e,o\}$ and $\eta _e(x)=1$ if $x$ is even, $\eta _e(x)=0$ if $x $ is odd ($\eta _o(x) =1- \eta _e(x)$ for all $x$). Further, $N_\Lambda= \sum_{x\in \Lambda} \sigma_\Lambda^{\eta_\gamma} (x)$ with $\sigma _\Lambda ^\eta$ as defined in (2.4) ; $Z_\gamma (\lambda ,\Lambda)$ is a normalization and for $\sigma \in \{0 ,1 \} ^{\Bbb Z ^d}$ $$ \chi(\sigma)= \cases 1 & \text{ if } \sigma (x) \sigma (y)=0 \text{ for all } x\sim y \\ 0 & \text{ otherwise} \\ \endcases \tag3.4 $$ Hence the model can be seen as a gas of hard (i.e. non overlapping) squares (see Fig.1) or diamonds with fugacity $\lambda$. Call $C(\eta)=\cup _{x:\eta (x)=1} Q(x)$ ($Q(x)=\{y \in \Bbb R ^d : \sum _{i=1} ^d \vert y_i -x _i\vert =1\}$) and denote by $C_0 (\eta )$ the connected component of $C(\eta )$ that contains the origin (two squares touching at a corner are connected, see Fig.1). There are clearly two different type of connected components of $C$ : the ones for which the squares are centered on even sites and the ones for which they are centered on odd sites. We will call them type $e$ and type $o$ clusters. For $\gamma , \delta \in \{ e ,o\}$ define $$ \theta_\lambda ^{\gamma} (\delta )= \mu ^\gamma _\lambda (\{ \eta :C_0(\eta) \text{ is of type } \delta \text{ and unbounded}\}) \tag3.5 $$ (3.5) is clearly the analogue of (2.7). The hard core model can be seen as a limit of the antiferromagnetic Ising model for $\beta \rightarrow \infty$ and $h\rightarrow 2dJ$ along $\beta (h-2dJ)=\text{cotan} \theta$ ($\theta \in (0, \pi)$). The phase diagram point $(2dJ,0)$ is highly degenerate, since there are infinitely many (in general nonperiodic) ground states. In these limits map $\sigma \in \{-1, +1 \}^{\Bbb Z ^d} \rightarrow \eta \in \{ 0,1 \}^{\Bbb Z ^d}$ by setting $\eta (x)=1$ if $\sigma (x)=-1$ and $\eta (x)=0$ if $\sigma (x)=+1$. We get a hard square model with activity $\lambda = \exp (-2 \text{cotan} \theta )$ (see Dobrushin {\it et al.} (1985) for details). This picture suggests that the critical fugacity (if it exists) should correspond to $\theta _c$ (see Fig.2). We rephrase $(2.8a)$, $(2.8b)$, $(2.8c)$ into \proclaim{Proposition 3.2} For the hard square model $$ \mu _\lambda ^e \not= \mu _\lambda ^o \Longrightarrow \theta^e _\lambda (e) >0 \tag3.6a $$ Moreover if $d=2$ $$ \theta^e _\lambda (e) >0 \Longleftrightarrow \mu _\lambda ^e \not= \mu _\lambda ^o \tag3.6b $$ and $$ \theta_\lambda ^e(o)=0 \tag3.6c $$ \endproclaim \demo{Proof} Define the partial order $\eta ^\prime \succ \eta$ if $\eta ^\prime (x) \ge \eta(x)$ for $x\in \Bbb Z^d _e$ and $\eta ^\prime (x) \le \eta(x)$ if $x \in \Bbb Z^d _o$. It is easy to check that $\mu _{\lambda, \Lambda}$ and its infinite volume limit are attractive. The proof is then the same of the one of Proposition 3.1. \qed \enddemo \remark{Remark} The motivation for Proposition 3.2 was provided by the work of C.K. Hu and K.S. Mak (Hu and Mak (1989) and (1990)) in which a similar result is conjectured on the basis of computer simulations. In Hu and Mak (1989) and (1990) they discuss also the case of hard core particles on a triangular lattice, the hard hexagon model. Our result extends easily to other bipartite lattices, such as the hexagonal one. The triangular lattice with nearest neighbor bonds is not bipartite, so our proof does not work. In view of Hu and Mak (1989) and (1990), one expects that Proposition 3.2 still holds for this model, but it is unclear to us, especially in view of the results of the Section 3.d on {\sl many layer} Ising models, whether Proposition 3.1 ($d=2$) holds for the whole domain of coexisting phases of the antiferromagnetic Ising model on the triangular lattice (in this case, for $h \in (0,6J)$, $g(H)$ contains three configuration). \endremark \subhead c) {\sl Many layer} models \endsubhead Given a model with configuration space $\{-1, +1\}^{\Bbb Z ^d}$ and Hamiltonian $H_1$ we can define a family of new models indexed by integers $Q\ge 2$. Take $S=\{-1 ,+1\}^Q$ so that the infinite volume configuration is of the form $\xi =(\sigma _1,\ldots ,\sigma _Q)\in S^{\Bbb Z^d}$ (to be identified with $\{-1, +1\}^{\Bbb Z^d} \times \ldots \times \{-1, +1\}^{\Bbb Z^d}$ ($Q$ copies)). Define the formal Hamiltonian as $$ H(\xi)=\sum _{i=1}^Q H_1(\sigma _i) \tag3.7 $$ so that for boundary conditions $\omega= (\eta _1 , \ldots ,\eta _Q)$, $\mu ^\omega _{\beta, \Lambda}= \mu^{\eta _1}_{1,\beta ,\Lambda} \times \ldots \times \mu^{\eta _Q}_{1,\beta ,\Lambda}$ (where the subscript 1 refers to the system with Hamiltonian $H_1$) is the finite volume Gibbs state with respect to $H$. Observe that $g(H)=(g(H_1))^Q$ (with the previous identification). \subsubhead Duplicated Ising model \endsubsubhead Take $Q=2$ and $H_1$ as in (2.1), characterized by $U(a,b)=ab$ and $h=0$. As observed before $g(H_1)=\{ \eta _+ ,\eta _-\}$ and so $g(H)=\{ \omega _{++}, \omega _{+-}, \omega _{-+}, \omega _{--}\}$, where $\omega _{++}= (\eta _+ ,\eta _+)$ and so on. \proclaim{Proposition 3.3} For the duplicated Ising model statements (2.8a), (2.8b) and (2.8c) hold. \endproclaim \demo{Proof} We want to estimate the expectation value of the sum of the spins at the origin given that $\xi (y)=(\sigma _1(y), \sigma _2(y))\not= (+1, +1)$ for $y \in \partial \Lambda$. In that case $\sigma _1 (y)+ \sigma _2(y) \le 0$ for $y\in \partial \Lambda$, so that by attractivity $$ \mu_\beta ^{\omega _{++}} (\sigma _1(0)+\sigma _2(0) \vert s_{\omega _{++}} (\xi)=0 \text{ on } \partial \Lambda ) \le $$ $$ \mu_\beta ^{\omega_{++}} (\sigma _1(0)+\sigma _2(0)\vert (\sigma _1, \sigma _2): \sigma _2(y)=-\sigma _1(y) \text{ for all } y \in \partial \Lambda) = \tag3.8 $$ $$ \sum_{\sigma\in \{-1,+1\}^{\partial \Lambda }} \Big\{ \left[\mu _{1, \beta} ^{\sigma} (\sigma _1(0))+ \mu _{1,\beta} ^{-\sigma} (\sigma _2(0))\right] \cdot $$ $$ \mu_\beta ^{\omega _{++}} (\xi(x)=(\sigma_1(x),-\sigma_1(x)), x\in\partial\Lambda \vert s_{\omega _{++}} (\xi) = 0 \text{ on } \partial \Lambda )\Big\} = 0 $$ On the other hand if $\mu_{1, \beta }^{\eta _+} \not= \mu _{1, \beta}^{\eta _-}$ $$ \mu ^{\omega _{++}}_\beta (\sigma _1(0)+\sigma _2 (0)) \equiv 2m^{\star }(\beta)>0 \tag3.9 $$ Apply Theorem 2 of Bricmont {\it et al.} (1987) to get $(2.8a)$. For what concerns $(2.8b)$ and $(2.8c)$, it is straightforward to see that $\nu ^{\omega _{++}}_\beta= \mu ^{\omega _{++}}_\beta s_{\omega _{++}} ^{-1}$ is reflection invariant, ergodic under translation and attractive. Hence we can apply Gandolfi {\it et al.} (1988) to get $(2.8b)$ and $(2.8c)$. \qed \enddemo \remark{Remark} Note that as the phase transition is second order ($\mu^{\eta _+}_{1,\beta}(\s(0))^+ \equiv m^{\star}(\beta)$ is continuous at $\beta=\beta_c$), the density of $(+1,+1)$ just below the critical temperature is only slightly above 1/4 and still the $(+,+)$ spins percolate in the $(+,+)-$ state\footnote{The threshold for Bernoulli site percolation on $\Bbb Z^2$ is about 0.59}. In the same way, the density of sites $x\in\Bbb Z^2$ where $(\s_1(x),\s_2(x))\neq(+,+)$ is there only slightly below 3/4 and still, in the $(+,+)-$ state, they do not percolate. \endremark The question therefore arises whether one can go arbitrarily far and construct examples where there is percolation for arbitrarily low densities or where there is no percolation no matter how large one makes the density. Such examples in fact exist (see for example Molchanov and Stepanov (1983)) but they are rather singular. It may well be that a minimum density for having percolation actually exists for {\it good} Markov fields. \proclaim{Conjecture} Given $\mu$ on the product space $S^{\Bbb Z^d}$ ($\vert S \vert =q <\infty$), a translation invariant pure Gibbs state for some isotropic nearest neighbor interaction, there is a constant $c(d)>0$ (independent of $q$) such that, if $\mu (\sigma (0)=a) \beta_c$ for which there is phase coexistence. There is $Q(\beta,d)$ such that for all $Q\ge Q(\beta,d)$ $$\theta ^{\omega}_\beta (1)=0 \tag3.10 $$ \endproclaim \demo{Proof} By direct computation $$ \max_{\xi^\prime} \mu^{\omega}_\beta ( \xi (x)=(1,...1)\vert \xi^\prime (y), y\sim x)= \left({1 \over 1+\exp(-4d\beta J)}\right)^Q \tag3.11 $$ we can now take $Q$ sufficiently large so that $$ \left( {1 \over 1+\exp (-4d\beta J)}\right)^Q 1/2$ and $\beta_c= (1/2J)\log(1+(\sqrt{5}/2))$ we get $Q(\beta_c,2)=25$). \qed \enddemo Hence we have an example in which the measure is attractive, but nevertheless phase coexistence does not imply percolation. Of course Proposition 3.4 also holds for the {\sl many layer} version of other Markov fields. \noindent \remark{Remark} If in $\mu^{\omega _{+-}}_\beta$ there is with probability one some circuit $\partial \Lambda $ around the origin on which the two coordinates agree, i.e. $\s_1(y)=\s_2(y), y\in \partial \Lambda $, then $\mu^{\eta _+}_{1,\beta}=\mu^{\eta_-}_{1,\beta}$ (the effect of the boundary will be clearly cancelled by conditioning on the circuit $\partial \Lambda$). In other words if $\mu^{\eta _+}_\beta\neq\mu^{\eta_-}_\beta$ (respectively $\mu ^{\eta _1}_{1,\beta} \neq \mu ^{\eta_2}_{1,\beta}$ for $\eta _1, \eta _2 \in \Omega$), there must be {\it disagreement percolation} in $\mu^{\omega _{++}}_\beta$ (respectively in $\mu^{\eta_1}_{1, \beta} \times \mu ^{\eta _2}_{1, \beta}$), as it is done in van den Berg (1993) (see also van den Berg and Maes (1994) for another coupling). {\it Disagreement percolation} means that there is an infinite cluster on which $\s_1(x)\neq\s_2(x)$. This is applied in van den Berg and Steif (1994) for the hard core model of above. They take independently two realizations $(\s_1,\s_2)$ according to the product coupling $\mu^e_\lambda\times\mu^o_\lambda$. A site $x \in \Bbb Z^2$ is a site of disagreement if $\s_1(x)\neq\s_2(x)$. They prove that $\mu^e_\lambda=\mu^o_\lambda$ if and only if $\mu^e_\lambda\times \mu^o_\lambda(\{(\s_1,\s_2)$ has an infinite path of disagreement$\})=0$. Using our general formulation we can strengthen this result somewhat : if $\mu^e_\lambda \neq\mu^o_\lambda $ not only will we get disagreement percolation in the above sense but in the state $\mu^e_\lambda \times\mu^o_\lambda $ this percolation will be over sites $x$ where $$\aligned (\s_1(x),\s_2(x)) & = (\eta_e(x),\eta_o(x))\\ & = (1,0) \text{ for } x \text{ even }\\ & = (0,1) \text{ for } x \text{ odd } \endaligned \tag3.13$$ The reason is that $\mu^e_\lambda \neq\mu^o_\lambda $ is equivalent to $\mu^e\times\mu^o\neq\mu^o\times\mu^e$ implying the stability in $\mu^e_\lambda \times\mu^o_\lambda $ of the configuration $(\eta_e,\eta_o)$ as given in (3.13). \endremark \subhead d) The q--state Potts model\endsubhead In this case $S=\{1, \ldots ,q\}$. The Hamiltonian (2.1) is specified by taking $J > 0, h=0$ and $U(a,b) = 0$ if $a=b$, $U(a,b)=-1$ if $a\neq b$. \newline It is straightforward to see that $g(H)=\{\eta _a :a \in S\}$ where $\eta _a(x)=a $ for all $x \in {\Bbb Z }^d$. A very useful way to analyze the Potts model is to take the FK--representation of Fortuin and Kasteleyn (1972). For that we let $\ell(b) = 0, 1$ be a bond configuration. A bond $b = $ is connecting nearest neighbors $x\sim y\in \Bbb Z^d$ and it can be {\it open} ($\ell(b)=1$) or {\it closed} ($\ell(b) = 0$). In $\Lambda\subset\Bbb Z^d$ we fix a bond configuration $\ell$ by assigning to all bonds $b=$ (connecting nearest neighbors $x\sim y$ at least one of which is inside $\Lambda$) the value 1 or 0. For bond percolation see the definitions in Grimmett (1994). We define the following expectation for local functions $f(\s_\Lambda)$ of the Potts model variables $\s_\Lambda$ in the volume $\Lambda$ with boundary conditions $\xi$ : $$ \langle f \rangle_{\Lambda}^{\xi}(\ell) = \frac1{q^{n_{\Lambda}(\ell)}} \sum_{\s_{\Lambda}} f(\s_{\Lambda}) \prod_{b=\cap\Lambda\neq\emptyset : \ell_b = 1} \delta(\s_\Lambda^\xi(x), \s_\Lambda^\xi(y))\tag3.14$$ % Here $\delta$ is the Kronecker delta, $n_{\Lambda}(\ell)$ is the number of connected $\ell$-clusters in the volume $\Lambda$ so that the expectation (3.7) is normalized. The configuration $\s_{\Lambda}^\xi$ is defined as in (2.4). The reason for introducing (3.7) is that the Potts model expectations in volume $\Lambda$ with boundary conditions $\xi$ can be written as $$ \mu_{\beta, \Lambda}^\xi (f)=\frac1{Z_\Lambda(\beta,\xi)} \sum_{\ell} \prod_{b\cap\Lambda\neq\emptyset} p^{\ell_b} (1-p)^{1-\ell_b} q^{n_{\Lambda}(\ell)} _{\Lambda}^\xi(\ell)\equiv \nu^{FK}_{q,\beta ,\Lambda} (_{\Lambda}^\xi(\cdot)) \tag3.15$$ % when we put $p=1-e^{-\beta J}$. Here $\nu^{FK}_{q,\beta ,\Lambda}$ denotes the finite volume Fortuin--Kasteleyn measure (or random cluster measure) on the bond configurations whose weights are defined by (3.15). The infinite volume measure will be denoted by $\nu ^{FK}_{q,\beta}$ (see Grimmett (1994) and (1994')). \proclaim{Proposition 3.5} For the Potts model (2.8a) holds. Moreover if $d=2$ and $\mu^\eta_\beta \neq \mu ^{\eta ^\prime}_\beta$ for $\eta , \eta ^\prime \in g(H)$ then $$ \theta_\beta ^\eta (0)=0 \tag3.16 $$ \endproclaim \demo{Proof} (2.8a) is easily proven by observing that $$\multline \mu ^{\eta _a}_{\beta, \Lambda} (\sigma (0)=a)=\\ {1\over q}+\left({q-1 \over q}\right) \nu ^{FK}_{q, \beta ,\Lambda}(\{\ell: 0 \text{ is connected to }\partial \Lambda \text { by a chain of open bonds}\}) \endmultline \tag 3.17 $$ and that given the coupling between $\sigma$ and $\ell$ implicit in (3.14) and (3.15), $\sigma (x)=1$ if $x$ belongs to one of the bonds in the infinite cluster of open bonds. By (3.17) the latter exists a.s. in the coexistence region (see Fortuin--Kasteleyn (1972) and Aizenman {\it et al.} (1988)).\newline To prove (3.16), let us consider the FK measure $\nu^{FK}_{q,\beta}$ associated to the extremal Potts measure $\mu_\beta ^{\eta_a}$. >From its construction (see Grimmett (1994) and (1994')), besides being translation and rotation invariant, it is ergodic under translations. Moreover it is known that this measure is also attractive (Fortuin and Kasteleyn (1972), Aizenmann {\it et al.} (1988), Grimmett (1994) and (1994')). A straightforward adaptation of Gandolfi {\it et al.} (1988) allows to conclude that if there is percolation of open bonds, outside any box containing the origin there is a circuit of open bonds surrounding the origin (a.s.). Again by the coupling between $\sigma $ and $\ell$, we conclude that if $\theta ^{\eta _a} (1)>0$, then $\mu ^{\eta _a} _{\beta}-$a.s. every point is surrounded by a circuit on which $\sigma(x)=a$. By $(2.8a)$ we conclude. \qed \enddemo \remark{Remark} Note that in $d=2$ for $q>4$ the ``magnetization'' $\mu^a[\s(0)=a]$ is believed to be discontinuous at $\beta=\beta_c$, see Baxter (1982). Therefore, in the two--dimensional Potts model the lowest density of a ground state configuration for which we know there is percolation is $1/4 + \epsilon$ (for arbitrary $\epsilon > 0$) and is obtained for $q=4$ in the corresponding Gibbs state just below the critical point\footnote{This is similar to the case of duplicated Ising variables, see example 3(c)}. \endremark \subhead e) Widom--Rowlinson lattice model\endsubhead The statements $(2.8a)$, $(2.8b)$ and $(2.8c)$ hold also for a class of models first introduced in Wheeler and Widom (1970). They are ``spin 1'' models with single site state space $S=\{-1,0,+1\}$ and Hamiltonian (2.1) determined by $U(a,b)=ab(1-ab), V(a)=a^2, 0\leq J <\infty$ and $h > 0$. This model was shown to have a phase transition by Lebowitz and Gallavotti (1971) and to be attractive by Lebowitz and Monroe (1972).\newline The detailed analysis of these models follows along standard lines. \head 4. Continuum model \endhead The continuum Widom--Rowlinson (WR-) model (Widom and Rowlinson (1970) consists of particles of type A and type B having positions in $\Bbb R ^d$ and fugacities $z_A$ and $z_B$ whose interaction consists of the hard core constraint that the centers of any two particles of different type must be at least distance $R$ from each other. In Cassandro {\it et al.} (1973) it is shown how this model can be obtained from a lattice model of the type described in Section 3(e) above. \newline More precisely, we take $\Lambda \subset \Bbb R^d$ a finite Borel set and let $x=(x_1,\ldots ,x_{N_A})$ (respectively $y=(y_1,\ldots ,y_{N_B})$ denote the position of particles A (respectively B), $x_i,y_i \in \Lambda$ for $i=1, \dots ,N_A$ and $j=1, \ldots, N_B$. Call $X$ the space of $\sigma-$finite integer valued measures over $\Lambda$ (and its Borel sets); our probability space will be $\Omega=X\times Y$ ($X=Y$ will have the topology of weak convergence, that characterizes its Borel sets). By separability, any element of $X$ can be written as $\sum_{i\in I} \delta _{x_i}$ ($I\subset\Bbb Z, \vert I \vert <\infty$) and so we will use the notation $N_A(\omega)$, $N_B(\omega)$, $x(\omega)$ and $y(\omega)$ for $\omega \in\Omega $ with obvious meaning. The constraint that is imposed is determined by the hard core length $R$: $$ \min_{i,j} \vert x_i -y_j\vert >R \tag 4.1 $$ Letting $I[x,y]$ denote the indicator function corresponding to (4.1), we put $$ Z^\Lambda_{N_A ,N_B}= \int_{\Lambda ^{N_A} \times \Lambda ^{N_B}} \roman{d} \lambda _{N_A}(x) \roman{d} \lambda _{N_B}(y) I[x,y] \tag4.2 $$ for $ \roman{d} \lambda _{N}(x) = \roman{d} ^d x_1 \ldots \roman{d}^d x_N $ the $N-$product Lebesgue measure. Fixing the fugacities $z_A , z_B>0$, the grand-canonical partition function of the WR-model is then $$ \Xi=\Xi(\Lambda ,z_A, z_B)= \sum _{N_A ,N_B} {z_A ^{N_A} z_B^{N_B} \over N_A ! N_B!} Z^\Lambda _{N_A ,N_B} \tag 4.3 $$ We will be mostly interested in the case $z_A=z_B=z$, for which we adopt the notation $\Xi (\Lambda, z)$.\newline So far we have not spoken of boundary conditions. Obviously we can fix the position of some particles introducing extra constraints (beyond (4.1)). For example, we speak of boundary conditions of type A if we replace $I[x,y]$ in (4.1) by $I_A[x,y]=I[x,y] I_A[y]$ where $I_A[y]$ is the indicator function corresponding to $$ \inf_{j,x\in \Lambda ^c} \vert y_j -x \vert >R \tag 4.4 $$ Analogous definitions and notations apply for boundary conditions of type B. The grand--canonical partition function is then changed into $\Xi_\gamma$, corresponding to the boundary conditions of type $\gamma=A,B$. \newline The finite volume Gibbs measure $\mu ^\gamma _\Lambda$ for boundary conditions $\gamma =A,B$ gives the probability of finding the particles in certain regions of $\Lambda$. If we condition on having $N_A$ type $A$ particles and $N_B$ type $B$ particles in $\Lambda$, the random field will have density $$ { \roman {d} \mu_{\Lambda}^\gamma (\cdot) \vert_{N_A(\omega)= N_A, N_B(\omega )=N_B} \over \roman{d} \lambda_{N_A} \times \roman{d} \lambda_{N_B} }(\omega)={1 \over Z^\Lambda _{N_A ,N_B} } {z_A^{N_A} z_B^{N_B} \over N_A ! N_B!} I_\gamma (\omega) \tag4.5 $$ in which $I_\gamma (\omega)=I_\gamma[x(\omega),y(\omega)]$ ($\omega \in \Omega$) and analogously for $I(\omega)$, see (4.1). The infinite volume measures are denoted by $\mu ^\gamma$ and can be obtained as limit from $\mu_\Lambda ^\gamma$ as $\Lambda \nearrow \Bbb R^d$ along suitable subsequences (for examples along spheres). The measure $\mu ^\gamma$ does depend on the boundary condition $\gamma$ if $z_A= z_B=z$ and $z$ is sufficiently large, Ruelle (1971). \definition{Clusters and percolation probability} Define the function $Sp:\Lambda \times \{\omega: I(\omega )=1\} \rightarrow \{A,B,W\}$ as $$ Sp(x)= Sp(x, \omega)= \cases A & \text{if dist}(\bold x (\omega),\{x\}) < R/2 \\ B & \text{if dist}(\bold y (\omega),\{x\}) < R/2 \\ W & \text{otherwise} \endcases \tag4.6 $$ We can imagine the function $Sp$ a coloring of $\Lambda $ in red (A), black (B) or white (W). From now on take $\gamma , \delta \in \{A, B\}$. The $\gamma$ cluster at the origin ($C^0_\gamma (\omega)$) will then be defined as the connected component of $Sp(\cdot ,\omega )^{-1}(\gamma)$ that contains the origin ($C^0_\gamma =\emptyset$ if $Sp(0) \not= \gamma$). The percolation probability is thus defined as $$ \theta ^\gamma (\delta)= \mu^\gamma ( \{ \omega : \text{diam}(C^0_\delta (\omega ))=\infty\}) \tag4.7 $$ \enddefinition \proclaim{Proposition 4.1} Using the notations and definitions above with $z=z_A=z_B$, $$ \mu ^A (Sp(0)=A)- \mu ^B (Sp(0)=A)= \theta ^A(A)-\theta^A(B) \tag4.8 $$ implying $$ \theta ^A(A) >0 \Longleftarrow \mu^A\neq\mu^B \tag4.9a $$ In $d=2$, $$ \theta^A(B)=0\tag4.9b$$ and $$ \theta ^A(A) >0 \Longleftrightarrow \mu^A\neq\mu^B \tag4.9c $$ \endproclaim \remark{Remark 1} (4.8) says that the particle clusters in the WR--model play a similar role as the random clusters in the FK--representation of the Potts model. We believe this to be the first example where such a direct relation between the `order parameter' and the cluster geometry is found. Note also that by attractivity (Lebowitz and Monroe (1972)) the left hand side of (4.8) is zero if and only if the $A$-phase is different from the $B$-phase. \endremark \remark{Remark 2} We believe that the results in the Proposition partially extend to the case where a hard core condition is added between alike particles. In that case the measures are not attractive but, as will become clear from the proof, that is not at all crucial here. Since there are extra technicalities and quite a bit of extra notation related to the existence of infinite volume limits if we don't have FKG, we prefer to give the proof only in the case of strict hard cores. \endremark \remark{Remark 3} So far we have assumed for simplicity that the fugacities of the two types of particles are equal, $z_A=z_B=z$. Note however that if say $z_A \geq z$, then $\mu^A_{z_A,z}$ stochastically dominates $\mu^A_{z,z}$ where we now explicitly indicate by subscripts the fugacities of $A$- respectively, $B$-type particles. This implies that if in $\mu^A_{z,z}$ there is percolation of $A$-type particles (as in the phase coexistence regime, $z>z_c$), then we get the same result for all $A$-particle fugacities $z_A \geq z$. Suppose we now integrate out the positions of the $B$-particles. A simple calculation shows that we get a new measure for the $A$-particles where now $z$ (previously the fugacity of the $B$-particles) plays the role of an inverse temperature. In that measure, for $z>z_c$, the $A$-particles percolate for all values $z_A \geq z$. \endremark \remark{Idea of the proof} Consider an $A$-cluster covering the origin. If it is bounded, then, by the hard core constraint, there is necessarily a white region surrounding it, see Fig. 3. This effectively screens the origin from the external boundary condition. Therefore this contribution to the probability of having an $A$-particle at the origin is the same in all states. What remains is the probability that the A--cluster extends infinitely far. This idea can be most easily implemented through a discretization of the space. \endremark \demo{Proof of Proposition 4.1} Take $\Lambda ^\prime \subset \Lambda \subset\Bbb R ^d$ two spheres. Define the {\sl finite volume} percolation probability as $ \theta ^\gamma (\delta; \Lambda ,\Lambda ^\prime)= \mu ^{\gamma}_\Lambda (C^0_\delta (\omega ) \cap \partial \Lambda ^\prime \not=\emptyset )$ and observe that $$ \lim _{\Lambda ^\prime \nearrow \Bbb R^2} \lim _{\Lambda \nearrow \Bbb R^2} \theta ^\gamma (\delta; \Lambda ,\Lambda ^\prime ) =\theta ^\gamma (\delta) \tag4.10 $$ Take $\epsilon > 0 $ small and cover $\Bbb R^d$ with a grid of spacing $\epsilon$. Disregarding boundary problems, this naturally defines a partition of $\Bbb R^d$ into squares of sidelength $\epsilon$ ($\epsilon-$squares).\newline To control the errors made by the space discretization we define $$ G(\epsilon)= \{\omega : \text{dist}(\bold x ,\bold y) >R+6d\epsilon, \ \vert \text{dist}(\bold x ,\partial \Lambda ^\prime )-R \vert> 6d\epsilon, \ \vert \text{dist}(\bold y ,\partial \Lambda ^\prime )-R \vert> 6d\epsilon, $$ $$ \vert \vert x_i -x_k \vert -R \vert >6d\epsilon \ i\not= k\in \{1, \ldots ,N_A(\omega)\} \} \tag4.11 $$ A simple argument by contradiction yields the existence of $\Delta_\Lambda (\epsilon) \geq 0$, vanishing as $\epsilon\downarrow 0$ and such that $$ \mu^{\gamma}_\Lambda (G(\epsilon))>1-\Delta_\Lambda (\epsilon) \tag4.12 $$ with $\gamma \in \{A, B\}$. The union of a finite number of $\epsilon-$squares is an $\epsilon-$cluster at the origin if the interior of this set is connected and if it contains the origin. Denote by ${\Cal C}_\epsilon $ the set of $\epsilon-$clusters at the origin ($\emptyset \in{\Cal C}_\epsilon$). \newline We are now going to define a subset of $\Omega$ characterized by having a certain element of ${\Cal C}_\epsilon$ as minimal $\epsilon-$covering of the corresponding maximal cluster at the origin. More precisely, given $C\in {\Cal C}_\epsilon$ define $$ \Cal A_\epsilon (C)=\{\omega: C^0_A(\omega ) \subset C \text{ and for all } C^\prime \in {\Cal C}_\epsilon, C^\prime \subsetneq C, C^0_A(\omega ) \not\subset C^\prime \} \tag4.13 $$ We have then by construction that the probability to find an $A$-particle at the origin is $$ \mu_\Lambda ^A (Sp(0)=A)= \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^A_\Lambda (\Cal A_\epsilon(C)) + \sum _{C\cap \partial \Lambda ^\prime \not=\emptyset} \mu ^A_\Lambda (\Cal A_\epsilon(C)) \tag4.14 $$ where the sums are over $C\in {\Cal C}(\epsilon)$. We deal with the two terms in the right hand side of (4.14) separately. First of all observe that by (4.12) $$ \left\vert \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^A_\Lambda (\Cal A_\epsilon(C))- \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^B_\Lambda (\Cal A_\epsilon(C)) \right\vert $$ $$ \le 2 \Delta_\Lambda (\epsilon)+ \left \vert \sum _C \left( \mu ^A_\Lambda (\Cal A_\epsilon(C)\cap G(\epsilon)) -\mu ^B_\Lambda (\Cal A_\epsilon(C)\cap G(\epsilon)) \right) \right\vert \tag4.15 $$ Define $\partial C_W$ to be that subset of $\Omega$ such that $Sp(x,\omega)=W$ for all $x$ contained in an $\epsilon-$square adjacent to the outer boundary of $C\in\Cal C_\epsilon $ (that is the external connected component of the boundary). Since $\Cal A_\epsilon (C)\cap G(\epsilon) \subset \Cal A_\epsilon (C)\cap \partial C_W$ we can continue (4.15) obtaining $$ \left\vert \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^A_\Lambda (\Cal A_\epsilon(C))- \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^B_\Lambda (\Cal A_\epsilon(C))\right\vert $$ $$ \le 4 \Delta_\Lambda (\epsilon)+ \sum_C \left\vert \mu ^A_\Lambda (\Cal A_\epsilon(C)\cap \partial C_W ) -\mu ^B_\Lambda (\Cal A_\epsilon(C)\cap \partial C_W ) \right\vert \tag4.16 $$ By the Markov property of the $\mu^\gamma_\Lambda $ the sum in the right hand side of (4.16) is zero. This is because by conditioning it is straightforward to see that every term in this sum is equal to $$ { \mu_\Lambda (\Cal A_\epsilon(C) \vert\partial C_W) \mu_\Lambda (I_A\vert\partial C_W) \mu_\Lambda (\partial C_W) \over \mu_\Lambda (I_A) } $$ $$ - { \mu_\Lambda (\Cal A_\epsilon(C) \vert\partial C_W) \mu_\Lambda (I_B\vert\partial C_W) \mu_\Lambda (\partial C_W) \over \mu_\Lambda (I_B) }=0 \tag4.17 $$ The last equality follows because of the symmetry under exchange $A\leftrightarrow B$. Hence $$ \vert \sum _{C\cap \partial \Lambda ^\prime =\emptyset} \mu ^A_\Lambda (\Cal A_\epsilon(C))- \sum _{C\cap \partial\Lambda ^\prime )^c=\emptyset} \mu ^B_\Lambda (\Cal A_\epsilon(C))\vert \le 4 \Delta_\Lambda (\epsilon) \tag4.18 $$ The absolute value of the difference between the second term in the right hand side of (4.14) and $\theta ^A(A;\Lambda ,\Lambda ^\prime)$ is by the definition (4.11) of $G(\epsilon)$ smaller than $\Delta _\Lambda (\epsilon)$.\newline Combining this with (4.18) and writing (4.14) also for the measure $\mu^B_\Lambda $, we get $$ \vert \mu _\Lambda ^A (Sp(0)=A)- \mu _\Lambda ^B (Sp(0)=A) +\theta^B(A;\Lambda ,\Lambda ^\prime)-\theta^A(A;\Lambda ,\Lambda ^\prime) \vert \le 6\Delta _\Lambda (\epsilon) \tag4.19 $$ Hence the left hand side of (4.19) is zero. The result follows by taking the limits as indicated in (4.10). In the case of two dimensions ($d=2$), we use a generalized version of Gandolfi {\it et al.} (1988) extending their result without difficulty to the continuum. This ensures that $\theta ^B(A)=0$ if $\theta ^A(A)>0$. \qed \enddemo \noindent{\bf Acknowledgments} : We are grateful to M. Aizenman, C.K. Hu, C.M. Newman, L. Russo and A. Sokal for fruitful discussions. This work is partially supported by NSF grant NSF--DMR92--13424 and by EC grant CHRX--CT93-0411. C.M. acknowledges support from the Belgian National Science Foundation and the hospitality of the Newton Institute in Cambridge. G.G. acknowledges the support of the C.N.R. and the hospitality of KU Leuven (Instituut voor Theoretische Fysica). \noindent {\bf Note:} after the completion of this work, we learnt that J.T. Chayes, L. Chayes and R. Koteck\'y have obtained results similar to those of the Section 4 of this paper (J.T. Chayes, L.Chayes, R. 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Sinai \paper Phase Diagrams for classical Lattice Systems \jour Theor. Math. Phys. \vol 25 \pages 1185--1192 %\vol 26 \pages 39--49 \yr 1976 \endref \ref \by D. Ruelle \paper Existence of a Phase Transition in a Continuous Classical System \jour Phys. Rev. Lett. \vol 27 \pages 1040 \yr1971 \endref \ref \by L. Russo \paper The Infinite Cluster Method in the Two--dimen\-si\-o\-nal Ising Model. \jour Comm. Math. Phys. \vol 67 \pages 251 \yr 1979 \endref \ref \by B. Simon \paper The Statistical Mechanics of Lattice Gases \jour Volume 1 \publ Princeton University Press, Princeton \yr 1993 \endref \ref \by Ya. G. Sinai \paper Theory of Phase Transitions : Rigorous Results \jour Pergamon Int. Series in Natural Philosophy \vol 8 \publ Pergamon Press, Oxford \yr 1982 \endref \ref \by J. Slawny \paper Low Temperature Properties of Classical Lattice Systems : Phase transitions and Phase Diagrams \jour Phase Transitions and Critical Phemomena \vol 5 \publ (C. Domb and J.L. Lebowitz, eds.), Academic Press, London and New York \yr 1986 \endref \ref \by J.C. Wheeler and B. Widom \paper Phase Equilibrium and Critical Behavior in a Two--Component Bethe--Lattice Gas or Three--Component Bethe--Lattice Solution \jour J. Chem. Phys. \vol 52 \pages 5334--5343 \yr 1970 \endref \ref \by B. Widom and J.S. Rowlinson \paper New Model for the Study of Liquid--Vapor Phase Transitions \jour J. Chem. Phys. \vol 52 \pages 1670 \yr 1970 \endref \ref \by M. Zahradn\'{\i}k \paper An alternate version of Pirogov--Sinai theory \jour Comm. Math. Phys. \vol 93 \pages 559--581 \yr 1984 \endref \endRefs \vfill \eject Caption for fig.1 \vskip 0.8 cm \noindent A portion of a configuration of the hard--square lattice gas. The squares centered on odd sites are painted darker than the ones centered on even sites. In this figure there are three odd clusters and two even clusters. \vskip 2 cm Caption for fig.2 \vskip 0.8 cm \noindent The expected phase diagram of the antiferromagnetic Ising model for $d=2$. The shadowed area is the region of phase transition (this is only a qualitative diagram). The hard square limit is obtained taking $J \rightarrow \infty$ along the line with slope $\tan (\theta)$ shown on the figure. The phase transition for the hard--square model is expected to happen at $\theta =\theta _c$. \vskip 2 cm Caption for fig.3 \vskip 0.8 cm \noindent A portion of a configuration of $A$ (darker) and $B$ (lighter) particles. There is an $A-$cluster at the origin and it is separated from the other clusters by a white layer. \vfill\eject \enddocument ENDBODY %!PS-Adobe-2.0 EPSF-1.2 %%DocumentFonts: Times-Roman %%Pages: 1 %%BoundingBox: 34 167 566 664 %%EndComments 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %%EndProlog %I Idraw 7 Grid 8 %%Page: 1 1 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.8 0 0 0.8 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.25 SetP %I t [ 1 0 0 1 173 174 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 114 24 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p < 77 bb ee dd 77 bb ee dd > -1 SetP %I t [ 1 0 0 1 -66 234 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 -67 85 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p < 77 bb ee dd 77 bb ee dd > -1 SetP %I t [ 1 0 0 1 83 263 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p < 77 bb ee dd 77 bb ee dd > -1 SetP %I t [ 1 0 0 1 -36 205 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p < 77 bb ee dd 77 bb ee dd > -1 SetP %I t [ 1 0 0 1 84 204 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 143 114 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p < 77 bb ee dd 77 bb ee dd > -1 SetP %I t [ 1 0 0 1 113 234 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 84 114 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 84 54 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 54 144 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 24 114 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I MLine %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 6 245 404 275 374 305 404 275 434 245 404 245 404 6 MLine End Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t u Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 693 643 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 613 693 613 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 583 693 583 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 1 -269 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 693 643 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 613 693 613 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 583 693 583 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 1 -179 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 693 643 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 613 693 613 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 583 693 583 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 1 -90 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 693 643 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 613 693 613 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 583 693 583 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 1 -358 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 693 643 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 613 693 613 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 583 693 583 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t u Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 448 0 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 359 0 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 270 0 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 180 0 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 90 0 ] concat Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 66 643 66 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 96 643 96 135 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 126 643 126 135 Line End End %I eop Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 664 135 713 643 Rect End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-medium-r-*-140-* /Times-Roman 14 SetF %I t [ 1 0 0 1 319 545 ] concat %I [ (0) ] Text End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 54 113 163 673 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 129 575 635 673 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 601 574 696 685 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 539 112 674 582 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 158 207 159 207 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 50 166 143 206 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 153 67 571 199 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 75 205 155 239 Rect End Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 158 158 551 228 Rect End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-medium-r-*-140-* /Times-Roman 14 SetF %I t [ 1 0 0 1 292 328 ] concat %I [ ( Fig. 1) ] Text End End %I eop showpage %%Trailer end %!PS-Adobe-2.0 EPSF-1.2 %%DocumentFonts: Times-Bold Courier-Bold Helvetica Times-Roman %%Pages: 1 %%BoundingBox: 32 172 576 632 %%EndComments 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %%EndProlog %I Idraw 7 Grid 8 %%Page: 1 1 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.8 0 0 0.8 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I CBSpl %I b 65535 1 1 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 -16 134 ] concat %I 15 215 245 255 374 305 444 335 464 375 474 414 474 454 464 484 444 524 374 564 245 594 165 395 85 185 165 215 245 215 245 15 CBSpl End Begin %I Poly none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -16 134 ] concat %I 6 185 245 185 85 594 85 594 245 185 245 185 245 6 Poly End Begin %I Line %I b 65535 1 1 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -16 134 ] concat %I 733 245 76 245 Line End Begin %I Line %I b 65535 1 1 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -16 134 ] concat %I 395 653 395 195 Line End Begin %I BSpl %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -16 134 ] concat %I 6 584 294 604 284 614 274 624 255 624 245 624 245 6 BSpl End Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 0 1 ] concat Begin %I Elli %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -16 134 ] concat %I 634 294 10 10 Elli End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -16 134 ] concat %I 624 294 644 294 Line End End %I eop Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-bold-r-*-140-* /Times-Bold 14 SetF %I t [ 1 0 0 1 359 369 ] concat %I [ (0) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-bold-r-*-140-* /Times-Bold 14 SetF %I t [ 1 0 0 1 349 777 ] concat %I [ (1/J) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-bold-r-*-140-* /Times-Bold 14 SetF %I t [ 1 0 0 1 697 369 ] concat %I [ (h) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-bold-r-*-140-* /Times-Bold 14 SetF %I t [ 1 0 0 1 349 638 ] concat %I [ (1/J) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-courier-bold-r-*-120-* /Courier-Bold 12 SetF %I t [ 1 0 0 1 369 628 ] concat %I [ (c) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 555 236 470 499 Line End Begin %I BSpl %I b 65535 1 1 0 [] 0 SetB %I cfg LtGray 0.762951 0.762951 0.762951 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 4 165 609 208 608 206 563 202 565 4 BSpl End Begin %I BSpl %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 5 544 269 561 268 577 254 584 237 583 237 5 BSpl End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-helvetica-medium-r-*-140-* /Helvetica 14 SetF %I t [ 1 0 0 1 556 441 ] concat %I [ (c) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -16 134 ] concat %I 564 245 644 464 Line End Begin %I Pict %I b u %I cfg u %I cbg u %I f *-helvetica-medium-r-*-140-* /Helvetica 14 SetF %I p u %I t [ 1 0 0 1 -71 18 ] concat Begin %I Elli %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -16 134 ] concat %I 634 294 10 10 Elli End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -16 134 ] concat %I 624 294 644 294 Line End End %I eop Begin %I Rect none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 50 445 263 640 Rect End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-medium-r-*-140-* /Times-Roman 14 SetF %I t [ 1 0 0 1 369 298 ] concat %I [ (Fig. 2) ] Text End End %I eop showpage %%Trailer end %!PS-Adobe-2.0 EPSF-1.2 %%DocumentFonts: Times-Roman %%Pages: 1 %%BoundingBox: 14 124 594 656 %%EndComments 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %%EndProlog %I Idraw 7 Grid 8 %%Page: 1 1 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.8 0 0 0.8 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Line %I b 65535 1 1 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -6 144 ] concat %I 355 673 355 15 Line End Begin %I Line %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -14 128 ] concat %I 36 384 753 384 Line End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 245 92 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 266 14 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 197 -138 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 226 -114 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 227 -6 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 196 -35 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 227 -77 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 307 24 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 300 -113 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 267 -142 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 270 54 ] concat %I 139 590 29 27 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 261 164 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 337 119 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 180 166 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 166 129 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 185 82 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 359 -179 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 308 144 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 242 -218 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 114 -71 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 93 -26 ] concat %I 122 587 34 31 Elli End Begin %I Elli none SetB %I b n %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0.75 SetP %I t [ 1 0 0 1 326 -41 ] concat %I 122 587 34 31 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 317 34 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 423 -141 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 93 -160 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 87 81 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 397 -42 ] concat %I 139 590 29 27 Elli End Begin %I Elli %I b 65535 1 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 402 -8 ] concat %I 139 590 29 27 Elli End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-medium-r-*-140-* /Times-Roman 14 SetF %I t [ 1 0 0 1 320 496 ] concat %I [ (0) ] Text End Begin %I CBSpl %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 -6 144 ] concat %I 79 379 294 388 280 410 270 424 272 442 283 446 297 447 302 463 305 472 315 477 324 480 341 474 354 463 363 448 365 438 365 424 359 411 340 407 339 401 350 410 373 403 387 397 395 388 398 375 402 376 410 389 412 399 420 405 430 415 428 428 433 440 440 457 439 473 443 481 452 487 457 495 470 497 485 492 501 479 510 470 510 457 511 449 509 440 524 424 531 424 548 418 559 406 565 392 570 376 568 363 559 357 549 357 535 359 521 366 514 381 506 382 494 387 483 379 471 359 471 344 463 339 449 338 443 319 437 308 424 307 403 317 392 330 378 341 379 336 364 343 349 337 340 315 330 306 315 310 295 329 276 360 278 376 298 380 294 380 294 79 CBSpl End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f *-times-medium-r-*-140-* /Times-Roman 14 SetF %I t [ 1 0 0 1 450 254 ] concat %I [ (Fig. 3) ] Text End End %I eop showpage %%Trailer end