ESI The Erwin Schr�odinger International Pasteurgasse 6/7Institute for Mathematical Physics A-1090 Wien, AustriaStrati�ed Low Temperature Phasesof Strati�ed Spin Models.A General Pirogov { Sinai ApproachPetr Holick�yMilo�s Zahradn��k

Vienna, Preprint ESI 346 (1996) June 18, 1996Supported by Federal Ministry of Science and Research, AustriaAvailable via http://www.esi.ac.at

STRATIFIED LOW TEMPERATURE PHASES OF STRATIFIEDSPIN MODELS. A GENERAL PIROGOV { SINAI APPROACHPetr Holick�y and Milo�s Zahradn��kFaculty of Mathematics and Physics, Charles University,Sokolovsk�a 83, 186 00 Prague, Czech RepublicJune 17, 1996Abstract. We adapt and improve the existing Pirogov { Sinai technology to obtaina general and unifying approach to the study of low temperature, \strati�ed" phasesfor classical spin models whose hamiltonianmay not even be translation invariant butis \strati�ed" i.e. invariant with respect to all \horizontal" shifts (not changing thelast coordinate). Examples are \strati�ed" versions of classical models like the Isingmodel with \vertically dependent" external �eld; models in halfspaces or layers andalso those translation invariant models where Dobrushin's phases with rigid interfaces(one or more) appear. Our method brings some clari�cation and sharpening evenwhen applied to the ordinary situations of the Pirogov { Sinai theory [S], [Z]. Ourmain result transcripts the question of characterizing the \strati�ed" Gibbs states ofthe given model to the question of �nding the ground states of some auxiliary onedimensional model with in�nite range but quickly decaying interactions.I. Introduction, Notes on the Developmentof the Problem and Some ExamplesThe rigorous study of Gibbs states having translation noninvariant structurewith a \rigid interface" goes back to the pioneering Dobrushin's paper [1]. Severalauthors continued this study; we note e.g. articles [HKZ] where an attempt tocombine basic Dobrushin's ideas with the power of Pirogov { Sinai theory wasmade.The leading idea in these investigations was to transcript the problem of descrip-tion of the structure of the rigid interface (between the two translation invariantphases above and below) to a suitable lower dimensional problem. In more concreteterms, using the expansion of the partition sums above and below the interface, thebehaviour of the \walls" of the interface between the + and � phases in the three1991 Mathematics Subject Classi�cation. 82A25.Key words and phrases. Low temperature Gibbs states, strati�ed hamiltonians and phases,interfaces, contours, Dobrushin's walls, Pirogov { Sinai theory, Peierls condition, contour func-tional, \metastable" submodels, inductively organized cluster expansions, ground states of onedimensional models, phase diagrams.Partially supported by: Commission of the EuropeanUnion under contracts CHRX-CT93-0411and CIPA-CT92-4016, Czech Republic grants �c . 202/96/0731 and �c. 96/272.Typeset by AMS-TEX1

dimensional Ising model can be viewed as a behaviour of contours of some auxiliarytwo dimensional perturbed Ising model.Recently, we applied a similar approach based on the reduction to a lower dimen-sional problem to the study of wetting phenomena and entropic repulsion in theIsing model in halfspace [HZ]. During our attempt to pursue the method to otherinteresting situations, like the study of \wetting layers" emerging in some phasesof the Blume { Capel model and also in the order { disorder { (other)order phasesappering in the Potts model below the critical temperature (the article [MZ] is un-der preparation) we found that the additional technical problems are forcing us tolook for a more appropriate method. Finally we were lead to a conclusion that the\dimensional reduction method" based on this particular kind of a partial exansionof the considered model should be abandoned. Instead, we found a modi�cationof the Pirogov { Sinai theory which applies directly to these \strati�ed" situations.We hope that the fact that our new version of the Pirogov { Sinai theory gives evensome new insight and simpli�cations into the traditional \translation invariant"Pirogov { Sinai theory con�rms that the method developed by us is adequate.Methodologically, our approach is based on the version [Z] of the Pirogov {Sinai theory but the concept of a \stable" (\small") contour and of a \metastableensemble" is now investigated in a greater depth. Moreover, the concept of a contourensemble now disappeared from our version of Pirogov { Sinai theory completely!The concept of a, suitably de�ned, \contour functional" F (�) (as compared to thecontour energy E(�)) remains as a very important testing quantity (allowing oneto decide whether the contour is \small" or not) but instead of the construction ofauxiliary contour models a central point of our approach is the idea of a succesivepartial expansion of the model based on an important new technical step whichis called recoloring of the contour here. Recoloring of a contour � in a partiallyexpanded model means that a new, \more expanded" model with the same partitionfunctions is constructed where � does not appear yet as a contour . We will showthat the \metastable" submodels of the given model (constructed for any strati�edboundary condition) can be expanded completely and that for the \stable" boundaryconditions, the corresponding metastable model will be identical with the original\physical" model. Roughly speaking, all the external contours will be \small"resp. \recolorable" (in the sense of [Z] resp. of this paper) in such a situation. Theorganization of our expansions will make unnecessary estimates like \Main Lemma"of [Z].Instead, we have now a more powerful method based on our Theorems 5 and 6.To summarize, we converted the Pirogov { Sinai theory just to a carefully organizedmethod of (succesive) expansion of some partition functions. The use of expansiontechniques is absolutely crucial in our situation and the construction of the expan-sions is a more delicate task than in the translation invariant situations studiedbefore. Namely, contours of the models studied so far were \crusted" in the sensethat the events outside and inside the contour were independent. This is not validhere in our new situation where contours can be also interpreted as \walls" (the ter-minology of [D]) of the interface and the events happening \inside" resp. \outside"of the wall cannot be tracted as independent ones.This problem was solved in [HKZ] by taking expansions \above" and \below"the interface and by replacing the walls by more complex \aggregates" of walls and2

clusters1 but for complicated phases with several interfaces such an approach is toocomplicated.Now we treat both the \crusted" contours and the \noncrusted" ones (walls) inthe same way. However, the fact that some contours \are not crusted" implies thatthe testing quantity F (�) called the \contour functional" of a given contour � mustbe now de�ned much more carefully2. We construct (succesively, by induction)the expansion of the whole metastable model, leaving out the previous idea (of[HKZ]) of the expansion in two di�erent steps (�rst the expansion of the ensembleof contours and then of the ensemble of the walls resp. aggregates).Let us mention some typical examples which can be treated by our method.(1) a) Models in halfspace Z�+ = ft 2 Z�; t� � 0g with \unstable" boundarycondition on the bottom (like the � boundary condition for the ferromag-netic Ising model with a negative external �eld (making + the only groundstate of the model)) Than the \Basuev states" (terminology of R. L. Do-brushin) with a weeting layer of minuses appear.b) Models in layers (like in [MS], [MDS])(2) a) Models of the Blume-Capel type with spins belonging to some �niteset Q 2 R and with the hamiltonian consisting of a quadratic (e.g.) pairinteraction and a potential V :(1.1) H(x) =X(t;s)(xt � xs)2 +Xt V (xt)where V has several \potential wells" (of the approximately same depth).If x+, x0, x� mark the bottoms of three adjancent wells of V then it mayhappen, for suitable choice of V , that both x+ and x� give rise to a stablephase while x0 is unstable. Then one should expect also the existence of aphase which \goes vertically from x+ to x� through a layer of a metastable0-th phase". The question is about the determination of the width of the0-th layer.b) Such a situation appears, in the Fortuin { Kasteleyn representation, alsofor the Potts model with large number of spins below the critical temper-ature, where phases of the type order { layer of disorder { another orderexist. (The paper [HMZ] which is under preparation will be devoted tothese questions.)(3) a) \Sedimentary Ising rock". Consider some ordinary translation invariantPirogov { Sinai type model and add to it a small perturbative hamiltonianwhich is invariant with respect to the Z��1 shifts (we identify here Z��1with the subspaceZ��1xf0g of Z�) i.e. depends on the last (\vertical") co-ordinate t� of t = (t1; : : : ; t�) 2 Z� only. Then, one should expect phaseswith a rich structure of (many) layers (of \stable or slightly instable trans-lation invariant phases of the unperturbed hamiltonian "). For example ifone adds, to a ferromagnetic Ising model, a small \horizontally invariant"external �eld with approximately zero mean over the vertical shifts, one1The whole situation was then projected to Z��1 which is the main idea of the paper [D].2Retaining its meaning, vaguely speaking, of the \work needed to install the given contour".3

should expect phases with in�nitely many layers (of changing � phases),and the problem is to compute the exact positions of the layers.b) We will see later that the class of \horizontally invariant models" �t-ting our scheme is much broader and many examples which are not smallperturbations of translation invariant models can be constructed.Our main result is given in part III, section 8:In the translation invariant Pirogov { Sinai theory, one constructs, for any referencecon�guration (i.e. for any \local ground state") a quantity called the \metastablefree energy". If the minimum of this quantity is attained in some con�gurationy, then the Gibbs state characterized as the \local perturbation of y" exists ([Z]).Here, our \reference con�gurations" are (all!) strati�ed con�gurations; instead ofquantities mentioned above we construct some auxiliary one dimensional model ofthe Ising type whose con�gurations correspond to various \horizontally invariantregimes" of the original model. The ground states of this one dimensional modelcorrespond to the di�erent strati�ed Gibbs states which we are looking for! This isour Main Theorem (section 8); the quantities ht(y) = ht� (y); t 2 Z� constructedthere give all the essential information about the model.These quantities are in principle computable as they are given by cluster expan-sion series (with complicated, but very quickly decaying terms). In the case wheny is the ground state of the corresponding one dimensional model (\stability of y")the quantities ht(y) have the physical interpretation of the \density of free energyof the y{th Gibbs state at the vertical level t�" .Note. We are concentrated, in this paper, in the investigation of a phase picturefor a �xed hamiltonian. The investigation of phase diagrams of particular models(notice that there are in principle in�nite parameters in the models like (3) above!)should be based on the study of the mapping(1.2) f hamiltonian 7�! the ground states of fht(y)g gusing theorems from the di�erential calculus of (in�nitely) many variables (like theimplicit function theorem). This may require a suitable technical modi�cation ofthe de�nition of the contour functional F and the quantity ht(y)3 to obtain as nicedi�erentiability (even local analyticity) properties of (1.2) as possible.Acknowledgements. The second author (M.Z.) thanks the Erwin Schr�odingerInstitute for hospitality during the time of the autumn (1995) semester \Gibbsrandom �elds and phase transitions".Unfortunately, the organizer of the semestr and our teacher R.L. Dobrushin couldnot already come. We dedicate this paper to his memory.II. General Description of the Considered Model.Transcription to an Abstract Pirogov { Sinai Type ModelGiven a con�guration space(2.1) X= SZ� ; � � 33Such a modi�cation could act on the nonground values of y only; the ground values of ht(y)have nontrivial physical interpretation and there can be no arbitrariness in their de�nition!4

where S is a �nite set(of \spins") we will consider a general \horizontally invariant"(\strati�ed") hamiltonian onX: The hamiltonian will be a �nite range one (in whatfollows we consider a suitable norm on Z� e.g. the l1 one)(2.2) H�(x�jx�c ) = XA\�6=;diamA�r�A(xA)where �A are some \interactions", i.e. functions on SA with values in R [ +1,which are\strati�ed" in the sense explained below.We will study the structure of (strati�ed) Gibbs states of the model, more pre-cisely of the probabilities4 which are given in �nite volumes � by formulas(2.3) P x�c� (x�) = Z(�; x�c )�1 exp(� 1T H�(x�jx�c ))where T is the \temperature" and the partition function Z(�; x�c) is(2.4) Z(�; x�c ) =Xx� exp(� 1T H�(x�jx�c)):Suitable in�nite volume limits will be constructed from these �nite volume Gibbsstates.Notes. 1. In fact, some other (more special than (2.4)) partition functions {namely so called (strictly) diluted partition functions will be important later andthe recurrent structure of the measures (2.3) { which is formulated by the DLRequations { will not be used explicitly in our later approach. More adequate forour later approach is the idea that � is some (large) volume which will be �xed inthe main part of our future considerations. (Only at the very end of the paper {when proving and interpreting our Main Theorem, section 3.8 { this \playground"� will be expanded to the whole Z� and the limit Gibbs states thus obtained willbe investigated.)2. In the following we will always put T = 1 i.e. we include the term 1T into thede�nition of �A and H. Thus the temperature will be just one of the parametersin the hamiltonian. We emphasize that in this paper we are interested only in theclari�cation of the phase picture for a given �xed hamiltonian. Doing this, one canstudy the change of this picture (and of relevant quantities like the free energies)when the parameters are changing. Our approach gives some basic tools for doingthat: namely we de�ne useful quantities calledmetastable free energies which reallygovern the behaviour of the phase diagram { see our Main Theorem. However, therelevant result on the behaviour of the phase diagram is not even formulated in ourpaper!3. One could be interested in the structure of Gibbs states, under suitable boundaryconditions, also for other in�nite volumes like the halfspace Z�+ = ft 2 Z�; t� � 0gor in a layer. It is not hard to see that such a situation could be modeled on � =Z�by choosing a suitable modi�cation of the hamiltonian: for example if we put�A(xA) = +14The probabilitiesPx�c� (�) are called �nite volume Gibbs states under boundary condition x�c .5

whenever A 6�Z�+ and xA 6= fxt = �xt; t 2 Ag we obtain a limit Gibbs state on Z�+under the boundary condition �x on Z�� =Z� nZ�+.4. In fact, sensible and nontrivial results (requiring the full strength of all theforthcoming constructions) can be formulated even for a �xed �nite volume �(imagine the cardinality j�j = 1027!), with suitable boundary conditions. However,in this case it is of course natural to study also a torus with periodic boundaryconditions. Though we do not work out here the (topological) modi�cations neededto carry our study from the caseZ� to the case of periodic boundary conditions, weexpect that only minor parts of the text should be adapted or replaced by anotherarguments (for example the parts of the text using the lexicographic ordering ofZ�). 1. Stratified configurations, hamiltonians and statesFor any u 2 Z��1 consider the shifts U � ft 7! t + (u; 0)g : Z� ! Z� andcorrespondingly de�ne the shiftsfx 7! U(x)g : SA ! SU(A)where U(x) = ~x has coordinates ~xt+u = xt, t 2 A and�A 7! U�U(A)where U�U(A)(U(xA)) = �A(xA). Say that a con�guration x is strati�ed (orhorizontally invariant ifU(x) = x for each u 2Z��1 = f(t1; : : : ; t��1; 0)g �Z� :De�nition. We denote by S�X the collection of all strati�ed con�gurations.Analogously we de�ne the notion of a strati�ed hamiltonian H = f�Ag and astrati�ed (Gibbs) measure � by requiringfU�Ag = f�Ag ; U(�) = �for each u 2Z��1.Notes. 1. These will be the \local ground states" of our model. Of course onlysome of these con�gurations will \deserve" this name. Analogously, in the tradi-tional Pirogov { Sinai situation, only some of the constant (resp. periodic) con�gu-rations \deserve" the name of \local ground state". However, it is often problematicto separate these \true local ground states" from the other horizontally invariant(analogously, translation invariant resp. periodic) con�gurations. We will see be-low that a nonexistence of a substantial energetic barrier between the \true localground states" and the remaining elements of Swould deteriorate the validity ofthe Peierls condition. The best solution in such a situation seems to be to choosethe reference family Sas big as above and to look for energetical barriers betweenSand con�gurations which are not strati�ed.55This is an interesting methodological point even for the ordinary Pirogov { Sinai theory. Wenow suggest to consider the family of all translation invariant con�gurations as the \referencefamily"of con�gurations in the ordinary Pirogov {Sinai setting. Such an approach leads, in fact,to a sharper and clearer formulation of the Peierls condition { see below.6

2. The framework when all local ground states of the model are assumed to bestrati�ed (analogously: translation invariant in the ordinary Pirogov { Sinai theory)seems at �rst sight to be too narrow in the situations (like Ising antiferromagnet)where periodical (local) ground states occur. However, it is easily seen that period-ical resp. horizontally periodical con�gurations can be converted to constant resp.strati�ed ones by taking blockspin transformation and so the setting we introducehere is su�ciently general.3. The fact that we are selecting several con�gurations (in fact the whole family S)as the \reference" ones { expecting that some of these con�gurations may (possi-bly, under suitable adjustment of the hamiltonian) give rise to corresponding Gibbsstates { suggests that our interest lies in the situations where phase transitionsmayoccur . Thus, the possible \degeneracy of the ground state"6 is the situation of ourinterest. Though in most situations we will have to deal with only one Gibbs statecorresponding to a given hamiltonian we want to have a theory dealing at the sametime with the situations of phase coexistence. This requirement distinguishes thePirogov { Sinai theory from the methods focused on the study of the unicity region.Recall that in the region of phase unicity other, well developed methods of study(based essentially on the Dobrushin' s unicity theorem and later investigations ofthe \complete analyticity" properties by [DSA]) are available.On the other hand, in the regions where phase coexistence is expected no seriousalternative to the Pirogov { Sinai theory exists reaching a comparable level ofgenerality and universality of its applications.2. Precontours and Admissible Systems of PrecontoursGiven a con�guration x 2 Xsay that a point t 2Z� is strati�ed point of x, moreprecisely y-strati�ed of x (where y 2 S) ifx~t = y~tholds for each ~t 2 Z� such that j~t� tj � r.Note. This is an analogy of the notion of a correct point of the ordinary Pirogov {Sinai theory; r is the range of interactions.Say that � � Z� is a standard volume if any t; t0 2 �c with the same lastcoordinate t� = t0� can be connected by a \horizontal" (keeping the last coordinateintact) connected path in �c , i.e. if all the sets Cn = ft 2 Z�; t� = ng n � areconnected. A con�guration x 2 X will be called y-diluted , for y 2 S, if there issome E �Z� with standard �nite connected components such that all points of Ecare y-strati�ed.This value y will be called the external colour of x. We will use the notationy = xext.6By a degeneracy of a (local) ground state one usually means the fact that several (local)ground states exist for a given hamiltonian. Here, we are looking for (local) ground states amongthe elements ofS i.e. for the con�gurations y 2 Ssuch thatPA(�A(xA)��A(yA)) > 0 wheneverx di�ers from y on a �nite set whose vertical size is \not too big").7

Precontours. For any diluted x denote by B(x) the collection of all its nonstrat-i�ed (i.e. strati�ed for no y 2 S) points. If C is a connected component of B(x)then the pair � = (C; xC)will be called the precontour of x and we will writeC = supp� :The pre�x pre- suggests that the notion of a precontour is a provisional one. Itwill be replaced below by a more elaborate notion of a contour, with more convenientproperties:Admissible systems of precontours. By an admissible system of precontourswe will mean any systemD = f�ig of precontours which is a collection of all precon-tours of some diluted con�guration x. The con�guration x is uniquely determinedby D only in all horizontal levels intersecting D , otherwise it will be given by thecontext (typically by the boundary conditions outside of the given �nite volume, inwhich we will be normally working) and it will be denoted by xD .Of course, this is a slight abuse of notations but we will see later that the quan-tities de�ned below as functions of xD will not depend, in fact, on this ambiguityin the choice of xD.We will also use the notion of an admissible system of precontours D in a given(standard or nonstandard) �nite volume �. In such a case, we will assume that acon�guration x = x�D can be de�ned in the whole lattice Z� having the followingproperties : i) D is the collection of all precontours of x = x�D and ii) all the pointsof �c are strati�ed points of the con�guration x = x�D.3. ContoursIn the ordinary Pirogov { Sinai theory, where elements of the \reference family"Sare constant con�gurations, one immediately realizes that precontours are \crusted"in the sense that the (only!) in�nite component of (supp�)c { called the exterior of� { satis�es the property that all its points are \y-correct" where y is the externalcolour of �. Even more importantly, the interior of � (the union of the remainingcomponents) is \disconnected" from the exterior of �. This enables to construct\telescopic equations" relating the diluted partition function in a given volume � tothe \crystallic" (see e.g. [S]) partition functions of the external contours appearingin �, and therefore again to the diluted partition functions of the interiors of thesecontours.This is not so here, where precontours can have the shape of Dobrushin's \walls"(see below) separating various types of \ ceilings " (i.e. various horizontally invari-ant con�gurations in our case) and the possible sense of the very notion of anexterior resp. interior of a precontour surely deserves a clari�cation.With such \noncrusted" objects, one has to be more careful when de�ning suit-able hierarchy between them; the usual concepts from the ordinary Pirogov { Sinaitheory like the concept of an \external contour" or the notion of a contour \inside ofthe other one" must be de�ned more cautiously and precontours are not the suitable8

objects to do that. Imagine various precontours like \�ngers" or \bumerangs" in-tersecting \interiors" of other precontours having the shape of Dobrushin's \walls"{ for example the case of two walls each having an \appendix, a �nger" touchingthe \interior" of the other wall { to be assured that the notion of an external orinternal contour requires a careful de�nition here.We should warn the reader that there will be no telescopic equations in ourapproach. However, the quantity A(�) constructed in section III.3 substitutesthese telescopic equations, in some sense.The de�nition of a wall suggested by Dobrushin (precontours \of the interface"are called \walls"in [D], and correspondingly also in [HKZ] and [HZ]) resolves theseproblems in the special situation of one interface, by considering the projectionof the situation appearing at the interface to the sublattice Z��1. However, thisconstruction can be hardly transferred to the situations where two or more parallelceilings appear.Thus, the proper de�nition of a contour in our situation (where general strati�edphases appear and where possibly many rigid interfaces appear in the consideredphases) cannot follow literally the concept of the above mentioned \wall". Wewill choose another aproach, which does not use an auxiliary transcription of thesituation to the dimension � � 1:The crucial notion in our approach will be that of the \exterior" (in Z�!) of anadmissible system of precontours ; alternatively the complementary notion of aninterior i.e. the volume \swallowed" by the given admissible system of precontours:(Notice that the de�nition below still follows essentially the original Dobrushin'sapproach.)De�nition. Let � = f�ig be an admissible system of precontours in Z� (orin a given standard volume �). We denote by ext� the collection of all pointsof (supp�)c which can be accessed from in�nity by some vertically constant (i.e.keeping the last coordinate intact), correct (in the sense that each point of the pathis a strati�ed one) connected path. We denote by(2.5) V (�) = (ext�)c :Note. This is just the intersection of all standard volumes containing supp�. Inthe rest of the paper { the exception will be the proof of our Main Theorem (in fact,this exception will be really relevant only for some explanatory notes interpretingin some detail the structure of the phases constructed by Main Theorem) we couldwork mainly with �nite standard volumes � and always with a boundary conditiony 2 S given on the boundary of the volume �c ; i.e. we will work with dilutedcon�gurations having a �nite number of precontours only.Now we come to the de�nition of contours:De�nition. Say that an admissible subcollection �0 � � is removable from the ad-missible collection � (we will also alternatively say that �0 is interior in �{ thoughthis characterization has not such a selfexplanatory meaning as in the ordinary,translation invariant, Pirogov { Sinai theory ) if(2.6) V (�0) \ supp(� n �0) = ;:and moreover if � n �0 is again an admissible system.9

Note. Imagine that the subsystem �0 was \replaced by its external colour" (in-duced by �0 inside V (�0)). Removability of �0 means just the possibility of such areplacement .De�nition. An admissible collection � of precontours in a given volume �, withV (�) � � and with no removable subfamilies will be called a contour in �.Note. Thus, contours are some \minimal " { in the sense that no subsystem can beremoved from them { collections of precontours which exist as admissible systemsin the given volume �. If all the horizontal sections of � are even simply connected(i.e. have exactly one component) then any contour in � is also a contour in thewhole latticeZ�. Otherwise, the property \being a contour" depends on the volume� and the boundary condition, too . For example, imagine a precontour having ashape of a \bumerang" with di�erent colours on its corners which are assumed tohave the same vertical level. This can be a contour in a suitable bigger \bumerang"� { which will be typically the interior of some other,\exterior" contour. However,it is not the contour in the whole lattice Z�.4. Representation theoremDe�nition. We introduce the structure of an oriented graph on the collection ofall contours more precisely the following \hierarchy" between the contours: Write�! �0 whenever � [�0 is admissible, V (�0) \ supp� 6= ; however � is removablefrom � [ �0.Note. The notation �! �0 will be frequently used, in an analogous sense, also forgeneral admissible systems (not only for contours) � and �0.To say that � is a contour of a given con�guration will not be so straightforwardas for the precontours (and as in the ordinary translation invariant situations ofthe Pirogov { Sinai theory). However, the following representation theorem is stillvalid:Theorem 1. Any diluted con�guration x 2 X having a �nite B(x) is uniquelyrepresented by a graph on some �nite subcollection of the collection of all contoursof the model; such a graph is always a \forest" of trees. By a tree we mean anoriented connected graph without loops i.e. without cycles of bonds (cycles of\arrows") of the type �1 ! �2 ! : : :�n ! �1. The external contours of theforest (namely those contours which are not the starting points of some arrowsof the graph) have mutually disjoint volumes, and more generally any subsystemof the forest constructed by the rule \if the end of the arrow is removed then thebeginning of the arrow is removed, too" is an admissible system inZ�. The mappingf con�guration! forest of trees g is one to one; in particular any forest correspondsto some con�guration of the original model. An analogous statement is also truefor any diluted con�guration in any �nite standard volume.7Notes. 0) The statement that any forest corresponds to some con�guration of theoriginal model will not be used in the following. What will be used only is the prop-erty that the considered collection of forests is horizontally translation invariant .7Once again: any x is uniquely determined by the family of its contours. This family is a\forest of trees" in the relation ! . Conversely, any such \forest of trees of contours" is anadmissible system i.e. it determines (uniquely) some con�guration.10

1) The generalization of the above result to general diluted con�gurations in in�-nite volumes with in�nite B(x) will not be considered here. When formulating theproperties of typical con�gurations of in�nite volume Gibbs states (constructed asthe consequence of our Main Theorem at the very end of the paper), this could bedone by saying that any such con�guration can be interpreted as an in�nite forestof �nite trees (connected graphs of contours without loops). However, one reallyneeds such formulations only for some explanatory notes commenting in more de-tail the structure of in�nite volume Gibbs states constructed by our Main Theorem.Otherwise, there will be no need for the consideration of con�gurations in in�nitevolumes in the rest of the paper.2) To reconstruct the con�guration x uniquely from the forest, we need to knowalso the external colour xext. Namely, the external colour of the forest can be onlypartially recovered from the contours of the forest. However, its value elsewherewill be usually given by the context (by the boundary conditions outside of the�nite volume, in which we will be actually working).De�nition.Removable contours of the forest will be called the internal contours of x. A contourwhich can appear as a single remaining contour after some succession of removalswill be called the external contour of x. By a removal of �0 from � we mean thereplacement of x� by x�n�0 .Proof of Theorem 1. It is based on the followingLemma 1. If � and �0 are two di�erent internal contours of an admissible systemD then(2.7) V (�) \ V (�0) = ; :Note. By an internal contour of D we mean here (we have not yet proven thetheorem!) any minimal possible removable subsystem of D .Proof of Lemma 1. Denote byC = V (�) \ V (�0) :We will show that the assumption C 6= ; would lead to the removability of thesystem of all precontours of � contained in C and this would be in contradictionwith the fact that � is contour. Denote by y resp. y0 the external colours � resp.�0: these con�gurations are de�ned locally for any nonempty sliceCn� = V (�) \Z�nresp. Cn�0 ; here we denote by Z�n = ft 2Z�; t� = ng. For any n 2 N such thatCn = Cn� \ Cn�0 6= ;de�ne y� as the external colour of Cn. Notice that all the points of @Cc arestrati�ed (because they belong either to V (�)c or V (�0)c) and thus we have eithery� = y or y� = y0 for any level n such that Cn 6= ;. Notice also that all points of11

@Cc can be accessed \from the in�nity" by a connected horizontal path belongingto Cc. (We will not prove here this obvious fact, saying that the intersection oftwo simply connected sets is again simply connected.) Now there are two possiblesituations:(1) If there is some n 2Zsuch thaty�n = y0n 6= ynthen we have the obvious relation (look at the level n !)supp� n C 6= ;and therefore the admissible system �� of all precontours of D contained in C mustnot contain all precontours of �. However �� is removable (just replace �� by itsexternal colour y� in C!) which is a contradiction with the fact that � is a contour.(2) If yn = y0n for all n (where both colours are uniquely de�ned) then it is easyto see also that y� = y = y0 on @Cc. Then we follow an analogous argument as in(1): The condition C 6= ; would mean that the collection �� is also a contour {and this is not a contradiction (with the fact that both � and �0 are contours i.e.with no removable subsystem ��) only if C = V (�) = V (�0). However, then e.g.�n�� must be empty because otherwise the intersection of �n�� and V (�0) wouldbe nonempty and �0 would not be an interior contour of the system.Proof of Theorem 1. We proved in Lemma 1 that interior contours of x areuniquely de�ned, with mutually disjoint volumes V (�). We can remove all theseinternal contours (in an arbitrary order!) thus obtaining some new con�guration�x. Then we determine all the internal contours of �x; after removing them from �xwe obtain another con�guration ��x etc. At the �nal step some collection of contourswhich are both internal and external remains; their removal has as its result thestrati�ed con�guration xext. The bonds of the forest representing x are all thepairs of the type � ! �0 where � is an internal contour of some intermediatecon�guration (x; �x; ��x; : : : ) and �0 appears as an internal contour after successiveremoval of all � such that supp� \ V (�0) 6= ;.Notes.1. Of course one can easily construct examples of \loops of contours"�1; : : : ;�n+1 =�1 such that �i ! �i+1 for each i = 1; : : : ; n but no such cycles can appear in therepresentation by Theorem 1. They form a single contour there!2. Analogously, one can formulate an analogous representation theorem for con�g-urations in �nite standard volumes.5. Connectivity of contoursTheorem 2. Contours are \halfconnected" in the sense that(2.8) j supp�j � 6481 con�where con� denotes the minimal possible cardinality of a connected set containingsupp�.(Compare also Theorem 2' in Section II.6.)Note. The notion of con� will be modi�ed in later sections, its new value (denotedby conn�) being actually smaller than con� (however without having a simplifyinge�ect on the proof below.) 12

Proof. We have to understand the structure of contours of our model { whichare de�ned as some complicated \conglomerates" of (connected!) precontours. Wepresent here only an outline of the proof leaving out the \topological" details.De�nition. Let f�ig be an (unadmissible) family of precontours. Say that the twopoints t; s 2 @([i supp�i)c with the same height t� = s� are in a con ict in f�igif their colours (induced by neighboring precontours) are di�erent and at the sametime t and s can be connected by some connected horizontal (keeping t� constant)path not intersecting [i supp�i. Say that � cures the con ict between t; s if t; sare no more in con ict in f�ig & f�g.Lemma. Let �1;�2; : : :�n be a sequence of precontours such that each �k curessome con ict in f�1;�2; : : :�k�1g. Then f�1;�2; : : :�ng is 89connected.Proof of Lemma. We will construct for each f�1;�2; : : :�kg a connected setMk � [ki=1 supp�i by the following inductive procedure: Connect each �k to the(yet constructed) \connected conglomerate" Mk�1 � [k�1i=1 supp�i in the followingway: there must be some horizontal sectionCnk = supp�k \Z�nsuch that in some internal component of (Cnk )c (taken in the lattice Z�n), somepoints of [k�1i=1 �i can be found. (Otherwise �k would cure no con ict.) Now takeMk as the union of Mk�1 and supp�k and of some shortest path connecting Cnkand Mk�1. Clearly, the lenght of such a shortest path is no more than 18 of thecardinality of Cn.De�nition. Say that an admissible collection f�1;�2; : : :�ng cures �1 if each �k,k � n cures some con ict f�1;�2; : : :�k�1g.(Recall that an admissible family of precontours in � is a family of precontourswhose \outside colours" are not in mutual con ict.)Proof of Theorem 2. Take some precontour �1 2 � and cure it successivelyto obtain some 89 connected subcontour ~�1 of �. Denote by �1 some maximal89 connected supersystem of ~�1 in �. Asssume that �1 = � n �1 is nonempty.(Otherwise, the proof of the theorem would be complete.) Take some �2 2 �1 andcure its con icts analogously as �1 was constructed, and so on.(Notice that theconnecting paths starting from the precontours curing some con icts in �2 etc. donot touch �1 yet!)Thus we obtain some decomposition(2.9) � = N[i=1�iwhere �i are 89 connected contours such that [Ni=k+1�i is nonremovable from[Ni=k�i. Take the admissible collection [N�1i=1 �i i.e. � n �N and denote byD 1 ; : : : ; Dm the internal contours of this truncated system. Notice that all V (D i )must be intersected by �N (those nonintersected by �N would be removable from�!) and therefore �N looks like a \bumerang" such that Di are like some \rings"entwining it. 13

Clearly, we can connect �N to any D i \at the expense of D i" by some shortestpath from �N \ V (D i ) having a lenght at most 18 of the cardinality of D i . Forany D i only one path will be constructed. This procedure of \making connectionsabove" can be replaced also for the bigger \bumerang" �N [ [iD i \at the expenseof the internal contours" of the system � n (�N [ [iD i ) etc. Assuming yet that wehave submerged supp�i into some connected set C�i such that(2.10) cardC�i � (1 + 18) card supp�iwe can submerge supp� into some connected set C�� such that C�� = C� [ P ,where P is the support of connected paths constructed above andcardC�� � (1 + 18) cardC� � 8164 card supp�:6. SupercontoursThe contours de�ned so far would still have some inconvenient features later.Remember that they are \not crusted" { in the sense that the internal contours �intersecting some other V (�0) need not to satisfy the condition V (�) � V (�0).8This is a fundamental obstacle { which apparently can not be remedied by some\better" de�nition of a contour. However, even the fact that the external contour�0 can be much \smaller " than � would be rather inconvenient in our followingconsiderations. (This will be so for pure technical reasons ; see part III, Theorem 5and also Theorem 7.) The latter inconveniency can be, however, remedied; one canrede�ne contours such that �0 is always \substantially bigger" than � if �! �0 :Below, we will \glue together some contours" to achieve this property, keeping stillthe validity of Theorem 2 (with a slightly smaller constant) for the newly de�nedconglomerates of contours. These conglomerates will be called supercontours andbelow (in Part III ) we will work exclusively with them (instead of working withcontours).De�nition. For any system of contours D introduce its \diameter"(2.11) diamD = max�2D diam�:Consider some (total) ordering � on the set of all systems of contours satisfyingthe following requirements : 1) if diamD < diamD0 then D � D0 ; 2) if D � D0then diamD � diamD0 ; 3) if D is a subsystem of D0 then D � D0 ; 4) if D = D0+ twhere t � 0 in the lexicographic order on Z� then D � D0.Notes. The particular choice of a norm on Z� is not relevant here. However, inview of the further usage of the notion of a diameter in part III let us make theagreement that from now on the l1 norm jtj = maxftig will be used everywhere inwhat follows.The lexicographic order on Z� is assumed to be �xed throughout the paper . Therequirements 1), 2), 3) and 4) clearly de�ne a partial order and we simply extendit to some total ordering of the family of all systems of contours.8Such a property was valid, it seems to us, in all the previous applications of the Pirogov {Sinai theory. 14

De�nition. Say that a contour of the \forest" of Theorem 1 has an index n > 0 ifit is an internal contour of the forest remaining after i) the removal of all the internalcontours of the forest (these contours will be said to have the index 1) and thenafter ii) the successive removal of all the contours having the index 2; 3; :::; n� 1.For any contour �0 having the index 2 �nd the biggest (in � ) contour � suchthat � ! �0. If � is bigger than �0 or the cardinality of supp� is bigger thanthe cardinality of supp�0 connect the contour �, by a shortest connected path, to�0. Moreover, if the cardinality of the support of the conglomerate thus formedby � and �0 (and their connecting path ) is still not at least twice bigger thanthe cardinality of supp�0 for any other interior �00 ! � repeat the procedure onceagain with �00 instead of �.Repeat the same process with the collections of all remaining interior contours(kept intact after the glueing procedures above) � ! �0 which are bigger (in thesense above) than some contour �0 having the index 3 . Then repeat the sameglueing procedure with contours having the index 4 etc.Finally remove all the remaining interior contours (not used in the glueing pro-cedures above for n = 2; 3; : : : ) and repeate the whole above contruction again andagain.Supercontours of the original forest are then de�ned as the connected (by pathsconstructed above) conglomerates of contours of the original forest. The notion ofa supercontour includes also all the contours left intact by the above procedure.We will use the short name supercontour for a supercontour of some forest. No-tice that the relation! between supercontours is de�ned in a natural way becauseglueing cancels the arrow � ! �0 but does not e�ect the remaining arrows of theforest.Theorem 2'. The new forest obtained by the construction above is uniquely deter-mined by the old forest. Any forest of supercontours whose external supercontoursform an admissible system can appear as the result of the construction above. Therelation � ! �0 is a subset of the relation � � �0 for all pairs of supercontoursof the given model. Moreover, the cardinality of supp�0 is always at least twicebigger than the cardinality of supp�. Supercontours satisfy again the statement ofTheorem 2, with the constant 64=81 replaced by some smaller constant, e.g.(2.8') j supp�j � 1=2 con�:Note. These properties will have no special importance in the rest of part II { butthey will be quite convenient later in Theorem 5 and also in Theorem 7.The proof is quite straightforward. To see that Theorem 2 remains valid noticethat the connecting path constructed in any step of the construction of a supercon-tour is done \at the expense" of the external contour �0; notice that any contour �0of the original forest is used at most once in such a construction, and to constructa connected path \inside" �0, one does not need more than 1=2 con�0 points (and81�264�3 < 12 ).In the rest of the paper, we will work exclusively with supercontours (of a givenadmissible system of precontours). We will omit the pre�x \super" in the following.(However, this will be important only later, in part III.)15

7. Expression of the hamiltonianFor any strati�ed con�guration y 2 Sde�ne its \density of energy" at t 2Z� :(2.12) et(y) =XA3t�A(yA)jAj�1 :Given a contour � one would like to de�ne also a quantity having the meaningof the \contour energy". One could think for example about the \energy excess"of H(x�), where x� denotes the con�guration having � as its only contour, withrespect to \something like H(xext)" where xext denotes the \external colour of �".However, such a straightforward approach to the de�nition of an energy of a contouris reasonable only in the very special cases { when the density of energy inside � isthe same as outside of �, at any horizontal level.Otherwise, it will be necessary (to keep the interpretation of a contour energy asa quantity which is \localized on supp� ") to replace the quantity H(x�)�H(xext)(which will be, of course, also very important later { see (2.19)) by the following,perhaps too formally de�ned at �rst sight, quantity: �rst extend the notation et(x)for any (even nonstrati�ed in t) x by putting(2.13) et(x) = et(x̂) ; t = (t1; t2; : : : ; t�)where x̂ is the strati�ed continuation of the vertical section fx(t1;:::;t��1;(�))g. Nowde�ne the con�guration x = xbest� minimizing the sum (notice that the terms ofthis in�nite sum are �xed outside supp�!)(2.14) Xt2Z� et(x)under the condition that x = x� on (supp�)c and also on the set@ supp� = ft;dist(t; (supp�)c) � rg :Put9(2.15) E(�) = H(x�)�Xt2Z� et(xbest� ) where H(x) = XA�Z� �A(xA):Then we have the following expression of the hamiltonian (� will be always a�nite set in the sequel) :Theorem 3. Let x be a diluted con�guration in �. Then(2.16) H(x� jx�c) =Xt2� et(xbestD ) +X�2DE(�)where D denotes the system of all contours of x and xbestD is de�ned as above (withD instead of �), by (2.14).9Of course this is again only a formal expression { but obviously the terms in the sums on theright hand side of the equation for E(�) can be reorganized such that a sum with only a �nitenumber of nonzero terms is obtained. 16

Proof. Immediate, if we notice that E(�) is a local quantity (depending on �only) and also an additive one:E(� [ �0) = E(�) +E(�0):Notice that whenever � is the unique contour of x then (2.16) follows directly from(2.15).8. The Peierls condition. The abstract Pirogov { Sinai modelIn the following we will assume that for any set G and any strati�ed con�gurationy 2 Sthe following (Peierls type) inequality holds with a su�ciently large constant� > 0:(2.17) X�: supp�=G & xext� =y exp(�E(�)) � exp(�� jGj)We recall that we include the inverse temperature into the hamiltonian and therefore� is of the order of the inverse temperature.Notes. 1.In practice, one usually establishes (2.17) through the inequalities(2.17*) E(�) � � �jGjwith suitable � � � � .2. In the following we will work exclusively with the expression (2.16). It isgenerally advisable to develop the Pirogov { Sinai theory in an abstract setting(2.16) { with the Peierls condition (2.17) established. The reformulation of theoriginal model to the language (2.16) can be considered as a suitable \preparation"of the given \physical model" { and the Pirogov { Sinai theory actually only startswith the setting (2.16) & (2.17).3. Such a preparation of the model (i.e. a conversion to the form (2.16) bya suitable de�nition of the notion of a contour) is often not unique i.e. it is not\naturally determined" by the given model . One can adapt it in various waysfor various concrete situations { by modifying the concept of a \strati�ed point",for example, or even by considering contours as objects di�erent from those weconstructed here. Remember, for example, that in the ordinary theory of the lowtemperature Ising model, contours are commonly de�ned as selfavoiding paths inthe dual lattice.9. Diluted and strictly diluted partition functionsIn addition to the volume V (�) there will be another (actually more importantin the following) modi�cation of this notion, denoted by Vl(�). The details of itsde�nition will be important only later. See also (3.19) in part III; Vl(�) will bede�ned here as a suitable \cone over V (�)", more precisely as follows:1010The \protecting zone" Vl(�) over V (�) constructed below is apparently unnecessarily big.This will make no harm, however. In later sections of part III, we will be more strict when tacklinganalogous di�culties as in the de�nition (3.19). The logarithmic height of the \caps" of Vl(�)over V (�) would be quite su�cient, in fact. 17

Conoidal sets. Say that a volume � is a conoidal volume (or conoid) if it contains,with each \horizontal set" B � Z�m \ � (where Z�m is the collection of points ofZ� with the �xed last coordinate t� = m) the whole \cone" ft 2 Z� : dist(t;B) �12 dist(t; @B)g where @B denotes the boundary of B taken in Z�m. If � is a contouror an admissible system we de�ne Vl(�) as the union of supp� and the smallestconoid containing V (�)n supp�. Equivalently, this is the union of V (�) and of thesmallest conoid containing all the \upper and lower ceilings of �" i.e. the at partsof the boundary of V (�) which are outside of supp�.We denote, from now on, by symbols Zy(�) resp. Zyl (�) the partition functions(2.18) Zy(�) =X exp(�H(x�jy�c ))where the sum is over all diluted con�gurations whose all contours satisfy thecondition(2.18') dist(V (�);�c) � 2resp. analogously for Zyl (�),(2.18") dist(Vl(�));�c) � 2 :Of course, for conoidal � the both partition functions (2.18') and (2.18") are thesame.The rest of this paper (and the essence of the Pirogov { Sinai theory in itspresented version) consists of the e�ort to expand the considered diluted partitionfunctions (3.18) (more precisely (3.18")) as far as it is possible or reasonable { inorder to deduce some useful corollaries from these expansions.The attempt to expand partition functions (2.18) can be based on the older ideaof a contour model [PS] (or of a metastable contour model [Z]). Though this notionin fact disappeared from the presented version of Pirogov { Sinai theory { insteadof speaking about suitable \metastable contour models" we will work, in fact, onlywith expansions of their partition functions (i.e. the partition functions of themetastable submodels of the given model) { it is perhaps useful to start with someintuitive arguments suggesting the introduction of the basic notion of a contourfunctional . This is just the introduction to the later, more technical constructions.We will see later that the very notion of a contour functional ([S]) \survived" inour approach (in contrary to the idea of a contour model) and it is still of a centralimportance! 10. The idea of a contour functionalThe basic task of the Pirogov { Sinai theory is to determine those con�gura-tions y among the \reference" ones (reference means strati�ed in our case) whichare stable in the sense that they give rise to Gibbs states whose almost all con-�gurations are some \local" perturbations of the considered reference (strati�ed)con�guration. More precisely, they are \y-diluted". A useful and intuitively appeal-ing tool to determine whether a given con�guration is \stable" is the construction18

of a \metastable model" [Z] (around the given reference con�guration). To de�nesuch a metastable model one introduces ([S], [Z]) an auxiliary quantity called the\contour functional" F (�) (the \work needed to install the given contour") which\tests" those contours whose appearance as of external contours of the metastablemodel is allowed. To get an idea of such a testing quantity let us start with its\zero temperature version": Put(2.19) F0(�) = H(x�)�H(xext� ) = E(�) �A0(�)where(2.20) A0(�) = Xt2V (�)(et(xext� )� et(xbest� ))This quantity is just a �rst approximation to the more relevant quantity given atthis moment only formally by(2.21) Fformal(�) = logZy(Z�)� logZ�(Z�)where y is the external colour of �(y = xext� ) and Z� denotes the partition function\over all con�gurations on Z� containing the contour �". Below we will de�ne, byrelations (3.21) & (3.22), a rigorous counterpart of this quantity, which will play avery important role in the sequel.For contours which are \not very big" the quantity F0 is a good approximationto Fformal. It enlightens somehow the concept of a small contour used below; theterm A0(�) typically satis�es an estimate like(2.22) A0(�) � CjV (�)jwith a constant C which is su�ciently smaller than � (imagine the Ising modelwith a small external �eld) and therefore if e.g.CjV (�)j � �2 j supp�j(this will hold for contours which are \not too big") we have, from the Peierlscondition (2.17*), the inequality(2.23) F (�) � E(�) � CjV (�)j � �2 j supp�j :We see that y = xext� is really a \local ground state" because installing of a \nottoo big" contour increases its energy.Unfortunately, it is not trivial to de�ne quantities like Fformal rigorously. Whilesuch a task is solved rather straightforwardly in other situations of the Pirogov{ Sinai theory (where contours are \crusted" { in the sense that there is no de-pendence between events inside and outside �), here the presence of \ceilings" ( athorizontal parts of boundaries of V (�) which do not belong to supp�) causes prob-lems! These problems lead to the necessity of considering of suitable expansions,and this is the main subject of the forthcoming part of the paper.19

III. The Concept of a Mixed (PartiallyExpanded) Model. Recoloring.This, the main and the �nal part of the paper is devoted to the constructionof suitable expansions of partition functions of models considered in part II. Herewe introduce the important technical notion of a \mixed" (or, partially expanded)model which serves as an intermediate construction between the original concept ofan \abstract Pirogov { Sinai model" and our �nal aim which is an utmost expansionof the partition functions of the considered model.Cluster expansions were always an important tool in the Pirogov { Sinai theory.However, in previous versions of this theory, the expansions were viewed merely assome auxiliary technique applied to the study of special polymer models (contourmodels) which were constructed �rst. One could think that the cluster expansionmethod could be replaced by \something else" giving \comparably nice" expressions(or, possibly, suitable bounds only) for the partition functions of the contour models.This is not so here where the idea of a partial expansion enters even our basicterminology, namely the concept of the mixed model. For example an analogy ofthe notion of a \metastable model" (see [Z]) can not be (apparently) de�ned herewithout the language of expansions; even the formulation of our Main Theoremuses this language.Of course the very idea of a \partial expansion" is not at all new. It was used(in various context also in situations close to the subject of the presented paper -see [I], [HKZ], [B], : : : ) by many authors but mostly as an important but auxiliarytool while in our formulation it is really the cornerstone of the theory.Our basic expansion step (Theorem 5, Theorem 4 and the Lemma preceding it){ called recoloring by us incorporates some of the usual cluster expansion ideology(based on expansion by power series and on the use of equations e.g. of Kirkwood{ Salsburg type) into the very construction of the contour functional.A consequence of our approach is that our use of cluster expansion technique isselfcontained and we need no references to the literature. We can, however, mention[M], [KP], [DZ] (as the papers having direct in uence on the present paper) fromthe numerous literature on the subject of cluster expansions.The construction of one \recoloring step" (Theorem 4) will not yet give therequired expansion of the model. It must be repeated (in�nitely) many times. Theiterative nature of our constructions cannot be hidden \somewhere into the proofs"but appears already at the level of the basic notions.We organize this part of the paper as follows. In section 1 we analyze the notionof a \cluster" (of supports of contours or, more generally, of another clusters); theclusters are then identi�ed with suitable graphs (without cycles) on Z�. Then, insection 2, we de�ne the central concept of a mixed model. This notion correspondsto an idea of a \partially expanded model" ; however it is useful to consider such aconcept in a broader sense.Section 3 describes an important procedure { the\recoloring"(i.e. removing of� from the model & adjusting of the new cluster series such that the partitionfunctions would not change) of a single interior contour �; in the context of ageneral mixed model. The important concept of a recolorable contour (or admissiblesubsystem) is introduced here: it corresponds, roughly speaking, to the validity of20

the Peierls condition for the contour functional F (�). Section 4 applies the resultof section 3 in such a way that recoloring of all the shifts of � is obtained; theresulting new mixed model is again a translation invariant one if the original mixedmodel satis�ed this property. Technically, sections 3 and 4 (and later section 7)form the core of our paper.Later sections 5,6,7 are then devoted to the problems of the succesive construc-tion of \more expanded" mixed models: An important intermediate result is The-orem 6 (section 6) giving a su�cient condition for the recolorability of an interiorsystem of contours in a general mixed model. Namely, to have more speci�c ex-amples of recolorable systems we introduce there a related but better controllablenotion of a small resp. extremally small system of contours (which is more usefulthan mere notion of a recolorable system).The message of the sections 5 to 7 is roughly speaking the following: once thereare some small contours in the mixed model then there is still \something left torecolor" i.e there are still also some recolorable subsystems in the model.The notion of an extremally small system is an elaboration of the older idea of a\small" or \stable" contour ([Z]). Notice that small resp. extremally small systemscan contain the \large" (\not extremally small") contours or admissible systems asits internal subsystems.Theorem 6 is proved with the help of Theorem 7; the latter is already somegeneral statement about the \connectivity constant" of some special (\tight") setsappearing in the study of extremally small systems.Only after �nishing the sequence of all the expansions (recolorings) organized byus we will be able to say what the metastable model is { in Section 8. This willbe the submodel of the original abstract Pirogov { Sinaimodel where only thosecon�gurations containing no \redundant" (i.e. surviving in the \fully expanded"model: by the fully expanded model we mean the �nal mixed model remaining atthe moment when the inductive procedure of its partial expansions was completed)external contour resp. admissible system will be admitted!Section 8 formulates then our main result, using the quantities called \metastablefree energies" just constructed by expansions. It turns out that that the minimalityof the metastable free energy of some y 2 S really means that there are no con-tours at all in the fully expanded model under such a boundary condition i.e. themetastable model corresponding to y gives an appropriate y{ th Gibbs state.The fact that under \stable" boundary conditions, \everything is recolorable"(i. e. the complete expansion of the partition functions is obtained) is the core ofthe proof of the main theorem. Having proved the preparatory Theorems 5,6, thisis now almost a tautology.Our new method based on Theorem 5 and Theorem 6 replaces the previouscoarser arguments from [Z] (which moreover cannot be used in these new situa-tions). However, even in the situation of [Z] our new method is simpler (at leastconceptionally: inequalities for the partition functions employed in [Z] are nowsystematically replaced by corresponding expansions whenever possible) and morepowerful. We plan to show the advantages of this new approach in the study offurther situations which are not covered by the usual variants of the Pirogov { Sinaitheory. 21

1. ClustersThis section prepares some technical notions and constructions needed for theproper formulation of the expansions which are used below.Cluster expansions of partition functions of polymer models are often written,in the literature, in the following form:(3.0) logZy(�) = XT�� kTwhere Zy(�) is the considered (diluted) partition function in volume � under aboundary condition y 2 Sand kT = kyT ; T �Z� are some local quantities (indexedby \connected clusters" T { see below for more details about this notion) whichare \quickly decaying" e.g. like (this will be the form used below by us)(3.1) jkT j � "connTwhere connT is something like the \cardinality of a minimal connected set con-taining the cluster T". ( See below in (3.6) for the de�nition of a quantity connTwhich will be used in our later considerations.)This will be our �nal goal : establishing of such expansions for a collection { aslarge as possible { of diluted partition functions of the given model.More complete information says that the quantities kT are in fact sums of quan-tities indexed by some \clusters of sets (resp. of contours) f�ig" (and having avalue which is a � product, over the cluster, of contour functionals exp(�F (�i)))having the given support T . While one can ignore the detailed description of thestructure of kT when applying the above expressions (e.g. in order to obtain usefulbounds for partition functions { which was the typical application of the clusterexpansions in most previous variants of the P. S. theory) here it will be necessaryto retain the more precise information because these expansions will be iteratedrepeatedly many times.Before de�ning the notion of a cluster formally, we start with the explanationof the notion of connT for the case when T is a set . Our de�nition relates such anotion to �nding of some \ shortest commensurately connected" superset containingT ; this will be important later in this section when analogous construction will beapplied also to a general cluster. We start in fact with the de�nition of a slightlychanged quantity denoted by ConnT which is de�ned in a more direct way.Note. The value of conT used in the de�nition of a contour in part II is inconvenienthere and cannot be reasonably used in what follows. However, our new valuedenoted by connT will be actually smaller than the corresponding quantity ofsection II and this will enable to transfer immediately the estimates of the type(2.8),(2.8') which were established in part II (and later combined with the Peierlscondition) to our present context.In the de�nition of connT (see De�nition 2 below) we will use the notion of anabstract tree, often also with a speci�ed root :22

De�nition. An abstract tree is de�ned as an equivalence class, with respect tothe isomorphisms of graphs, of unoriented graphs without cycles. (By a cycle of agraph G we mean a collection of the type ft1; t2g; : : : ; ftn; t1g composed of bondsof G.) If we want to specify also the root of such a tree (i.e. mark one point of thegraph) then such an object can be de�ned also in a recursive way, just by specifyingthe collection of all subtrees, with marked roots, emerging if the root of the tree isremoved.Note. The identi�cation of a cluster with a suitable tree { which will be given below{ suggests that the following idea of the summation of cluster expansion series willbe developed: instead of estimating the number of various clusters with the samelength we rather employ here the idea of the summation over the trees (based onthe recursive summation over the outer bonds of the tree).It seems that this method gives good estimates. Therefore, we are following thismethod here, in spite of the fact that the treatment given below is maybe too muchgeneral for the purposes of the forthcoming text. Namely, a weaker version couldbe apparently also made which would be closer to our later approach of section 7 {which is based on the notion of a tight set; see the proof of Theorem 7. Neverthelesswe keep the method of summation over trees here, also as a suitable reference forpossible further applications of the method (like the paper [COZ] which is underpreparation).De�nition 1. By a commensurate tree on Z� we mean the following object:(1) It is a pair T = (G;�) consisting of an abstract tree G and a mapping �of this abstract tree G to Z�; the mapping can be constructed, after �xingof some root of the given abstract tree, also recursively (according to therecursive de�nition of an abstract tree given above): the image of the newlyadded root is speci�ed at each stage of the construction. The vertices of theabstract tree G are mapped (generally not one to one) to some subset ofZ� which will be called the support of the tree (denoted by suppT ). Noticethat possibly several vertices of the given abstract tree G can be mappedto the same t 2 Z�. Then these vertices of the tree T will be sometimesdenoted by symbols t0, t00, t000 : : : to distinguish them.(2) The bonds of the tree T constructed in (1) are (unordered) pairs of the typeft; sg ; s = t+ 2k~eiwhere k 2 N ; t 2 2kZ� and where ~ei is either zero or a vector of the canon-ical base of Z�. More precisely we consider bonds of the type f�(a); �(b)g(where �(a) 2 ft0; t00; : : : g and �(b) 2 fs0; s00 : : : g) which are images under �of the corresponding bonds of the abstract tree G. (We put no limitationson the number of such bonds per a given pair ft; sg.)(3) The commensurability is meant here in the sense that if fA = �(a); B =�(b)g and fA = �(a); C = �(c)g are two bonds then(3.2) 12�(A;B) � �(A;C) � 2�(A;B)where the distance �(A;B) between A 2 ft0; t00; : : : g and B 2 fs0; s00; : : : g isde�ned as 2k resp 1 according to whether s = t+2k~ei or s = t in the aboverelation. 23

(4) we de�ne the length of such a tree as the number of its bonds excluding allthe bonds (\loops") of the type ft0; t00g.Note. The usage of the lattices 2kZ� and our very notion of a commensurabilitywill be quite important in the following. The choice of the factor 2 in (3.2) ismore or less arbitrary but convenient later. We should notice that later, in theproof of Theorem 6 below, the notion of commensurability will be transcriptedto an alternate language based on the usage of the unit cubes from lattices 2kZ�(considered as cubes from the original lattice Z�) instead of the employment of thebonds of the type above.De�nition 2. Given any set T � Z� we assign to it a shortest possible commen-surate tree containing for any t 2 T at least one bond of the type ft0; t00g. We willdenote such a tree (it is often not determined uniquely, even if its root is alreadyselected) as T = T (T ) . We recall that the length of the tree was de�ned by (4)above and therefore the loops ft0; t00g are not contributing to the length of the tree;the condition that all such loops are in the considered tree can be replaced by re-quiring that any t 2 T belongs to some bond of the tree having the length at most2. De�ne the auxiliary quantity ConnT as the length of the tree T (T ). In thefollowing, it will be more useful to have a modi�ed version of this quantity, denotedby connT and de�ned as follows:(3.4) connT = ConnT + [3� log2 diamT ]:Note. (3.4) will be a more adequate quantity than ConnT in what follows; seeProposition below. Namely, the clusters of sets will be de�ned below in a recursiveway as collections of objects (sets or contours) \whose diameters are not smallerthan their distance to other objects of the collection"; and when constructing addi-tional commensurate path connecting a given set T with a point in distance diamTone requires an additional amount of � log diamT steps:Lemma 1. Let �(t; s) = d. Then there is a commensurate path starting in the loopft; tg and ending in the loop fs; sg having the length at most equal to [ 3� log2 d ].Proof of Lemma 1. It follows easily from the following considerations: �rst noticethat it su�ces to consider the case of the dimension � = 1. Consider now the pathon Zwith steps having the lengths1; 2; 4; : : : ; 2k; : : : ; 4; 2; 1which overcomes the distance d = 3 � 2k � 2. The length of this path is 2k � 1 �2 log2 d. If 3 2k � 3 � d0 � 6 2k � 3then it is possible to construct a commensurate path overcoming the distance d0simply by doubling some of the steps in the sequence above or possibly by triplingthe middle step. We need at most 2k� 1+ k+2 � 3 log2 d0 steps which completesthe proof.Now we come to the de�nition of a cluster :24

De�nition 3. The notion of a cluster of sets (only some sets will be employed inthe construction of clusters, see below) is de�ned recursively, retaining the letter Tfor the notation of clusters, as follows:i) Any set T = supp�where � is a contour or an admissible system (to be speci�ed below; wewill consider below only some special, \recolorable" systems � which willbe de�ned later) is a cluster.ii) If Ti are some clusters and T0 is from i) such that(3.5) dist(suppT0; suppTi) � minfdiamsuppT0; 2 diam(suppTi)gholds for each i � 1 then the collectionT = (T0; fTig)is again a cluster. We denotesuppT = suppT0 [ [i suppTi :The set T0 will be called the core of the cluster T .Note. The condition (3.5) is a technical one; its adequateness (with respect toour actual constructions) will be seen later in Theorem 4. The appearance of theadditional \logdiam" term (compared to ConnT ) in the de�nition of connT will beseen to be related to our formulation of the condition (3.5). See Proposition below.De�nition 4. We assign, to any cluster T , a commensurate tree T as follows: Ifthe trees T0 and Ti are already constructed { by De�nition 2 and the inductionassumption for Ti (recall that ConnT0 = jT0j where T0 is a shortest commensuratelyconnected tree whose support contains T0) then we de�ne T as the shortest possiblecommensurately connected tree containing (as mutually disjoint subtrees) all thetrees T0 and Ti and such that all branches of T n T0 start with some loop of thetype ft0; t00g. To have an idea about the length of T consider a treeT 0 = T0 [ [i(Ti [ Pi)where Pi are some shortest possible commensurate paths, each of them starting insome loop fsi; s0ig of T0 and ending in some loop fti; t0ig of Ti.In analogy to De�nition 2, the quantity ConnT is now de�ned as the length of thetree T and we put(3.6) connT = ConnT + [3� log2 diam suppT ]:In order to reconstruct back the original cluster T from a given tree T thefollowing notion will be useful: 25

De�nition 5. Assume that some total ordering � on the collection of all subsetsof Z� is de�ned, extending both the lexicographic order between the shifts of agiven set as well as the relation A � B if A � B. Say that a cluster T = (T0; fTig)is a standard one if S � T0 holds for any set S used in the recurrent de�nition ofthe clusters Ti.Notes. 1) Notice that ConnT is not greater than jT 0j i.e.(3.6') connT � jConnT0j+Xi jConnTij+Xi li + [3� log2 diamsuppT ]:In fact, in Theorem 6 we will show that for all clusters considered later by us,the quantities connT and j suppT j will be of the same order . Moreover, one couldrewrite the present section for this (narrower) setting in the spirit analogous tothat of the later Theorem 6, without employing the bothering (but small!) logdiamterms. We prefer the more general exposition here in view of wider applicability ofthe estimates obtained here also to other situations.2) The mapping from clusters to trees constructed above is not one to one. However,if we assign to any bond of T a \ ag" i.e. mark it by a value 0 or 1 and interpretthe components of T marked by 0 resp. 1 as the sets used in the de�nition of Tresp. as the connecting paths (connecting T0 with Ti etc.) then the core of thecluster T (and the cores of Ti etc.) can be recognized just by looking at the biggest(at �) 0 { component of T . Thus, one has a crude bound 2n for the number ofstandard clusters T with the same tree T of the cardinality n. In the forthcomingapplications, all clusters constructed by us will be standard ones and so we will notdiscuss the possible modi�cations of the estimates discussed below which would beneeded if also nonstandard clusters would appear.The following estimate will be used later in Theorems 5 and 5' (though in aslightly changed form). It says that having established a slightly stronger versionof the estimate (3.1) for the sets T one obtains (3.1) also for all clusters T if thequantities kT are given by the recurrent formulas below. The quantity ConnT doesnot seem to have comparably nice properties ; the additional \ logdiam" term inour de�nition of connT seems to be essential here.Next we formulate two auxiliary results: The �rst one will be directly used later(in a slightly di�erent form not changing its essence { see the proof of Theorem 5).On the other hand, the second result is its corollary which we formulate in a moregeneral setting { which will be possibly interesting also in other situations whereour method can be applied. This latter result resembles the classical Meyer method(see Ruelle's book [R1]).Proposition 1. Assume that the quantities kT are de�ned recursively by formulas(3.7) kT = kT0Yi kTi:Assume that for the sets T0 the following stronger variant of (3.1) is valid:(3.8) jkT0j � "connT0+6� log2(diamT0+6):26

Then the estimate(3.9) jkT j � "connTholds also for all the clusters T = fT0; fTigg, with connT de�ned by the precedingde�nition, assuming that it is already valid for all clusters Ti; i 6= 0 in (3.7) fromwhich the clusters T were formed.Proof. It su�ces to prove that(3.10) ConnT0 + 6� log2(diamT0 + 6) +Xi connTi � connT:Notice �rst the following simple estimate. De�ne the support suppT and thediameter diamT of a cluster T = fTig recursively by putting suppT = suppT0 \suppTi and diamT = diamsuppT .Lemma 2. Let Tj be the longest of all clusters Ti (maximizing its diameter). Thenlog2 diamT � log2(diamT0 + 6) + log2 diamTjassuming that the right hand side is greater or equal to 6.Proof of Lemma 2. It is straightforward: notice that the condition (3.5) implies thebound diamT � diamT0 + (1 + 2 + 2 + 1) diamTj :Then we use the inequality log2(x+6y) � log2(x+6)+ log2 y which is surely validif x � 1 and y � 1.Notice that it su�ces now to establish the following bound (from which therequired bound (3.9) is obtained by summing with the bound of Lemma 2):ConnT0 + 3� log2(diamT0 + 6) +Xi6=j connTi +ConnTj � ConnTwhich is surely valid because then we can rewrite it (notice that lj � 3� log2(diamT0+6) by Lemma 1 and omit the number 6! ) in a stronger formConnT0 +Xi (ConnTi + li) � ConnTwhere li denotes the length of the path Pi used in the de�nition of the auxiliarytree T 0 (see De�nition 4). Namely, the bound lj � 3� log2 diamT0 follows from thecondition (3.5). The validity of the last inequality follows from the very de�nitionof ConnT (see (3.6')) and this completes the proof of Proposition.Notations. In the following we will usually write, for clusters T ,t 2 T; T � �;dist(T;�); : : :instead of the more precise notationst 2 suppT; suppT � �;dist(suppT;�) : : : :Writing G 2 T we will mark the situation when the set G was used in the recursivede�nition of T (as the \core" of some intermediate cluster used in the construction)of the cluster T .Finally we formulate one consequence of the condition (3.1), to be used later inthe formulation of our main result. 27

Proposition 2. If there is a small " such that for each cluster T ,(3.11) jkT j � "connTthen the cluster series with the terms kT quickly converge in the following sense:for any t 2Zand for any d 2 N we have(3.12) XT ;t2T;connT�d jkT j � (C")dand analogously for connT � d replaced by jT j � d.First notice that instead of connT it su�ces to prove an analogous result for thesmaller and \more natural" quantity ConnT . We recall that our introduction ofthe quantity connT was motivated by the necessity to derive (3.1) for all clustersfrom something like (3.9) which should be assumed to be valid for all sets T . Oncewe have (3.12) for all clusters we can forget the quantity connT and replace it byConnT if the convergence of the cluster expansion is investigated.We will prove Proposition 2 in the following broader setting (which is closelyrelated to usual estimates in the theory of the Mayer expansions { see the book[R1]). It is easy to understand that Proposition 3 below actually generalizes thestatement of Proposition 2: Recall that we identi�ed any cluster T with somecommensurate tree T on Z� and the number of standard clusters corresponding toT is at most 2jT j. Having this in mind the forthcoming result can be formulatedfor quantities kT indexed by commensurate trees T on Z�.Proposition 3. Let the quantities kT be given as products of some quantities de-noted by kft;sg or kb (see the commentary below)kT =Yb k bwhere the product is over all the \bonds" b = ffA;Bg; �g of the commensurate treeT = fG;�g. Recall that the \bonds" fA;Bg of an abstract tree G are mapped by� to unoriented pairs ft = �(A); s = �(B)g of points of Z�. The notation kb isused instead of a more explicit notation kft;sg for b = ffA;Bg; �g such that ft =�(A); s = �(B)g. Assume that these quantities kb are nonnegative and kft;tg = 1for each t. Let for any unordered pair b = (t; s) (a slight abuse of notations) wehave the estimate(3.13) X k b0 � qwhere the summation in (3.13) is over all unordered pairs b0 = (s; u); s 6= u whichare commensurate with b and q is some small (e.g. q < 1=4) positive constant.Then for any pair b we have also the bound(3.14) X kT � kb � q028

where the summation in (3.14) is over all commensurate trees T containing the\bond" b as its \extremal bond" and having a length at least 2. By the extremalityof b = ft = �(A); s = �(B)g we mean here that one vertice of the pair fA;Bgremains \free",\endvertice" in the original abstract tree; this must hold at least forone of the bonds fA;Bg which are mapped to the given pair ft; sg. The quantity q0can be chosen like 3q.Note. More precisely, the optimal value of q0 can be found from the equation (seethe end of the proof below) exp(q00) = 1 + q0where q00 denotes the supremum, over all bonds b, of the sums analogous to that inthe left hand side of (3.13) but with modi�ed termsk00b0 = kb0(1 + q0)1� kb0(1 + q0)where q0 has its previously established value. Namely if all kb are small and wealready have established the smallness of q0 then we have the crude bound k00b0 <2kb(1 + q0); hence we have also the inequality q00 < 2q(1 + q0). See the end of theproof below.Proof. Apply the method of induction. Denote by P<nT kT the subsum of PT kTtaken over all trees T which \have the level at most n" in the sense that the longestpath to be found in T starting in b contains at most n bonds. Removing of a bondb from such a tree and cutting of all the connections going through the \nonfree"vertex of b results in an appearance of certain (nonzero) number { say m { ofsubtrees of the level at most n� 1. (These subtrees are \sticked" to that vertex ofthe bond b which is not \free").Now, considering the induction assumption for P<n�1T kT i.e. the inequalityP<n�1T kT � q0 one gets the following estimate. Let us start with the situationwhere only one bond b0 (possibly a \multiple one" if looking at the image � ofthe original abstract tree G in Z� i.e. several f�(A); �(B)g with the same ft =�(A); s = �(B)g are possible) sticks to b at a given vertice t of b. Let T 0 denotethe subtree \continuing" the bond b0. Summation over all possible T 0 gives thefollowing contribution (the subtree T 00 resulting from the removal of b0 from T 0may be already empty or may possibly contain some more bonds; for the sum overthe latter situation we already have the induction asssumption with the bound q0):kb0(1 + q0) if only one subtree sticks to t. If we are counting also the cases whenseveral (k � 1) subtrees starting with the same bond b0 stick to b we have thecontribution X(kb0(1 + q0))k = kb0(1 + q0)1� kb0(1 + q0) :Now perform the summation over all more complicated possibilities when m � 1di�erent types of bonds b0 stick to b; each of them can have some multiplicityk � 1 and each such bond b0 is the starting bond of some { empty or nonempty {subtree. (Contributions of these situations were just counted above.) We get the29

�nal estimate <nXT kT <Xm (q00)m=m! � exp(q00)which concludes the inductive step of the proof of Proposition 3.Finally, it is easy to see that the condition of the type (3.1) implies the validityof the conditions (3.13),(3.13') for suitably chosen new kT = kT = Qb kb (namelywith kb � ") majorizing the original values kT . Then also the constant q in (3.13)can be taken as q � N" where N denotes the upper bound for the number of bondscommensurate to b. The constant C in Proposition 2 depends then only on thedimension (and on our choice of the constant 2 in the de�nition of commensuratebonds) if " is su�ciently small.2. The Concept of a Mixed Model.The con�guration space of the mixed model in a volume � (typically a standardone) under a boundary condition y 2 Sgiven on �c, will be given as a suitable subsetof the con�guration space of (the strictly diluted con�gurations of) the original,abstract Pirogov { Sinai model. Thus, we consider a collection of pairs(x�; f�ig)where x� [ y�c is assumed to be strati�ed for any given point of � n ([i supp�i)and has a \vertically best extension" to [i supp�i (in the sense of minimal possiblevalue ofP et(x)). Here, � = f�ig is an admissible collection of contours satisfyingthe condition dist(Vl(�);�c) � 2 and giving a noncon icting precription, togetherwith y, for the value of x�.Later, we will often assume (but notable exception is the forthcoming section 3!)that the con�guration space is horizontally translation invariant in the sense that ahorizontal (& strictly diluted in a given volume �) shift of an allowed con�gurationis again an allowed con�guration.Notes.0) The volume � may be often considered to be a standard and even a conoidalone, e.g. a large cube. We will not consider at all any possible relations betweenthe models in di�erent volumes! In particular, we will not derive any analogy of the\telescopic equations" for diluted partition functions (which are usually formulatedin such situations). In what follows, � is just a suitable, �xed, �nite substitute forthe whole lattice Z� ; the thermodynamic limit � ! Z� being performed only atthe very end of our considerations, in the last section of the paper.1) In all the con�gurations (x; f�ig) considered below (in the forthcoming sectionspreceding the statement of Main Theorem) we will have x = xbestf�ig. We will usuallyomit the superscript \best" in the following.2) Sometimes (like in the study of the probabilities of external contours in our MainTheorem) we will be interested also in nonstandard volumes � (like the complement,in some standard �0, of some set [ supp�0i (where f�0ig is an admissible system30

of contours) and we could in principle consider also those boundary conditions ywhich are only locally from S. However, we will not need this.11The hamiltonian of the mixed model will be given (compare with (2.16)!) asfollows: Consider any con�guration (x~�;D) of the abstract Pirogov { Sinai modelwhich is given on a set � � Z� such that D = f�ig is an admissible system in �and x� is compatible with D i.e. strati�ed outside of the set suppD = [i supp�iand \best" inside suppD. (See (2.14). Denote the \external" value of x as xext.(For nonstandard � it is de�ned at least on @�c). It is from S(maybe only locallyif � is a nonstandard volume). De�ne, for (x~�;D) where D = f�ig, the quantityH(D; xext), more precisely denoted by(3.15) H((x� ;D); xext) =Xi E(�i) +Xt2� et(x) � XT��nsuppD kT (x)where E(�) and et are as before (we recall that both E(�) and et(y) are quantitieswhich are horizontally translation invariant only) and where kT (x) are some new,quickly decaying quantities which can be viewed as an \additional cluster �eld"and which are required to satisfy the following conditions:(1) (Horizontal translation invariance of kT ) The quantities kT = kT (x) arelocal in the sense that they depend only on xt;dist(t; T ) � diamT . Theymoreover ful�l the relation(3.16) kT (x) = kUT (Ux)whenever both quantities are nonzero (see Note 3 below), for any horizontalshift U such that both T and UT su�ciently \inside" � n suppD :dist(T; suppD [ �cnon at) � diamTdist(UT; suppD [ �cnon at) � diamT :where �cnon at denotes the collection of those points t 2 �c which do notsatisfy the property (\horizontal translation invariancy of � at s") that thevertical sections �ftg = �\f(t1; : : : ; t��1; (:)g commute with U (U(�ftg) =�fUtg)for horizontal shifts U of unit length.(2) (Quick decay of kT ) There is a small " > 0 such that for any T ,(3.17) jkT j � " connT :Notes.1. We will later glue together { in our \recoloring procedures" { contours � andclusters T such that dist(T; supp�) � 2 diamT { to form new clusters (of some newmixed model); this is one of the reasons why we added, in the preceding section, the11If the volume � is a standard one i.e. if all the \horizontal sections" of � have simplyconnected components then the phrases \y is locally resp. globally from S on @�c " have thesame meaning. 31

\safety constant" 3 log2 diamT to the quantity ConnT in the de�nition of connTto keep the control over the connectivity properties of the new clusters formed bysuch (recursive) procedures.2. The collection of allowed contours (and of allowed con�gurations, see below)will vary from one mixed model to another. Typically the allowed set of contourswill be some subset of the original collection of contours (of some given abstractPirogov { Sinai model) { and this subset will become even smaller after applyingfurther expansions (recolorings) to the given model.On the contrary, the collection of nonzero kT will always grow with such anexpansion. See the forthcoming section for more details.3. The restriction of the assumption (3.16) to nonzero products kT kUT is re-lated to the fact that, in the forthcoming section, we will work with (\slightly")translation noninvariant models; the new cluster quantities kT will be constructedsuccessively in the lexicographic order, through an in�nite sequence of intermediate(noninvariant) mixed models. However, if the condition (3.16) is complemented bythe assumption that both kT and kUT are nonzero if at least one of them is nonzeroand if the con�guration space is horizontally invariant (in the sense of what ad-missible systems are allowed in Z�) we will speak about the translation invariantmixed model.4. Having speci�ed the collection of allowed contours of the mixed model we donot even require that all admissible collections of allowed contours are allowed con-�gurations of the mixed model. At this moment we impose no special requirementson what collections of contours are really allowed in our model; see the forthcom-ing sections 3,4,7 for a more concrete information about the actual choice of thecon�guration space.The partition functions of the mixed model in a given volume � (strictly dilutedones; we can forget the notion of a diluted partition function for most of part III;however our volumes � will be often the standard and moreover the conoidal ones)will be given as Z�l (�) = XD���Z�D(�)where � is a boundary condition on @�c (which is locally from S) and D is anadmissible family of contours; the notation D �� � will mean, everywhere in thefollowing, that dist(Vl(�));�c) � 2 for any contour � on D. By (3.15) we de�ne(3.18) Z�D(�) = exp(�Xi E(�i)) exp(�Xt2� et(�)) exp( XT��nsuppD kT (�))where � denotes the con�guration(�[ @D)best.Note. It seems unnatural to use the symbol Z for the \mere Gibbs factor" (3.18).However, the case kT = 0 is not a typical example here. In a more general case,the considered mixed model corresponds actually to some partial expansion of themodel (2.18). Then (3.18) is really some partition function, corresponding to anevent \D is the collection of (still) nonexpanded contours of the original model(2.18)". 32

3. Recoloring of a single internal admissible subsystem �This is a central construction of our approach, replacing (together with theforthcoming constructions of later sections) the concept of a (metastable) contourmodel used in the previous versions of the Pirogov { Sinai theory.In this section we describe some abstract, \algebraic" aspects of one recoloringstep (of an arbitrary mixed model). An invariant (conserving the horizontal in-variancy) modi�cation of this construction will be given in the forthcoming section4. A suitable sequence of (translation invariant) recoloring steps { yielding as its�nal result the \total" expansion of a given Pirogov { Sinai abstract model { willbe discussed later, starting from section 5.Recoloring will be just one step towards the desired \total expansion" of themodel, and this step is described in detail in Lemma and Theorem 4 below.Roughly speaking, recoloring of � will just mean a replacement of a given mixedmodel by another mixed model where � will not already be allowed as a contourand where some new quantities kT (for some new clusters T containing G as itscore) will appear. The remainder of the model will be kept intact and the crucialfact will be that the diluted partition functions of both the original and recoloredmodel will be required to be the same for all �nite volumes.Let us start with the de�nition of the following important quantity A(�) which\measures the instability of xext� in V (�)" and which will play a key role later whende�ning a rigorous substitute to (2.21); see also the remark below De�nition 2.There will be several variants of the quantity A(�) { see below { and the technicaldi�erence between their de�nitions (essentially the decision what volume will beused instead of V (�)) will be quite important in this section, in spite of the factthat the values of all these quantities will be roughly the same. Let us introduceagain (it will be quite indispensable in what follows) the de�nition of a suitable\protecting zone" of the volume V (�):De�nition 1. De�ne �rst the \conoidal" volume(3.19) Vl(�) = V (�) [ V 0(�)where V 0(�) is the collection of all points t 2 (V (�))c satisfying the bound(3.20) dist(t; V (�)) � 12 dist(t; supp�) :Denote by A(�) , more precisely by A(�; xVl(�)) the quantity(3.21) A(�; xVl(�)) = Xt2V (�)(et(y)�et(x))+ XT�Vl(�)T\supp�=; kT (xVl(�))� XT�Vl(�)kT (y)where x = (@�)best (the con�guration minimizing the hamiltonian under the con-dition @�) and y = xextVl(�).Note. Thus, to be able to de�ne A(�) we must know the con�guration x on thewhole volume Vl(�) (because kT (x) resp. kT (y) depend on the values of x resp. y33

on T and we do not assume T � V (�)). However, the value of x on Vl(�) will benormally determined by the context in which the contour (or admissible system) �will appear and so we will use the shorter notation A(�) instead of the more precisenotation A(�; xVl(�)) without any ambiguity.We will see that the quantity(3.22) F (�) = E(�) �A(�)is a useful exact substitute for the formal quantity Fformal from (2.21). The choice ofthe set Vl(�) will guarantee (among other convenient properties) that the clustersconstructed below having the \core" � will be su�ciently \tight"(which would notbe the case if we would take mere V (�) here).However, if � is an interior subsystem of some bigger admissible collection �&Dof all contours of some con�guration (x�;�&D) in a �nite volume �, the followingmodi�cations of the quantity A(�) will be sometimes considered later, too: (Wewill be interested below only in the cases when moreover Vl(�)\D = ; i.e when �is not \too tightly attached" to D.)De�nition 2. In analogy to (3.21) de�ne also the modi�ed quantities Aloc(�),Afull(�), Afull;D;�(�) as in the relation (3.21) but with the volume Vl(�) in thesecond and the third sum on the right hand side of (3.21) being replaced successivelyby volumes V (�), Z�, � n suppD. Corespondingly de�ne, by (3.22), the quantitiesFloc(�), Ffull(�), Ffull;D;�(�) and also F0(�)) (taking A0(�) from (2.20)).Note. Assuming the existence of cluster expansion for the partition functions inthe expression (2.21) for Fformal one sees that the true analogy of the quantityFformal is Ffull(�), not F (�). However, Ffull(�) is a nonlocal quantity (though avery quickly converging sum of local quantities kT ) and this would be inpractical inthe expansions constructed below. However, we return to the value Ffull(�) in latersections. On the other hand, Floc(�) is a perfectly local quantity, but sometimesit is a \too crude" (and therefore never used below) approximation to Fformal(�);such a situation happens in the cases where there are very big12 at \ceilings" onthe boundary of V (�); a situation having no analogy in the translation invariantsituation where the choice of Floc(�) would be O.K. because contours are \crusted"in that case.The quantity F (�) is a reasonable compromise because it approximates Ffull(�)with a great accuracy (� "j supp�j) and at the same time it is \local" in a rea-sonable sense. One could take even smaller sets V 0(�) to retain this accuracybut our choice will have additional advantages below in Theorem 5. The quantityFfull;D;�(�) is just a temporary notation used in the proof of Lemma below.The forthcoming lemma is an essential step in the procedure called \recoloringof an internal contour". This is further developed by Theorem 4, concluding thee�ort of Section 3.12A general \philosophical" remark: sometimes, one is �ghting severe technical problems inthe Pirogov { Sinai theory which however start to be relevant only in volumes which are reallyastronomically large; for example the problem mentioned above is hardly of much relevance involumes of a size, say 1027 ! 34

Lemma. Assume that we have a mixed model satisfying (3.1). Let � � Z�. LetD & � �� � be an admissible system such that � is its removable subsystem,satisfying moreover the condition Vl(�)\D\�c = ;. Let � be a boundary conditionon @�c which is in conformity with D & � (such that there exist a con�gurationxD&� = (�[@D[@�)best for which all points of (suppD[supp�)c\� are strati�ed).Then(3.23) Z�D&�(�) = Z�D(�) exp(�F (�)) exp(X knewT )where knewT depends on both xD&�=T and xD=T and the summation is over allT � � such that T 6� Vl(�); T \ V (�) 6= ;; T \ suppD = ; and dist(T; supp�) �2 diamT . The quantities knewT are given by formulas (3.26), (3.27) below and theysatisfy the estimate(3.24) jknewT j � 2 "connTfor each T .Proof. Write � resp. � instead of xD&� resp. xD. Write Z�D&�(�) as(3.25)exp(�E(�) �E(D)) exp(�Xt2� et( �)) exp( XT��T\(suppD[supp�)=; kT ( �)) == exp(�E(D)) exp(�Xt2� et(�)) exp(�Ffull;D;�(�)) exp( XT��T\(suppD)=; kT (�)) == Z�D(�) exp(�F (�)) exp( XT��T 6�Vl(�)T\(suppD[supp�)=; kT ( �) � XT��T 6�Vl(�)T\suppD=; kT (�))which proves (3.23) with the following choice of the quantities knewT (recall that the\old" quantities kT satisfy (3.1), hence we have (3.24) ):(3.26) knewT = kT ( �) � kT (�)if T � � ; T 6� Vl(�) ; T \ (suppD [ supp�) = ;; T \ V (�) 6= ;, resp.(3.27) knewT = �kT (�)if T � � ; T 6� Vl(�) ; T \ D = ;; T \ supp� 6= ;. The condition T \ V (�) 6= ;(in (3.26); contrast it to the condition T 6� Vl(�)!) follows from the observationthat T 6� Vl(�) & dist(T; supp�) � 2 diamT ) T \ V (�) = ;(3.28) ) ( �)T = �T ) knewT = 0 :(Compare the de�nition of Vl(�); the condition T \ V (�) 6= ; & T 6� Vl(�) wouldimply that there is some t 2 T n Vl(�) with dist(t; V (�)) � 12 dist(T; supp�). )35

Notation. An interior subsystem � of an admissible system �0, satisfying also thecondition Vl(�) \ (�0 n �) = ; will be called a strictly interior subsystem of �0.Theorem 4. Assume that we have a mixed model satisfying the condition (3.1).Consider the partition functions ((3.18))Z�D;G(�) = X�:supp�=GZ�D&�(�)and(3.29) Z�D;[G ](�) = Z�D(�) +Z�D;G(�):where D&� is an admissible system such that � is its strictly interior subsystem,satisfying moreover the condition Vl(�) \ �c = ;. (The partition functions abovecorrespond to the events \ D & � appears" resp. \ D & possibly � appears" ; � isa contour such that supp� = G.) These partition functions can be expressed as(3.30) Z�D;G(�) = ( XT :G2T k+T )Z�D(�)resp.(3.31) Z�D;[G](�) = exp( XT :G2T k�T )Z�D(�)where G 2 T means that G is the core of the cluster T and k+T resp. k�T aresome new cluster terms, described in detail in the proof. The leading new quantityk�G = k+G is equal to(3.32) k+G = k+G;� = X�:supp�=G exp(�F (�)):The remaining quantities k+T obey the bounds, for any cluster of the type T =(G; fTig)(where Ti ; i = 1; :::;m are clusters of the given mixed model)(3.33) jk+T j � k+G " Pmi=1 connTiand analogous bounds are valid for k�T . See the proof ((3.41)) for the more preciseform of the estimate on the right hand side; also for more complicated clusters T .The quantities k+G; k+T ; k�T do not already depend on �; they depend only on Gand on the values �G13 and are translation invariant in the sense of (3.15).13See the preceding Lemma; the quantity k+G does not already depend on the interior of � incontrast to the quantity exp(�F (�)). 36

Moreover, if k+G satis�es (3.7) then the validity of the bound (3.1) in the givenmixed model implies its validity also for the new quantities k+T and k�T .14The procedure called \recoloring of a contour" described by formula (3.31) willbe used later repeatedly many times for all the \smallest possible" strictly interiorsubsystems �. The new cluster quantities k�T will play later the same role as the\old" quantities kT ); and therefore it will be crucial to ensure the validity of theestimate (3.1) for them: Comparing (3.33),(3.32) with (3.7) (Proposition of Section1) one sees that the su�cient smallness of k�G will be guaranteed by the followingestimate :De�nition. We will say that an admissible system � is recolorable if(3.34) X�:supp�=G exp(�F (�)) � exp(�� 0 connG)holds with a su�ciently large � 0 (to be speci�ed in the Corollary below; see (3.35)).The constant � 0 will be chosen below roughly as �12� .Notes. 1. Notice that this is the requirement on the set G, not on a particularcontour � { though practically this is closely related to saying that for each � withthe same support G, one has a bound(3.34") F (�) � � 00 conn�with some other large � 00.2. One should always have in mind that the property \to be recolorable" is rathersensitive (for contours of very large size, of course) with respect to the boundaryconditions. The mere knowledge of the external colour of � (at the horizontal levelof V (�)) may be insu�cient to decide the recolorability. In general, the externalcolour of the whole Vl(�) must be known.Corollary. Let � be recolorable, with � 0 satisfying the inequality(3.35) exp(�� 0 connG) � "ConnT+6� log2(diamT+6):Then the validity of the bound (3.1) (with connT de�ned by (3.6)) in the givenmixed model implies its validity also for the new quantities k+T and k�T .Notes. This means that the new mixed model constructed by the formula (3.31)(which has a new, richer family of cluster �elds fkTg&fk�Tg but which does not yetallow � as an interior subsystem of its con�guration) is of the same type as before,satisfying again the estimate (3.1). The bound (3.35) relates the appropriate choicesof constants "; � 0 in (3.1) and (3.34). The latter condition is a Peierls type conditionfor the quantity F (�) , and one has to choose a suitable � 0 � � there, to obtain auseful notion. (� is from (2.17).) This will be discussed later and we will see thatwhen taking � 0 = c� the convenient choice of the constant will be c = 112� .14In the next section, we will formulate a \horizontally invariant" modi�cation of Theorem 4,namely Theorem 5. 37

One should also notice that later, when performing our successive process of\recoloring of all (recolorable) �", our procedure will be organized in such a way(see the forthcoming section) that quantities k�T with new clusters T (nonexistentwith nonzero kT in the previous mixed model) will appear at each stage of theconstruction.Proof of Theorem 4. By (3.23) we haveZ�D;G(�) = X�:supp�=GZ�D&�(�) = Z�D(�)X� exp(�F (�)) exp(XT knewT ) :Writing exp(�F (�)) = ��k+G and expanding the exponential this can be written as(3.36)Z�D;G(�) = Z�D(�)�X� �� 1Xk=0 X(T1;:::;Tk)(n1;:::;nk) kYi=1 1ni! (knewTi )ni )�k+G = Z�D(�)(XT k+T )where the new values of knewT are denoted here as15(3.37) k+T = X�:supp�=G ��k+G kYi=1 1ni! (knewTi )ni :Here, the clusters T are de�ned, for ni = 1, as(3.38) T = (G; fTig):(including also the empty collection fTig); otherwise the cluster T contains thecorresponding number ni of copies of Ti.Notice that the expression of k+T by values knewT is not exactly as (3.7), Propo-sition of Section 1. However, it is straightforward to adapt the correspondingestimates noticing thatP� �� = 1. Thus, one obtains (3.33) and then, from (3.34)and (3.35), also (3.1) for the quantity k+T . (If some ni > 1 then the estimate is evena stronger one).The expansion (3.31) is obtained by taking logarithms:(3.39) logZ�D;[G](�) = logZ�D(�) + log(1 + XT :G2T k+T ) = logZ�D(�) + XT :G2T k�Twhere(3.40) k�T = (�1)n�1n nYi=1k+Ti15It is perhaps worth mentioning that whereas F (�) depend on the values �, in particularon the \interior colour of �", the quantity k+T already depends only on G and on the \externalcolour" of �. 38

for any cluster T = (T1; T2; : : : ; Tn). (Again,one has to modify correspondingly thisformula if multiple copies of one cluster Ti appear in T ).Strictly speaking, clusters of such a type were not de�ned yet. However, if e. g.three clusters T1 , T2 and T2 have the same core G then we identify the orderedtriple (T1; T2; T3) for the clusters Ti given as Ti = fG;T �i g ( where T �i denotes somecollection of clusters) with the clusterfG;T �1 ; fG;T �2 ; fG;T �3 ggg:Now, for any such cluster T = (T1; T2; : : : ; Tn) (recall that G 2 Ti for each i!)one obtains, after some inspection, the desired bound(3.41) jk�T j � (k+G)n "Pni=1 conn+(Ti) � "connTwhere conn+(T ) denotes the quantity Pj connTj for T = (G; fTjg); the last in-equality follows from (3.34) and (3.35) by a similar argument as we used in theestimate of (3.37) above. This concludes the proof of Theorem 4 and of its Corol-lary. 4. Recoloring: towards a new stratified mixed modelThe aim of this section is to formulate a procedure (based on Theorem 4) whichconverts a given horizontally translation invariant mixed model into a new hori-zontally translation invariant model, having the same \diluted "partition functionsbut with a smaller set of allowed con�gurations (and with a richer set of clustersT having nonzero contributions kT ; the \old" nonzero values kT being kept at thesame value as before). Such a transformation of the model could be characterizedas the \removal, from the model, of all con�gurations which have a shift of � amongits strictly interior subsystems".Recall the ordering � of systems of contours we introduced in section II (see(2.11) and the text below it). We will say that a recolorable system � of contours issmallest recolorable system of the given mixed model if it is strictly interior (in theconsidered volume �), recolorable and moreover there is no smaller strictly interiorrecolorable (smaller in �) �0 which would appear in some con�guration of the givenmixed model.Note. Below we will use the recoloring step formulated by Theorem 5 successively,according to the growing \size" of the smallest recolorable contours � which haveto be recolored. Moreover, the mixed models studied by us later will appear asthe result of successive recolorings applied to some given P. S. abstract model; thecon�guration spaces of the mixed models thus obtained will be de�ned in termsof requirements on the size of the smallest interior recolorable subsystems of thecon�guration. See the forthcoming sections for details.Equivalent mixed models. Two mixed models will be said to be equivalent if alltheir strictly diluted partition functions are the same. (Usually, the con�gurationspace of one of these two mixed models will be a suitable subset of the con�gurationspace of the other model.) 39

Theorem 5. Assume that we have a horizontally translation invariant mixed modelsatisfying the condition (3.1). Let � be a recolorable16 subsystem whose horizontalshift can appear as a strictly interior subsystem of some con�guration of the model.Assume that kT = 0 holds for all clusters T containing a shift of supp� (\contain-ing" in the sense that supp� was used in the recursive construction of the clusterT ).Then there is an equivalent mixed model having the following properties :(1) Its con�guration space is the collection of all con�gurations of the originalmixed model which do not contain a horizontal shift of � among its (smallestpossible) strictly interior recolorable subsystems.(2) If T is a cluster not containing � then the value of kT in the new mixedmodel is the same as before.(3) If T contains � then the new value of kT satis�es the condition (3.1), too,assuming that " and � 0 are such that (3.35) holds.Proof. This follows from Theorem 4 if we use it successively in the following way:take the (completely ordered in � ) sequence of all shifts of �. Given t 2 Z�consider an intermediate \t { th model" which has the con�guration space de�nedby the requirement that exactly those congigurations of the original mixed modelare allowed for which no interior subsystems �+ t0 such that t0 � t exist. If s is thenearest greater point to t (t � s) in the given volume � (remember that we alwayswork in �nite volumes) then we de�ne the transition to the \s { th model" by thevery procedure described in Theorem 4.It is straightforward to check the translation invariance (3.16) and unicity of thede�nition (not depending on the actual volume � if T is distant from its complement{ in the sense of (3.16)) of the new quantities kT thus obtained.Note. There are also other methods to prove Theorem 5. Namely, it is possible(and it is, in fact, apparently a more standard way how to deal with these clusterexpansions) to reformulate the Lemma and Theorem 4 above for a simultaneousrecoloring of all the shifts of � at once. We do not follow such a (more direct, butwith slightly more complicated formulas) approach here, in this paper. Such anapproach is used also in the recent lectures [ZRO] (in a simplest possible form, webelieve) and in future, we plan to replace the arguments based on the successsiveuse of the lexicographic order by this more standard approach.5. Small and extremally small systems of contoursWe are still working with a general mixed model. Only later we will explain therelevant choice of a mixed model in various concrete situations ; this choice willalways be given as some partial expansion of the original abstract Pirogov { Sinaimodel which we are investigating.The successive application of the recoloring procedures constructed in the pre-ceding chapter will �nally lead to a family (indexed, in any volume, by elements ofS) of mixed models where no recolorable systems will be left! The reader is advised16smallest possible (in a geometrical sense); this will be the case needed in the applications ofTheorem 5 below 40

to skip brie y to the section 8 and to look at the formulation and the proof of MainTheorem { to see the important consequences of this fact.In the meantime, in sections 5 to 7, we will investigate the notion of a recolorablesubsystem { and the related notion of a small resp. extremally small subsystemintroduced in this section { in more depth, to obtain a useful supplementary \topo-logical" result (Theorems 6 and 7) needed in the proof of the Main Theorem. Ourdiscussion of the forthcoming notions of a \small" and an \extremally small" sys-tem, and the notion of a \skeleton" introduced below (notice that we are introducingthere another testing quantity A�(�) { as a more careful alternative to A(�)) isperhaps slightly more detailed than absolutely necessary (if a shortest possible proofof Main Theorem is required). However, we are keeping it here as we expect ourmore detailed exposition to be useful not only here (giving more information andestimates with better constants) but also in the investigation of the \metastability"problem and in the study of the completeness of the phase picture constructed byour Main Theorem.The reader interested in acquiring the idea of the proof of MainTheorem can skip now directly to section 8 { omitting even the very notion of asmall contour (given below) but �nally realizing that some variant of such a notion(and the topological Theorem 7) is needed there!Recall the de�nition of A(�), Afull(�), Aloc(�) and of the corresponding quanti-ties F (�) from section 3, de�nition 2. When checking the condition (3.34) (through(3.34")) one needs inequalities of the typeA(�) � ~� j supp�jwhere ~� is a suitable, \not too large" constant (like �2 ).In fact, there is quite a freedom in the choice of the constants ~�; � 0; � 00 (from(3.34) resp. from the relation (3.49) below) and the di�erence between the variousvariants of A(�) resp. F (�) (which was so important in Section 3) will be quiteirrelevant here17. Namely, we obviously have the following bound.Proposition.(3.42) jA(�) �Afull(�)j � "0j supp�jwhere the constant "0 is of the order "n, n being the cardinality of a smallest possiblecluster appearing in (3.21) and " = C exp(�� ) for a suitable constant C = C(�).The quantities Afull(�) (and, even more importantly, the quantities A�full(�)introduced below) will be more convenient in these �nal sections than A(�) andthe bounds of the type A(�) � ~� j supp�j will be studied for them instead of A(�).Then we supplement these bounds by (3.42).18The meaning of the quantities A(�) is that they give some information aboutthe \volume gain of the free energy" caused by the fact that inside �, possibly some\more stable" regime is found. One could ask this question in a more precise way:17except of the choice of Aloc(�) { which would be too rough in some situations where some\reallly big" ceilings appear.18The quantities A(�) remain, of course, in (3.22) but their test \whether they are dangerouslybig" will now be done through the closely related but \more nicely looking" quantities Afull(�).41

whether the regime which resides inside � is the \best" possible and also what isthe \energetically optimal realization" of such a contour. Fortunately, one does notneed to investigate these questions in more detail, in particular the question \whatis the optimal shape of a contour" is quite irrelevant here.On the other hand, the question \what is the best regime to be found inside �"will be important in the investigations below and we will approach it as follows:We rewrite, from now on, the quantity Afull(�) in a more concise way, replacing thesum over cluster quantities kT by a more nicely looking (and more exible) sumof suitable point quantities. These latter quantities are however nonlocal (but veryquickly converging limits of local quantities). Introduce the following notations.De�nition. For any mixed model and any strati�ed con�guration y de�ne thequantity(3.43) ft(y) = et(y) � XT :t2T kT (y)jT j :For an arbitrary nonstrati�ed con�guration x, de�ne ft(x) = ft(xhort ) where xhort isthe horizontally invariant extension of the con�guration x(t1;:::;t��1;(:)).Note. These quantities will be very important in the sequel. However, in spiteof their \physical" meaning which we discover below (they will be interpreted asthe \density, at t, of the free energy of the metastable state constructed aroundy") there is still some arbitrariness in their de�nition: For example, the modi�edquantity(3.44) f̂t(y) = et(y) �Xt2T kT (y)where the sum is over all clusters T such that t is the �rst point of suppT in thelexicographic order could be used in the same way.19Agreement. Here and below we need to work with con�gurations y 2 Sde�ned onthe whole lattice Z�. Let us make an agreement that whenever we have a strati�edcon�guration de�ned at the moment only in a partial way (typical situation: theexternal colour of some con�guration de�ned in some �nite volume, or in the interiorof some bigger contour) then we extend it20 in some prescribed (�xed for once) wayto a con�guration on the whole Z�. The details of the extension will be irrelevant.Proposition. The quantity Afull(�) can be expressed also by the formula(3.45) Afull(�) = Xt2Z� (ft(xext� )� ft(x�)) + �(�) where j�(�)j � "0j supp�j:Proof. This follows from the observation that any T \not touching supp�" whichis counted in the de�nition of Afull(�) is counted also (exactly once!) in the above19Namely, the physically important quantities like Pt(ft(z) � ft(y)) are the same for bothalternatives, whenever y and z are strati�ed and di�ering only on some layer of a �nite width.20In fact, this is a comparable act of arbitrariness like that we used in our choice of the setsVl(�). 42

sumPt2Z� (ft(xext� )�ft(x�)). The corrections to this observation (they are neededonly for clusters touching supp�) form, when summed together, the very term �(�)which therefore satis�es (being the sum of small and quickly decaying quantitieskT jT j�1) the bound above.The forthcoming notion will be useful for understanding what would be the \bestpossible gain in free energy" inside a given contour � :De�nition. Given a con�guration x which is strati�ed outside of some volume Vdenote by xbestV the con�gurationminimizing, at each vertical section (t1; : : : ; t��1; (:)),the value Pt�2Z ft(x0) under the condition that x0 = x outside of V . We will usu-ally consider such a con�guration x in a volume V = V (�), where � is a contouror an admissible system.Notice that here, in comparison to the formulation of the Peierls condition inPart II, we use the quantity ft(x0) instead of et(x0). However, ft is roughly equalto et and the sum of the terms (ft � et) over supp� is again (like in (3.42)) of theorder "0j supp�j which is a quantity quite negligible when checking the validity ofthe Peierls condition.Now we are able to de�ne an alternative (with the same intuitive meaning) toA(�) { which will be more exible in the fortcoming estimates.De�nition. Given any �nite volume21 V and any y 2 S introduce the quantityA�(V ), more precisely A�full(V; y) :(3.47) A�full(V; y) = maxfXt2Z� (ft(y) � ft(z))g = Xt2Z� (ft(y) � ft(ybestV ))where the maximum is taken over all z which are equal to y outside of V and whereybestV is the con�guration realizing this maximum.Notes. 0. Compare (3.47) with (3.45) (for V = V (�)). We have, of course, therelation(3.48) jA�full(V; y) �A(�)j � "0j supp�jwhenever V = V (�) and y is the external colour of �.1. Our preference of this notion (to an alternative Afull(V ) which was de�ned withthe help of summation over clusters kT ) is mainly an aesthetical one. Namely,the sums of point quantities will be more convenient in later estimates. Noticethat we do not require any strati�cation of z (and of ybestV ) inside V . Later (seethe section Skeleton) we will introduce, for technical reasons, some new, arti�cial\contours" of the model. These new contours will have the shape of a (large) cube� living inside of some strati�ed regime of the actual \physical" con�guration; their\energy" will be de�ned (typically, this energy will be of the order diam� only)just to compensate the \volume gain inside � " (steming from the fact that thecon�guration inside of such an arti�cial contour will be assumed to \jump freelyto some better regime inside � "). The interplay between these formal notions and21We will use later this quantity not only for volumes V = V (�) but also for cubes V .43

between the behaviour of the real contours of the model can be best studied in thelanguage of the quantities A�full .2. Recall that the quantity A�full requires the knowledge of y in the whole Z�.Thus, there is still some arbitrariness in the de�nition of A�full(V; y) because our yis usually given only in some �nite volume. This arbitrariness is compensated bythe more transparent form of the right hand side of (3.47) (compared to (3.21)).Thus, when estimating the size of the quantities A(�) we will work, from nowon, with the more convenient quantities Afull(�). or even A�full(V (�)). In fact, it isadvisable22 to restrict the discussion of the \dangerously big value of A(�)" to thesuperordinated cubes only:De�nition. A cube � will be called small with respect to a con�guration y 2 Sifthe following inequality23 holds :(3.49) A�(�; y) � � 0 diam�where � 0 is something like � 0 = � � "0 and "0 is from (3.45),(3.48). If y is givenonly partially (on some neighborhood of �; y is, in fact, always given in some �nitevolume only) then � will be called small with respect to y if it is small for somestrati�ed extension of y.Say that a cube � in Z� is the covering cube of a volume S � Z� resp. of acontour � if � is the smallest possible cube (smallest in the usual ordering on thecollection of cubes which is de�ned as an extension of both the lexicographic orderof all the shifts of a single cube as well as the inclusion relation between cubes)which is a superset of S resp. of Vl(�). (Notice that we take Vl(�) instead of mereV (�) here, the latter being equivalent to supp� ). We will denote the coveringcube � by a symbol �(S) resp. �(�). If � is the covering cube of � and y is theexternal colour of � then we will say that � is small if � is small with respect toy. We will say that a strictly interior subsystem � of some admissible system D in�, under some boundary condition y given of @�c, is small in � if � is small forthe exterior colour of � induced by D and y.Notes. 1. Recall that we take, here and everywhere in part III, the norm jtj =maxfjtijg.2. The property \to be small" is formulated with the help of an (arbitrarily chosen,but �xed) extension of the external colour y of � (extension to the whole Z�). Itwill not be formulated for subsystems which are not strictly interior in �. It is easyto see that for any small � we have the inequality, with y denoting the externalcolour of � (extended to the whole Z�, as mentioned above)(3.50) A�full(V (�); y) � A�full(Vl(�); y) � A�full(�(�)) � � 0 diam�:22They are some technical subtleties in this recommendation. They will be more clear later,after de�ning the notion of an extremally small system, in the proof of Theorem 6. See (3.55).23Do not care about the particular choice of the constant � � here. Any su�ciently bigconstant would do the job. On the other hand, the advisibility of our very choice of diam� willbe clear only later. We mention that the choice of @� instead of diam� here (such an alternativecould maybe look more natural as the quantities j supp�j appear otherwise anywhere wheneverthe energy of a contour is considered) would cause di�culties later.44

Complement this with (3.42) and (3.45)! The idea now is, roughly speaking, thatall small contours resp. admissible systems should be recolorable. This is obviouslytrue for contours because we have from (3.50) and (3.42) the following inequalities(see (2.17*), (2.8) and the Proposition in part II, section 6)(3.51) F (�) = E(�) �A(�) � (� � � ")j supp�j � � 0 diam� � � �2 j supp�jand the last term is greater than, say � �=4 con� because contours are (as we knowfrom Theorem 2, part II) halfconnected.However, we will often need to \recolor" also some more complicated interioradmissible systems � with unclear apriori relation between j supp�j and con�(resp. conn�). In such a case, the corresponding more general argument (valid forany admissible system �) will be developed in (3.54) below. However, the notion of\smallness" has to be modi�ed here and the arguments (3.51) should be replacedby the more detailed bounds given below (in (3.60)).De�nition. We will say that a small, strictly interior subsystem � of a con�gu-ration (x;D) is extremally small in D (more precisely in (x;D) ) if it is small andmoreover if the following recursive requirement for � is satis�ed: for no strictlyinterior subsystem �0 $ �, �0 is already extremally small. The recursion starts forthose � for which there are no strictly interior �0 $ � at all.Notes. 1. This is the point where our interpretation of contours as supercontours(recall that we replaced the original notion of a contour by a more elaborate notionof a supercontour already in part II, section 7) �nally becomes useful. We can claimnow that any admissible system D has the following property: after the removal ofa removable subsystem � from D, no removable subsystem D0 of D remains suchthat D0 � � but � ! D0. This property will be highly desirable when recoloringthe extremally small subsystems (which is something which we will do later, whenproving that extremal smallness implies recolorability). Otherwise we could not useTheorem 5!2. The adjective \small" resp.\extremally small" has only a loose relation to theactual size of supp�. It is in accordance with the usage of this term (and also of therelated, perhaps even more confusingly sounding term \stable") in [Z]. There areother adjectives used to describe such a property in the literature { like \damped"in [K].3. A typical example of an extremally small system D is a collection of the typeD = �ext&f�i : �i ! �extg where the contours �i are not small but the wholesystem D is small. The case f�ig = ; is the most common one, of course.4. One should again emphasize that there is always some freedom in the de�nitionsof these notions. For example the large quantities �; � 0 in the de�nition of a smallresp. recolorable � can be changed { they can have even an \ individual value " ��(� 1) for any particular system � etc. These ambiguitites are more important thanthe arbitrariness of the choice of y in (3.49) but still have no \physical" meaningas they will not a�ect the (physically meaningful) notion of a stable phase used inMain Theorem.Proposition. A con�guration (x;D) of a mixed model whose all strictly externalsubsystems are small contains at least one extremally small subsystem.45

Proof. Consider the decomposition of D into minimal strictly external (by a strictexternality of � in D we mean that Vl(�) \ supp(D n � = ;) subsystems. Denotethem as f�i ; i = 1; 2; : : : g. Take the smallest, in the geometrical sense of � , ex-ternal subsystem, say �1, of this con�guration. Now, if the collection of extremallysmall subsystems (equivalently, by induction, the collection of small subsystems)�0 $ �1 of �1 would be empty then �1 itself would be extremally small!The importance of the notion of a small resp. extremally small subsystem stemsfrom the fact that extremally small subsystems provide practically the only relevantexample of recolorable subsystems.24 The following result gives such a statement.It is a crucial step (together with Theorem 5 above) in the proof of the forthcomingMain Theorem.Theorem 6. If � 0 = �12� then any extremally small subsystem is recolorable.(We mean the values from (2.17) and (3.34')).Theorem 6 will imply that after the completion of the recoloring procedures ofTheorem 5 (applied to the original P.S. model), no small contours or subsystemswill be left in the �nal mixed model. This leads to the Main Theorem, see section8. 6. The proof of Theorem 6Say that � is tight if any its removable subcollection D satis�es the bound(3.52) dist(�(D);� n D) � diam�(D):(Recall that we take the l1 norm everyhere.)We will show that the proof of Theorem 6 can be reduced to the case of extremallysmall tight systems �. Then the quantity A(�) in F (�) = E(�)�A(�) will become\safely small" with respect to E(�) (the quantity A�(�) will be even nonpositive inthe most important case of the \stable" external colour of �!) and Theorem 6 willbe simply some combinatorial statement relating the values j supp�j and conn�.See Theorem 7 below.However, the case of systems � which are not tight is the more characteristicand important for the proof. The quantities A(�) (more speci�cally, the quantitiesA�full(�)) will then play an important role in this reduction to tight systems.The skeleton of D.Consider the following auxiliary construction in any volume �, for any (locally)strati�ed con�guration y on �. We will actually use it below with the special choice� = Vl(D) n suppD (for a contour or admissible system D) and with y being thelocal colour induced by D on V (D).24One can construct, of course, examples of recolorable but not extremally small systems.They correspond, however, just to some marginal cases not covered by the particular choice of theconstants �; � 0; � (�) in the de�nition of a small contour �.46

De�nition. Given a strati�ed con�guration y and a cube � say that � is minimalnonsmall if A�(�; y) > � 0(diam�) (see (3.49)) and no smaller cube which is at thesame time a subcube of � satis�es such a condition.25Given a volume �, let us �nd some smallest possible (in �) minimal nonsmallcube � � � (if there is one). Take all the adjacent (having distance 1 to �) cubes�0 � � which are horizontal shifts of �, then take all the adjacent, horizontallyshifted cubes to the cubes just constructed etc. Thus we obtain some \layer" (onlypartially �lled in �; cubes which would go outside of � are excluded!) of cubesinside �.Construct also other possible partial layers not touching those constructed be-fore, according to the rule that a layer with a smallest possible diameter of its\paving blocks" is constructed in each step.The exact meaning of the statement that the layers would not mutually \touch"is that (e.g.) the vertical distance between any two adjacent layers is bigger thanthe logarithm of the thickness of both layers. Why we require this will be expainedbelow.26The collection of all minimal cubes of � thus constructed will be called theskeleton of �. The same construction can be de�ned, in analogous way, in any(generally nonstandard) volume with a given locally strati�ed con�guration. Inparticular, a skeleton of the interior V (D) n suppD of any extremally small systemD can be thus constructed.Note. The fact that skeleton has a \smallest possible grain" is maybe slightly super- uous here but it will surely be useful not only below but also in some other, moredetailed estimates (like those used in the study of the completeness of the phasepicture constructed by Main Theorem). In the situation where � = �0 n suppD, Dextremally small, the smallest possible grain of the skeleton of D guarantees thatthe size of any interior (and therefore nonsmall) � � D is at least as big as the sizeof the nearest neighboring cube from the skeleton (if there is one).We conclude: for any extremally small D in a volume � with a strati�ed bound-ary condition y given on the boundary of the complement of �, we constructedthe \skeleton" of the volume V (�) n supp� which is composed of nonsmall cubes.These cubes are \densely packed" as formulated above.Rearrangement of the energy of D.Let us de�ne the following \rearrangement" of the energy of a given extremallysmall con�guration D: Imagine that the cubes of the skeleton are just some new\contours".The idea is to show that such an \enrichened" system D� of \contours" is tightin the sense (3.52), and the value of its contour functional F (D�) (see (3.54) below)is smaller than that of F (D); however connD� is apparently bigger than connD.Therefore, by checking the recolorability of the enriched system we will also prove25Sometimes, for \stable" y (de�ned below in Main Theorem) such a cube will not exist;however this is not the case of a typical nontrivial situation below.26The reason is to keep A�full roughly additive as a function de�ned on the union of cubes ofthe skeleton. We will see below that the logarithmic distance will assure this { because of theexponential decay of the terms kT in the sums (3.43) used in the de�nition of the quantities ft .See Lemma below (3.54). 47

the recolorability of the original system. This will give the desired generalizationof the argument already given in (3.51) for the case when D = � is a single contour.Let us show this in more detail: Denote, as announced, by D� the collectionof all contours of D and also of all the cubes of the skeleton of D. Let us makean agreement that for any cube � = � we put (just to unify the notations in theformula (3.54) below!)(3.53) j supp�j = diam�:With this notation, using the Peierls condition (2.17) and the de�nition (3.22) ofF (D) we can prove (3.34') by showing the following inequalities: Recall that(3.49') A�(�; x) � � 0 diam�holds for any cube of the skeleton. See (3.49); x denotes here the con�gurationinduced by D on � (and extended somehow to the whole Z�). Now we have thefollowing relation between the contour functionals of the original extremally smallsystem D and the (\enrichened", by cubes of the skeleton) system D�:Proposition. We have the relation(3.54) � j suppDj �A(D) � � 0j suppDj �A�full(D) � � 0j suppD�j �A�full(D�):To prove this, notice that the �rst inequality just informs us about the approxi-mation of A by A�full (compare (3.42),(3.45) and (3.47)) while the second inequalitywill be shown now to be a consequence of the very de�nition of the skeleton. Write(3.49') as(3.49") � 0j supp�j �A�full(�) � 0for any new \contour" � = � of the skeleton of D.27The idea of the proof of (3.54) is that the terms F (D) resp. more precisely� 0j suppDj �A�full(D) are essentially additive (as functions of the components � ofD). The additivity of the functions of the type E(D) =PE(�) (where the sum isover all contours of D) is of course trivial. Concerning the approximate additivityof the function A�full we have the following auxiliary result.28Lemma. Let D� = C [ S be a compatible collection of contours C = f�ig andmutually disjoint cubes S = f�jg such that the cubes from S are not intersectingthe contours of the system C. Let the horizontal distance between any two cubesfrom S be greater or equal than the logarithm of the diameter of the smaller cube.27What follows will be just a suitable play with the quantities A�full j�j and � 0jdiam�j (wherethe �rst quantity is replaced by the second one for any cube of the skeleton). We can interpretthis replacement as an \installing of an arti�cial contour �".28The quantity A(V ) is of course exactly additive for disjoint volumes V but it would havesome other, more severe disadvantages (than A�full) when the \surface tension" along the verticalsides of the cubes would be discussed. 48

Then (we take all the quantities A�full with respect to the corresponding externalcolour induced by C)(3.55) A�full(D�) = A�full(C) +X�2SA�full(�S) +D(D�)where the correction term D satis�es the bound (with some large ~�)(3.55D) jD(D�)j �X�2S(diam�)�~� :Proof. Consider two such collections D� which di�er just by one cube � i.e. letD�� = D�&�. Assume that the cube � is no greater than any cube of D�.It is now su�cient to prove the bound(3.55D') jDj = jA�full(D��) �A�full(D�)�A�full(�)j � (diam�)�~� :Notice that (3.43) can be written also as the sum over intervals I(3.43I) ft(y) = et(y) � XI�Z : t2I kIwhere kI is the sum of all the contributions to (3.43) having a �xed projection I ofT to the last coordinate axis. Of course, we have a bound, for suitable large �̂jkI j � exp(��̂ jIj):Imagine that in the expression on the left hand side of (3.55I) we ignore (whensubstituting these quantities, for any vertical section of � and any t, into (3.47))all I intersecting both � and some other cube resp. contour of D�. Then therelation(3.55I) A�ignore(D��) �A�ignore(D�)�A�ignore(�) = 0is exact , as simple inspection shows. The correction (due to the quantities kI justignored) is then obviously of the order exp(��̂ d) where d is the distance of � andthe cubes from D�. This proves (3.55I), and therefore also (3.55D').Let us continue now the investigation of the right hand side of (3.54): By (3.50),the right hand side of the relation (3.54) is greater than (we use here the very29smallness of the cube �(D�) !)(3.56) � 0j suppD�j �A�full(�(��)) � � 0(j suppD�j � diam�(D�))which is surely greater (notice that diam�(D�) = diam�(D)!) than, say, �2 j suppD�j.Thus it su�ces to show now that the quantity �2 j supp��j (and, therefore, alsothe right hand side of (3.57)) is greater than, say, �12� conn�.We will prove this by proving the following result (Theorem 7). First generalizethe notion of a tight system � to any subset of Z�:29Notice that our use of the squares in the de�nition of smallness is rather important techni-cally. Namely, our method of the proof relies quite heavily here on the fact that the enriched (bycubes of the skeleton) system D� of any small D is again a small system!49

Tight sets.Say that S � T is isolated in T (T �Z�) if(3.52') dist(�(Vl(S)); T n S) � diam�(Vl(S)):Say that T is tight if there are no isolated subsets of T .Notes. 1. The choice of �(Vl(S)) above is of course motivated by our de�nitionof �(�) and our emphasis on the notion of a strict interiority and diluteness.\Topologically", there is not much di�erence between the choice above or the choiceof the cube �(S).2. Notice that the system supp�� is already tight because the de�nition of askeleton of � gives no room for subsets of the type (3.52) for the enriched setT = supp�� !Theorem 7. If T is tight then(3.57) connT � 6�jT j:Let us start the proof of Theorem 7 by the simple observation (used in Lemma 1,section 7 below) that any contour of �� can be assumed to have a a cardinality atleast 1024 = 210. Really, it is rather straightforward to see the validity of a bound(3.57') connD � 9j suppDjfor any interior tight subsystem D of � whose cardinality is d � 1024. To checkthis bound remember that our contours (supercontours in the sense of Chapter 2;this is the second moment { after the proof of Theorem 5 { where we pro�t fromtheir properties) are such that �! �0 implies that the cardinality of supp�0 is atleast twice bigger than, say the cardinality of supp�. Thus, the longest branch ofthe forest D has at most 9 sites.Now take any tree of D. Retain the notation D for it, and for any subtree E �D construct a suitable commensurate path connecting E to some superordinatedcontour of the remainder of D. The length of any such path can be surely chosensmaller than, say, the cardinality of the support of E:Namely, consider the projection to Z��1 : then all the projections of contours of Dare connected. Construct �rst such a connecting a path in the projection to Z��1,with \horizontal" steps of length 1 (surely, less than d = diamE steps are needed),and then add suitable vertical components to the already constructed horizontalsteps to keep the successive steps commensurate and to guarantee that the path(not only its projection) really starts in E and ends in D n E. The vertical distanceto be overcomed is also smaller than d = diam E and clearly, a suitable choice of dcommensurate vertical steps (which can be understood as vertical components ofthe horizontal unit steps constructed above) overcoming the given vertical distanceis also possible.Now, when comparing the total length P l(P ) of the these paths P = P (E),connecting any interior subtree E � D to the remainder of the corresponding tree ofD, with the sum of the cardinalities jEj of the subtrees (which surely have at least as50

many points as is the length of the path P (E)!) we see thatP l(P (E)) < 9j suppDjbecause each point of D was used at most 9 times in the above consideration.(This statement is actually some \weaker" analogy of Theorem 7 for systems havingcontours with a cardinality less than 1024.)Moreover, by the same argument one can connect any such \small sized" D tothe superordinated tree of the remainder of � by a path containing no more thanj suppDj of additional steps. The conclusion is that 10j suppDj is surely the upperbound for the number of points needed to make each such D commensuratelyconnected and connected also to remainder � n D. Thus we can really restrictourselves to the case when the smallest interior components of � have a cardinalityat least 1024.Note. Notice that after the removal of any such \small sized" subtree D, the re-mainder is still extremally small (if the original admissible system was).Thus it su�ces to prove the desired inequality(3.58) conn�� � 6�j supp��jat the assumption that all the contours of �� have already a (halfconnected: recallTheorem 2 and Proposition in section 2.7 ) support having a cardinality at least1024. If we assume contours to be moreover connected then it su�ces to show (inslightly more general setting) that the inequality(3.59) connS � 3�jSjholds for any S �Z� at the assumption that the components of S have cardinalityat least 1024.Apparently, the proof of such a geometrical statement will �nish also the proofof Theorem 6. Namely, then we can conclude the arguments of (3.54) and (3.55)as follows:� j suppDj �A(D) � � 0j suppD�j �A�full(D�) � � 0j suppD�j � � 0 diam�(D�)i.e.(3.60) � j suppDj �A(D) � �2 j suppD�j � �12� connD:Proof of (3.59). Commensurately connected collections of cubes.It will be useful to reformulate �rst the notion of a commensurately connectedgraph in an alternate language { based on the employment of cubes fromZ� insteadof the bonds from 2kZ� :De�nition. Say that the two cubes �;�0 �Z� are commensurate if(3.61) � \�0 6= ; and j log2 diam�� log2 diam�0j � 2:(Notice that the constant 21 in (3.2) was replaced by 22 in (3.61). This is forpurely technical reasons and will be convenient below.)51

Proposition 1. If G is a commensurately connected graph then the collectionf�(b); b 2 Gg of covering cubes of bonds of G is commensurately connected in thesense above.This is immediate, by comparing (3.2) and (3.61).The opposite relation (that any commensurately connected collection of cubescan be \approximated" by a commensurately connected graph) can be also estab-lished: Introduce �rst another auxiliary geometrical notion.De�nition. If � is a cube with a diameter 2k � diam� < 2k+1 then the lexi-cographically �rst point of � \ 2kZ� will be called the anchor of �, denoted bya(�).Proposition 2. Let S = f�ig be a commensurately connected collection of cubesfrom Z�. Then there is a commensurately connected tree T such that all the anchorsa(�i) are among the (possibly multiple) vertices of T and(3.62) jT j � 3�jSj:Proof. We may assume that S is already a tree. Take any commensurate bondf�;�0g 2 S. Write [log2 diam�] = k; [log2 diam�0] = k0; we may assume thatk0 2 fk; k+1; k+2g. A straightforward inspection shows that it su�ces to considerthe case � = 1; k = 1, and a(�0) = 0. Notice that then the following path froma = a(�) to a0 = a(�0) can always be constructed :(3.62) a0 = a+ v1 + v2 + v3where the vectors vi having the lengths 2li ; li 2 N satisfy the following requirements:k � l1 � k + 1; l1 � 1 � l2 � l1 + 1; l2 � 1 � l3 � l2 + 1; k0 � l3 � k0 + 1:It is clear that the tree de�ned by all the bonds fa; a+v1g; fa+v1; a+v1+v2g; fa+v1+v2; a0g (where a; a0 vary over all commensurate pairs �;�0 and the triple of thetype above is actually repeated in the direction of any coordinate axis for � � 1) iscommensurately connected.Corollary. Let S � Z�. Let conn� denote the cardinality of a smallest commen-surately connected collection of cubes satisfying the following requirement: if allpoints of S are added (we identify the points of Z� as cubes of diameter 1) then thewhole collection is commensurately connected. Then(3.63) conn� S � 13� (connS):We will now prove Theorem 7 by showing that the inequality(3.64) conn� S � jSjholds for any tight set S whose components have a cardinality at least 1024.52

Second covering cube.De�ne now a suitable collection of cubes having a size 2k where k = 1; 2; 3; : : :such that any cube in Z� can be \packed", with a reasonable \accuracy", by somecube of the collection:De�nition. Denote by Kk the collection of all cubes in Z� which are shifts, bysuitable values from the lattice 2k�1Z�, of the unit cube [0; 2k]� in the lattice2kZ�. Write K = [kKk where k = 1; 2; : : : . We have the natural ordering �on K extending the ordering by size resp. the lexicographic ordering of shiftsof one particular cube; this ordering can be extended to suitable total orderingof all cubes in Z� which is in accordance with the inclusion relation as well aswith the lexicographic order of mutually shifted cubes, and we denote by b� the(lexicographically �rst) cube from Kk, k smallest possible, containing �. This willbe called the second covering cube of � resp. of a set S such that � = �(S).Notice the following fact : if � is the covering cube of S then the second coveringcube of S contains � and has a diameter at most four times bigger than �.7. The Proof of (3.64)Let us make an agreement that an explicit choice of the appropriate constantshere will be given below only for the case of the dimension � = 3. Apparently, for� > 3 the �nal constant in Theorem 7 is even better { but we do not care yet.Black and grey cubes of a set S � Z�. Say that a cube � is a black cube ofS �Z� if �\S contains at least 4(diam�) 12 points, resp. at least diam� points ifits diameter is smaller than 16. Any cube from K which is the second covering cubeof some black cube of S will be called the grey cube of S. (It obviously contains atleast 2(d0) 12 points of S where d0 � 4d denotes the diameter of the correspondinggrey resp. black cube.) We will show the following statements :Lemma 1. If T is tight and its connected components have a diameter at least 16then the collection of its black cubes is commensurately connected.Lemma 2. The number of grey cubes of a size at least 1024 of any set S �Z� isno greater than 1=2 jSj.Lemma 3. If S = f�ig is a commensurately collected collection (of black cubes ofsome set S) then the collection bS = fc�ig (of grey cubes of S) is contained in somecommensurately connected collection S 0 such that jS 0j < 2j bSj.Proof of Lemma 1. We will proceed by the induction over the number of points inT . Say that S is a nice subset of T if it has the following property : for any t 2 Sthere is a commensurately connected collection of black cubes f�ig of S which isconcentric (i.e. t 2 �i for each i ), starts in the covering cube of S and ends in t.Take some maximal nice subset S of T . We will show that S = T if T is tight.Really, if N = T n S is nonempty then either there is some isolated subset M ofN or N is tight. In the former case take M as the smallest possible isolated (andtherefore tight) subset of N . Then M (or N itself, in the latter case) is also niceby the induction assumption. Take the covering cube �(M) of M .53

We claim now that there is some black cube �0 of S such that(3.65) dist(�(M);�0) < diam�(M) and j log2 diam�(M) � log2 diam�0j � 1:This follows from the fact that M cannot be isolated in S[M (otherwiseM wouldbe isolated also in T ). Therefore, there is some t 2 S whose distance from �(M) isno greater than diam�(M) and we take an appropriately large black cube �0 3 t.(Its existence is guaranteed by the \nicety" of S.)Now, if �� is any black supercube of �0 (black in S ) then the supercube of�(M) [ �� { denoted as ���{will be shown to be again a black cube (of thewhole set T ) and this would mean that S [N would be nice, as simple inspectionshows.(Check that there is now a commensurate path from t to �(M) and anycommensurate chain of cubes going \up" from �(M) through cubes of the type�� can be modi�ed by going through corresponding cubes ���. Thus S = T . Themodi�ed chain is clearly also a commensurate one, containing �(M).)The observation that ��� is black in T follows from the following more generalstatement :Lemma. If �0;�00 (�00 = �(M) in the above application)are two cubes which areblack cubes of some sets T 0; T 00 ; T 0\ T 00 = ; and such that dist(�0;�00) � diam�0and j log2 diam�00�log2 diam�0j � 1 then the covering cube �(�0[�00) is the blackcube of the set T = T 0 [ T 00.This is easily seen (it su�ces to consider the case of the dimension � = 1!) fromthe inequality (d denotes the distance between two cubes of diameters 1 (�0) resp.x (�00))(3.66) d � 1 & 1=2 � x � 2) (1 + x + d) 12 � 1 + x 12 :Proof of Lemma 2. We will give the proof only for the case � = 3. For the purposesof this proof modify the cubes a+[0; 2k]� from Kk to the following form: a+[0; 2k�1]� . Then the system Kk can be decomposed into 8 pavings of Z� by disjoint sets.Take the sum(3.70) 82 1X10 12 k2 < 7=16and imagine that any point of t 2 S transfers the 122�k2 { th portion of its \unitmass" (3.70) to any cube of Kk containing t. By the de�nition of a grey cube,the total mass thus transferred to any grey cube of S is at least 1 and, therefore,the cardinality of the set of all grey cubes of S (of a diameter bigger than 1024) issmaller than 7=16 jSj � 1=2 jSj.Proof of Lemma 3. This easily follows from the following observation : if �;�0are commensurate cubes then either their second covering cubes b�;c�0 are alsocommensurate or one of these latter two cubes can be replaced by an auxiliary,twice bigger supercube from K such that the �rst statement is true.54

This concludes the proof of Theorem 7 if we moreover notice, that to connectall the components of S to their black supercubes we can construct commensuratepaths with less than, say, 1=2 jSj steps.Thus, also Theorem 6 is proven.Note. Theorem 6 is a stronger and better replacement for the \Main Lemma" of[Z].One could use its analogy also in the translation invariant situation of [Z]. Thenit can have (e.g.) the following form:If �i are mutually external \large contours" such that ajV (�i) > � j supp�ij andif we denote by ext = � n [V (�i) then(3.71) aj ext j > � Conn(�; f�ig)where the integer on the right hand side denotes the cardinality of a smallest possibleset whose union with �c and all supp�i is connected.With this lemma, one can rewrite the ordinary P.S. theory in a way analogousto that used here without an explicit construction of the contour models. See thelectures notes [ZRO].8. The Metastable Model. The Main Theorem.Theorems 5 and 6 imply that the process of recoloring does not stop up tothe very moment when there are no recolorable removable subsystems available inthe �nal mixed model. This �nal mixed model will be called the (totally) expandedPirogov { Sinai model , corresponding to the original abstract Pirogov { Sinai modelwith the hamiltonian (2.16) satisfying (2.17). Thus we have the following result.Corollary. In the expanded model, only those con�gurations of contours remainwhich contain no strictly interior recolorable subsystems. In other words, which arerepresented (Theorem 1) by a graph (without loops) whose any complete subgraph(complete in the relation � ! �0 : if the end of the arrow is in the subgraphthen the whole arrow is also in the subgraph) which is moreover strictly interior isnonrecolorable (therefore also nonsmall).De�nition. Con�gurations of contours of the type above will be called redundant .Note. An equivalent characterization of redundancy in terms of the related notionof metastability will be given below. Notice that the extremal smallness of �means just the smallness of � & redundancy of all strictly interior subsystems of�. Speaking about the nonsmallness of some � in Corollary above one has in mindthe nonsmallness of � in the \provisional" mixed model, constructed up to themoment when the extremal smallness of � is checked before applying Theorem 5.In this provisional mixed model, there are already no extremally small systemswhich would be \geometrically smaller than �" (in �).By the way, nonsmallness of a contour (or admissible system) � in the �nal, fullyexpanded model and in the provisional one would mean almost the same. Moreprecisely we note that for cubes � of a comparable (or smaller) size as � thequantities A�(�) and A�(�) are practically the same for both the provisional model(expanded \up to the size of �") and the �nal expanded model. (The quantitiesA(V ) are exactly the same for volumes V � V (�), in both models.) Namely, the55

clusters T having a di�erent value in both considered mixed models (the provisionalone and the �nal one) have a size at least diam� and the di�erence between thecorresponding values of A(�) is thus of the order "diam� (a tiny quantity comparedto � j supp�j).This fact (together with the fact that that A� and A are almost the same inany mixed model) will be used later in the proof of Main Theorem (relation (3.81)below) and also in (3.90).The metastable models. Say that a subcollection D of an admissible system ~Dis an external one if ~D n D can be decomposed into disjoint collections D0, �1, �2,: : : such that (V (D) [i V (�i)) \ V (D0) = ; and moreover �i !D for any i � 1.A con�guration x = (xbest;�) which is y { diluted (i.e. equal to y 2 S outsideof some set having �nite components which are standard sets) will be called y{thmetastable (shortly, metastable) if no redundant external subsystem D of � exists.The restriction of the original abstract Pirogov { Sinai model (with the hamiltonian(2.16)) to all y{th metastable con�gurations will be called the y{th metastablemodel. If the abstract Pirogov { Sinai model was constructed as a representationof an original model (2.2) then we de�ne the metastable y{ th submodel of (2.2)as the restriction of the hamiltonian (2.2) to the con�guration space Xymeta of alldiluted con�gurations x 2 Xwhose representations x = (xbest;�) in terms of theircontours are y{th metastable in the sense above. Now we can also say that anadmissible system D is redundant if and only if it has no removable metastablesubsystems.Notation. Recall the quantities ft(y), y 2 S { see (3.43) { which were constructedfor any mixed model. Consider now these quantities for the case of the totallyexpanded model whose construction was just �nished. For the expanded modelcorresponding to the original Pirogov { Sinai abstract model, the quantity ft(y)will be denoted as(3.72) ft(y) = ht(y):Now we are able to formulate the basic result of the paper:Stable elements of S. A strati�ed con�guration y 2 S for which there is noredundant contour or admissible system � such that (x�)ext = y will be calledstable.In other words, y is stable if the collection of con�gurations x 6= y having the valuey outside � is empty in the fully expanded model.Main Theorem. Consider an abstract Pirogov { Sinai model de�ned by (2.16) and(2.17), with � su�ciently large. Then the quantities ht(y) constructed by (3.72) arethe free energies of the corresponding y{ th metastable models. The con�gurationsy 2 Swhich are stable correspond precisely to those con�gurations from Swhich arethe ground states of the quantity(3.74) Xt02[ t ] ht0(y)56

where [ t ] denotes the collection of all t0 = t+(0; 0; : : : ; 0; t�); t� 2Z. The \groundstate" is understood in the sense that we always have(3.75) Xt02[ t ](ht0(~y) � ht0(y)) � 0if ~y 2 S di�ers from y on a layer of a �nite width. For any stable y there existsa probability measure P y on the con�guration space of the given abstract Pirogov {Sinaimodel whose almost all con�gurations are y-th metastable and moreover, theconditioned probabilities P q� of P y, being taken with respect to all con�gurationswhich are y { diluted resp. strongly diluted in � correspond to the diluted resp.strictly diluted ensembles (2.18).Corollary. If the considered abstract Pirogov { Sinai model represents a \physical"model given by hamiltonian (2.2) then for any stable y there exists a Gibbs state (ofthe model (2.2)) on X= SZ� whose support can be identi�ed as a suitable subset ofthe collection Xqmeta of all y-th metastable con�gurations.By a support of a probability measure we mean a Borel (more precisely countablycompact) set having measure 1.Notes. 0. Clearly, the families of con�gurations Xymeta are mutually disjoint fordi�erent y 2 S. We do not study here in much detail the structure of a typicalcon�guration (of a q { th Gibbs state) here. See, however, the �nal section 9 forsome information. (This problem deserves a more full treatment. However, it seemsreasonable to do this in connection with some future investigation of other relatedquestions { like the completeness of the phase picture constructed here. We planto devote a separate paper to these questions.)1. There are no other strati�ed Gibbs states of such an abstract model. We arenot giving here the proof of such a completeness of our phase picture (character-ized by the stable values of y 2 S).It can be done similarly as in [Z]. See also somecomments in section Concluding Notes below. However, we plan a more systematictreatment of this and related questions in a separate paper.2. By the phrase \the y { th Gibbs state can be identi�ed with the correspondingy { th metastable model" we mean that almost all con�gurations of this Gibbs stateare y { diluted and moreover the \islands"(the components of Vl(�) where � is thecollection of all contours of the the considered con�guration x ; this covers the setof all points of x which are not y { strati�ed) are typically \small" and \rare"(butdistributed with a uniform density throughoutZ�). For a more complete statement,see the section 9 below.3. Having in mind that the quantities ht(y) can be e�ectively computed from expan-sions (3.43) (within a given precision; of course this is in full a horribly complicatedsum { but its terms are converging very quickly, indeed), our Main Theorem givesin fact a constructive criterion for �nding the stable values y 2 S.Practically, one may suggest an \approximate �nding" of stable values of y fromsome \M{expanded" model (M is some square, for example) where only thoseextremally small subsystems whose size does not exceed the size of M are alreadyrecolored. 57

In fact, even for squaresM quite small some useful approximations can be found,often enabling already to distinguish between the stable and nonstable y. This isbecause the series (3.43) are really very quickly converging and moreover we oftenhave some additional symmetry in the special cases of interest { like the +=� sym-metry in some special cases of the Blume Capel models. (For Blume Capel models,even the smallest size 2�2 of the squareM can be useful { see [BS]. Namely, consid-ering only �rst two or three terms in (3.43) a correct conclusion about the stabilityof 0 on one side and +=� on the other side can be made.)4. In fact, in �nite volumes � there is no noticeable di�erence between the be-haviour of the stable y {th phase and another ~y {th phase if the quantitya = Xt2[ t ]ht(~y)� ht(y)is such that, say, a�1 > (diam�)��1. Quite straightforward estimate of the quan-tities(3.76) A(�) � p(V (�)) � � diam� � � (diam�)��1p(V (�))where p denotes the orthogonal projection on Z��1 � Z� shows that equationsof the type (3.49) cannot be violated in such a small volume.(\Small" can have ameaning \having a diameter of the order 1027" here if a is very small; namely thedi�erence between various hy is only of the order exp(�F (�)) ) if ey are the sameand � is the smallest contour \which makes the di�erence" between two di�erenty.)5. Though the question of the existence of at least one stable y 2 S is not the abso-lutely crucial one (as the preceding note shows) one should mention, nevertheless,that at least one stable y 2 S really does exist :Say that y 2 S is N-ground if for any ~y 2 S such that ~y = y for jt� j � N wehave the inequality (3.75). Now, if the con�gurations yN 2 S are N-ground thena suitable subsequence of fyNg must converge to some ground (\stable") valuey 2 S. (We use the compactness of the space S in this argument as well as thequick convergence of the cluster series for the quantities ht(y)).6. For some models, like the Ising model with strati�ed random external �eld, thecollection of all (almost) stable y 2 S can be very rich and the phase diagram { asthe function of all (vertically dependent) values of the �eld { extremely complicated.We plan to study this particular case in some later paper.7. The latter example case shows that it is not very reasonable to try to formulateresults about the shape of the phase diagram in full generality here { becausepossibly in�nite parameters are present in the hamiltonians of the strati�ed type.However, if we call by a phase diagram of the model the very mapping(3.79) f y 2 S 7�! fht(y)g gthen the information about the actual phase diagram, its dependence on the param-eters in the Hamiltonian etc. can be deduced from (3.79) ; just by using suitablevariants of the implicit function theorem (possibly with in�nitely many variables).58

However, this is not a paper on analysis of manifolds and so we omit these questionscompletely.It is worth noticing here that, in order to get a best possible smoothness of themapping (3.79) (and of the mappings derived from it by implicit function theorems),it may be reasonable to modify suitably the de�nitions of extremally small contoursetc. { to obtain the best available di�erentiability (even local analyticity) propertiesof this mapping. This question also deserves a separate study, like in [ZA].Proof of Main Theorem. The key statements were already Theorems 5 and 6 above{ which guarantee the very existence of the fully expanded model and therefore thepossibility of the very formulation (based on the existence of the quantities ht) ofour result. Noticing this, one has to add now only a few additional observations :1) If y satis�es (3.75) then for any admissible system � such that (x�)ext = ywe have, from the very de�nition of the quantity A�(�), � = �(�) (see (3.37) andthe commentary below ) the inequality(3.80) A�(�) < "diam�:Really, one could take even zero on the right hand side if the appropriate par-tially expanded model (namely the mixed model studied at the moment when thesmallness of � was discussed)could be taken here. However, we have a di�erentmixed model now (at the end of the recoloring procedures) { the fully expandedone { and we have to use (3.36)) to notice that(3.81) jA�partially expanded(�) �A�expanded (�)j � "0jdiam�jwhere "0 is very small, of the size "diam� ! Thus we see, that the smallness of any� with the external colour y satisfying (3.75) is almost a tautology and therefore,under such boundary condition y, there is no di�erence between the original Pirogov{ Sinaimodel and the metastable one.Moreover one has quickly converging expansions of partition functions with theboundary condition y in any volume, and this implies the validity of the propertiesof the \y { th Gibbs state" stated in the Main Theorem. One could prove also theexponential decay of correlations in any Gibbs state thus constructed. See the lastsection 9 below for some relations (namely (3.90)) proving (or, at least, preparinga ground for the proof) of these facts.2) On the other hand, if y is not a ground state then there is some ~y di�eringfrom y on some layer L of a �nite width { say d { such that the vertical sum is(3.82) Xt02[t] ht0(y) � ht0(~y) � �for suitable � > 0. Take a very large box B �Z��1 such that(3.83) � j@Bj << �jBjand consider the volume (which has the form of a \desk")� = ft 2 ~L; t̂ 2 Bg59

where ~L is another (thicker) layer containing L \su�ciently inside". Take thecon�guration x = y outside �, x = ~y inside �. Then, if we compute the quantityA�(�) for the volume � = (@B �Z)\ Lwe have, according to (3.45), the bound from below:(3.84) A�(�) =Xt2�(ht(y) � ht(~y)) � �jBj � "j supp�j >> �dj@Bj >> � jdiam�jaccording to (3.49) which shows (compute A�(�) for � � � : this is even biggerthan A�(�) !) that the \contour � encircling the cylinder �" is not a small contour!Strictly speaking, this argument requires some commentary { because we do notknow that such a � is a contour of the model. This can be done precise simply byallowing also contours with the weight +1 in our abstract Pirogov { Sinai model;such an assumption does not change its \physical" properties but allows to workwith a richer family of contours (which is quite indispensable here, as we see in(3.84)).Note. This is only one of the arguments why it is generally advisable to work withan abstract Pirogov { Sinai model instead of the original spin model.9. Properties of Typical Configurations. Concluding NotesWe are still in a situation of an abstract Pirogov { Sinai model. Denote byXymeta the con�guration space of all con�gurations of the y -th metastable model.More precisely denote byXymeta(�) the con�guration space of all x = (x�best;�) suchthat � is strictly diluted in � i.e. Vl(�) � �. We have the probability measureP ymeta;� on Xymeta(�), the corresponding partition function Zmeta(�; y) being givenby the Gibbs density exp(�H(x� jy)) (and H(x�jy) being given by (2.16)). WriteP y� instead of P ymeta;� if y is stable. Having de�ned these \strictly diluted" Gibbsconditioned probabilities P ymeta;�, the question now is whether a suitable limit over�!Z�(3.85) P ymeta(E) = lim�!Z� P ymeta;�(E)exists for a su�ciently rich collection of events E (\su�ciently rich" would mean,in the spin model, e.g. the collection of all cylindrical events i.e. events dependingon a �nite number of coordinates; in the abstract Pirogov { Sinaimodel a simplestexample of such an event is as below in (3.86)) and gives, for y stable, a Gibbsmeasure P y on the whole X. Let us show for example that the limit(3.86) P y(E(t)) = lim�!Z� P y�(E(t))exists for any (even nonstable) y and for any t 2 Z�, for the event E(t) de�ned asfollows: \t =2 Vl(�), where � denotes the collection of all contours of x" i.e. for theevent \t is stricly exterior point of the given con�guration x".60

We have obviously the formula(3.87) P y�(E(t)) = (Zyl (�))�1Zyl (� n t) exp(�et):We can expand, by (3.0) (in the fully expanded model) both the partition functionson the right hand side of (3.87). The possibility of such an expansion follows fromthe very notion of stability of y! (resp. from the de�nition of the metastableensemble, if y is unstable and we compute the probability P y�;meta). We have, fordist(�; t) ! 1, the following expression: (Analogous, slightly more cumbersomeexpressions obtained in any �nite volume � are omitted here.)(3.88) logP y(E(t)) =X(kextT � kT )where the quantities kT resp. kextT correspond to the expansion of Zyl (�)) resp.Zyl (� n t)) and the sum is over those (quickly decaying by (3.1)!) quantities kTresp. kextT only which touch t (contain or have a distance at most 1).Clearly, for su�ciently small " we have from (3.1) and (3.88) the approximaterelation(3.89) 1� P y(E(t)) � ":This suggests that the \islands" of a typical con�guration (x;�) (interpreted hereas connected components of Vl(�), but see below for a more elaborate notion of an\island") are really typically \small and rare" because they do not typically intersecta given (arbitrarily chosen) point t 2 Z�. To make this intuitive description of atypical con�guration (which is quite characteristic for the Pirogov { Sinai theoryand the phase picture this theory gives) more detailed, de�ne below the \islands"of a given con�guration (x;�) in a di�erent, more detailed way grasping also someimportant features (namely the appearance of redundant contours) of the regimeappearing inside Vl(�) and thus allowing also precise expansion formulas for theevent \a given island appears":De�nition. An admissible subsystem � � D of a con�guration (x;D) is called anisland if there is a strictly external (in D) superset �0 � � such that V (�0 n�)\� =; and �0 n � contains no redundant subsystems. (In other words, a successiverecoloring \around �" can be applied, deleting all the elements of �0 n�. The case�0 = � is typical, of course, and � is most commonly either a single contour or acollection of one external contour and some interior redundant ones.) Denote byP y[�] the probability (in P y) of the event \� is an island of a given con�guration".Proposition.(3.90) P y[�] = exp(�F1(�)) exp(XT (kT � kextT ))where F1 is de�ned as in (3.22),(3.21) but with respect to the fully expanded model(not the temporary one, used in the moment when � was recolored!) and the sum61

is over those T only which touch supp�. The quantities kT resp. kextT corrrespondto the quantities kT ( �) resp. kT (�) in (3.25).Note. 1. Notice that F1 is a horizontally translation invariant quantity.2. Notice that we do not formulate here the probabilities of the events dependingalso on the interior of �. Having determined all possible P y� this is already (in thestrictly diluted ensemble) a straightforward task, using the properties of conditionedGibbs distributions in �nite volumes.Proof. This is just an application of (3.25), for D = ;. It is not exactly the sameargument as the one which was used in section 3 for the recoloring of �. Namely,a (slightly) di�erent mixed model is used here, also with clusters \bigger than �";however the di�erence between F (�) and F1(�) is extremely small, of the order"diam�.Corollary. The probability P y[�] of an island � satis�es an estimate, with " =exp(�� 0) where � 0 is something like �13�(3.92) P y[�] � "conn� :The probability of the event \there is an island around t having a diameter � d" canbe estimated as const "d. The mean relative area, occupied by the islands � resp.by the \interiors" Vl(�) of a typical con�guration is smaller than some "0 = "0(")resp. "00 = "00("), and independent of the particular choice of the con�guration.There is an exponential decay of correlations in the probability P y.We do not prove these general (but more or less straighforward once (3.89),(3.90)was established) facts here.30De�nition. Say that a volume � �Z� is balanced or conoidal if for any admissiblesystem in �, V (�) � �) Vl(�) � �:To have examples of balanced sets, take the sets (\rectangular cup & cap gluedtogether"; compare (3.19) and (3.20))ft 2Z� : dist(t; @�) � 1=2 diam�gwhere � is a cube in Z��1 and @� denotes its boundary in Z��1 �Z�.For balanced volumes, diluted and strictly diluted partition functions (and cor-responding Gibbs measures) are obviously the same and the fact that the limit ofdiluted Gibbs measures is (for �!Z�) a Gibbs measure on X is obvious.3130The above arguments give also only an outline of the full proof that P y constructed by (3.86)gives really a probability measure on Xwith the properties stated above. To interpret furtherthis measure even as a Gibbs measure on Xi.e. as a Gibbs measure of the original hamiltonian(2.2) it would be useful to have another general statement, whose proof is omitted in this paper(however, we do not know a suitable reference): A limit, for � ! Z�, of strictly diluted Gibbsmeasures in � is a Gibbs measure on X. Instead of such a general statement we consider belowonly the more special limit over conoidal �!Z�.31The general case of an arbirary volume � is not considered here. It requires some more careconcerning the \stability" of considered �nite volume measures with respect to various boundaryconditions. Anyway, these and related questions must be investigated in more detail also whenevera completeness of the phase picture constructed by Main Theorem is studied. We postpone thisdiscussion to a forthcoming paper. 62

We expect that the new method presented in this paper will be applicable alsoin other situations (even nonstrati�ed ones) where \noncrusted" contours appear.Notice also that the method is applicable to situations where one starts (aftersuitable preparation of the given \physical" model) with some mixed model insteadof the abstract P.S. model. This is the case of models with continuous spins (studied in [DZ]) having several \potential wells", for example. In these models,the expansion around positive mass gaussians (approximating the regimes of thepotential wells ) of restricted ensembles (of con�gurations living in the vicinityof the potential wells) yields a mixed model of the type studied here, and thenthe analysis developed in part III of this paper could be applied to these models,possibly also for the wells which are not so \deep" such that the previous analysis(like [DZ]) could be applied to them.References[S] Ya. G. Sinai, Theory of Phase Transitions. Rigorous Results, Pergamon Press, 1982.[PS] S. A. Pirogov and Ya. G. Sinai, Phase Diagrams of Classical Lattice Systems, Theor.Math. Phys. 25, 26 (1975, 1976), 1185 { 1192, 39 { 49.[Z] M. Zahradn��k, An alternate version of Pirogov { Sinai theory, Comm. Math. Phys. 93(1984), 559{581.[D] R. L. Dobrushin, Gibbs states describing coexistence of phases for a three-dimensionalIsing model, Theor. Prob. Appl. 17, 582{600.[FR] J. Fr�ohlich,Mathematical aspects of the physics of disordered systems, Critical Phenomena,Random Systems, Gauge theories (Les Houches 84; ed. by K. Osterwalder and R. Stora),North Holland, 1986, pp. II, 725 { 893.[BMF] J. Bricmont, A. El Mellouki and J. Fr�ohlich, Journal Stat.Phys. 42 (1986), 743.[ZG] M. Zahradn��k, Low Temperature Gibbs States and the Interfaces between Them, Proc.Conf.Stat. Mech. and Field Theory, Groningen 1985, Lecture Notes Phys. 257, 1986, pp. 53 -74.[HKZ] P. Holick�y, R.Koteck�y and M. Zahradn��k, Rigid interfaces for lattice models at low tem-peratures, Journal Stat.Phys. 50 (1988), 755 { 812.[HZ] P. Holick�y, M. Zahradn��k, On entropic repulsion in low temperature Ising model, Pro-ceedings of the Workshop on Cellular Automata and Cooperative Systems (Les Houches,June-July 1992), vol. 396 NATO ASI SERIES, Kluwer, 1993, pp. 275 -289.[M] I. Melicher�c��k, Entropic repulsion of two interfaces in Ising model (in Czech), Diplomawork, Charles Univ. Prague 1991.[DSA] R. L. Dobrushin and S. B. Shlosman, Completely analytic interactions. Constructive de-scription., J. Stat. Phys. 46 (1987), 983 { 1014.[DZ] R. L. Dobrushin, M. Zahradn��k, Phase Diagrams of Contin.Spin Systems, Math.Problemsof Stat. Phys. and Dynamics (R. L. Dobrushin, ed.), Reidel, 1986, pp. 1{123.[ZA] M. Zahradn��k, Analyticity of Low-Temperature Phase Diagrams of Lattice Spin Models,Journ. Stat. Phys. 47 (1988), 725{755.[KPZ] F. Koukiou, D. Petritis, M. Zahradn��k, Low temperature phase transitions on Penrose typelattices, Proceedings of the Workshop on Cellular Automata and Cooperative Systems (LesHouches, June-July 1992), vol. 396 NATO ASI SERIES, Kluwer, 1993, pp. 275 -289.63

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