Zahradník, Miloš

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Institute of Mathematics AS CR, v. v. i.

WSAA 11Miloš ZahradníkAn alternate approach to the Pirogov – Sinai theory, not employing the modelswith a parameterIn: Zdeněk Frolík (ed.): Proceedings of the 11th Winter School on Abstract Analysis. CircoloMatematico di Palermo, Palermo, 1984. Rendiconti del Circolo Matematico di Palermo, Serie II,Supplemento No. 3. pp. [397]--401.Persistent URL: http://dml.cz/dmlcz/701781Terms of use:© Circolo Matematico di Palermo, 1984Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czAN ALTERNATB APPROACH TO THE PIROGOV - SINAI THEOBTf NOT .EMPLOYINO THE MODBLS WITH A PARAMETER. Miloš Zahradník This is a very short exposition of another approach to the Pirogov - Sinai theory of phase diagrams*The detailed version will be published [ 1 ] .We outline here the main features of our approach onlyfdiffering from the "classical" P.S. theory and also from the approach of Kotecltf - Preiss published in this volume. We do not introduce the contour models with a parameter. Insteadfwe define the contour functionals explicitly,by (4) and introduce the notion of a "truncated functional1* (5),which is a T •functional.This leads to the construction of "truncated" contour models. These truncated models are equal to the usual contour models if the corresponding ground state is stable (7). We distinguish the stable and unstable ground states by using the Hparameters of nonstabilityH a/ which appear.in our approachtin another,perhaps more transparent meaning (8).We use these parameters also for a classification of contours on HsmallH and Hlarge" ones (the latter appearing in a nonstable phases only). Some comments about the methods of proofs: Our main technical tool is the method of cluster expansion,widely used in the literature.lt is very suitable for the investigation of contour models such that the corresponding contour functional is a 1"-functional. For the investigation of the behaviour of HnonstableH phases we use the following method.by fixing the large contours(resp.exter-nal large contours) of a configuration and Hsweeping outH the re-maining contours,we define an auxilliary model,having a configu-rations with large (reap, external large) contours only,and an appropriately changed hamiltoniaa.We develop a staple method of study of such models (see ["IJ t$ 2)fa byproduct of the method of cluster expansion,too. 398 If. ZAHRADNfK Basic notions. We will follow the notation of [4] (and [ 2] ) ,if possible. Given any configuration on J^ we distinguish its correct (mo-re precisely ^-correct,for an appropriate Of ) points .We defi-ne the contours P^ and "contour hamiltonians" $ (Pt) -see £4]. Formally,we write the hamiltonian as ft . 1 / M V 1 M + I I ? cn^; (1) where A ̂  is the union of all <y*correct points and { r<*\ is the family of all <y -contours of a given configuration. Assume that § (P*) satisfy the Peierls condition f(rr) yr /supp p* / (2) for a sufficiently large T # Remind the notion of a "crystallic" partition function 2(P r,#0 .Denote further by Jl <™ d (r+) the set of all confi-gurations on C (F*)» supp Pri/int P*- which satisfy the con-dition that,being extended to the whole TL? by the constant value y ^ fno contours of this extended configuration have a distance ^ A from the set supp n r (we assume that th* ground states are constant configurations,for the simplicity). Put y* SL**d(Pr) and define the contour functional associated with #£ Ftfr^)« log *2.red(r^ir£) - log 2 (r+,H) (4) ("the work needed to install the contour /»t in place of previously cf*-correct configuration " ) . Define also the truncated contour functional F^(r^) * max ( F(r*) , ̂ T / s u p p P * / ) (5) (the choice of •§ is quite arbitrary) . Denote by ~ A y * - lia J/vr< log Z t ( A , f\) the free energy of the contour ;*odel with a contour functional F % Put Ay * A(\fr) - AT . (6) Say that a % - th ground state i s stable i f Aicf0 * **£ i A r \ . (7) Denote by CL^ the ••parameters of nonstability" defined by ALTERNATE APPROACH TO PIROGOV-SINAI THEORY 399 &r m <h<r - ^ rr (8) Note. It is shown in [ 1 J that /A^ia the free- energy of the original mo del. The parameters Ou^ are not in general (if drt 0 ) equal to the parameters /fry of [ 2 J or [ 4 J • Basic results. Theorem. For a stable <y , Far s F<y and there is a Gibbs state, which is a perturbation of the Of -th ground state such that the probability distributions of its externa/! contours are identical wi£h the distributions of external contours in the contour model, defined by the contour functional F<y Proofs It is based on the following technical result : Definition • A contour P^ is called stable if Lemma • The following estimates are true : i) nonstable contours satisfy the inequality dcy I CCr*) I* J/dupp r+ I • (9) Especially,there are no unstable contours f* ̂  if (y ia stable* ii) Zo.M/^J > **p(-A>r 1*1) ^(-ClUl) (io) and "*> ? t (AiVt) * «f-(~Ar,lAl) Mf.(Cim) (ID for some universal constant C * C CT) . Proof . See £ 1 J The proof proceeds by induction,combining i)fii),iii). The induction step for i) is an elementary consequence of ii),iii)« The relation (lo) is proved by standard methods of cluster expan-sion. The only nontrivial step is the induction step in iii). We introduce the following useful notion: Definition. A contour P r i s called small i f i t i s stable and moreover there i s no unstable r v such that aupp P^ c. int r V . It i s proven in C 1 J that 400 M. ZAHRADNfK "z-t M , * ) * I "pt-«<riBxi\) TT^(-§(r?)+d\»^r*fi x - ^ ( - A ^ M l ) AA^(cidAi) (u) with the same C as in (10,11) and &' = c' f<M , the sum being taken over all possible famillies { J1**} of external large contours in A atnd w/fh Ex4r » A \ l/cfr.t) . It is shown in § 2 of [ 1 ] that (12) implies the desired estimate iii). We omit the construction of phase diagram (see FlJ »[2 ] • [ 3 ]) , [ 4 3 )*In the last part of this note we formulate a result concerning the completeness of the phase diagram constructed by Pirogov-Sinai. There are some partial results of Martirosjan (see [ 5 1 ).The full answer is given in [ 1 ] The completeness was also proved,independently,by another methods, by D.Preiss. We formulate here a simplified version,containing,however,all the important details. Definition. Given any configuration <f 6 JLy (A) denote by o£ » <£ f<f) the system of all large contours which satisfy the condition that supp P d int r for no small contour p of y . Define a canonical configuration y * associated to (f such that ct is exactly the contour system of ^ ^ (i.e. we remove all the remaining contours). Define a stable domain of y e--f*LM) as the union of all the compo-nents of A N U st/pf>P which have a stable value In the canonical configuration <f£ .The points of a stable domain will be ca-lled stable points of a configuration \f • Proposition. Denote by Jl£ (A) the subset of -fly (A) consisting of confi-gurations with at least ty unstable points.Denote by i ^ (At^i) the corresponding partition function. Then the following estima-te holds,with the same constant as in (10,11) • ALTERNATE APPROACH TO PIROGOV - SINAI THEORY 401 ^% (A,K) £ expC-^IAl) exp (- | W) exp ( C I 3 A/) #(13) where cv = min £ d<y : <k<y £ o"} Note.Roughly speaking,if lAl » I & A \ then for a large fraction of points + € A there is a large probability that there is "stable component" C of A ^ \J supp P such that diat( i , Cc ) is large.The statement on completeness now follows using the assumed translation invariancy of a Gibbs state. See again £ 1 D f o r the details. References. 1 M. Zahradnik . An alternate version of the Pirogov -Sinai theory and the completeness of their diagram. To be published. 2 Sinai Ya.G. Theory of phase transition: rigorous results Akadlmiai Kiado, Budapest 1982 • 3 Pirogov S. A.,Sinai Ya.G Phase diagrams of classical lattice systems I , II Teor.Mat.Fiz. 2§(1975),358-369,26(1976),61-76. 4 Koteck^ R. ,Pre i88 D. An induct ive approach to Pirogov -Sinai theory t h i s volume 5 Mar t i ros ian D.G. Teor.Mat.Fiz22(1975),No.3,page 335 Mar t i ros ian D.G. Uspechi Mat. Nauk 1975, t .30 ,No.6 ,p . l81 Dept. of Mathematical Analysie,Faculty of Mathematics and Physics ,Char les Univ.,Sokolovské 83,186 00 Prague, Czechoslovakie. 

An alternate approach to the Pirogov – Sinai theory, not employing the models with a parameter

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An alternate approach to the Pirogov – Sinai theory, not employing the models with a parameter

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Institute of Mathematics AS CR, v. v. i.