2 Notations, Denitions. 2. Conguration space. We consider the regular 2?dimensional lattice 2 and denote by L := f; jj < g the set of nite non-empty s

Transcription

1 Potentials for One-Dimensional Restrictions of Gibbs Measures Christian MAES ;4, Frank REDIG 2;4, and Annelies VAN MOFFAERT 3;4 () Onderzoeksleider FWO, Flanders (2) Post-doctoraal onderzoeker FWO, Flanders (3) Aspirant FWO, Flanders (4) Instituut voor Theoretische Fysica, K.U.Leuven Celestijnenlaan 200D, B-300 Leuven, Belgium Abstract: We discuss restrictions of two-dimensional translation-invariant Gibbs measures to a one-dimensional layer. We prove that there exists a translation invariant a.s. absolutely convergent potential making these restrictions into weakly Gibbsian measures. We discuss the existence of the thermodynamic functions for this potential and the variational principle for the weakly Gibbsian measures. Keywords: non-gibbsian states, variational principle, restrictions, weakly Gibbsian measures. Introduction In this paper we study the restriction of a certain class of translation invariant Gibbs measures to a one-dimensional sublayer. These measures are in general non-gibbsian (see [8] for the Ising model and [6] for more general cases). Following suggestions of Dobrushin ([3]) we show that the Gibbsian character of these restricted measures can be restored on a set of measure one. The potential which we associate to the restricted measures is the so-called Kozlov ([0]) or telescoping ([5]) potential, which is directly constructed from the conditional probabilities. Using a percolation estimate we show that the conditional probabilities have a right-continuous version and that the potential is absolutely convergent on a set of measure one. Using some additional properties of the potential (exponential tails) we prove existence of energy density for a set of \decent" boundary conditions. For the particular case of the restriction of the plus phase of the two-dimensional Ising model, we obtain a variational principle in a restricted class of \tempered measures", as in the case of unbounded spin systems (see []). An important motivation to study restricted Gibbs measures comes from probabilistic cellular automata, or their continuous time analogues, interacting particle systems. It is well-known that, under very general conditions, the space-time measure of a PCA is a Gibbs measure, and thus the stationary measure is the restriction of a Gibbs measure to a layer (see [2]). Another question is the following: if one starts a PCA (or a spin system) from a Gibbs measure, is the measure at time t > 0 still a Gibbs measure, or a weakly Gibbsian measure? The measure at time t > 0 is the restriction to a layer of a Gibbs measure in a strip. Whereas one expects that such time-evolved measures are non-gibbsian as soon as the system exhibits some (hidden or not hidden) phase-transition (see [5]), one also expects that they are generically weakly Gibbsian. The paper is organized as follows: Section contains basic notation; in Section 2 we state our basic assumptions and show that the Kozlov potential for the restricted measures is a.s. convergent. In Section 3 we discuss the existence of the energy density and the variational principle for the restricion of the Ising model, and we indicate possible generalizations.

2 2 Notations, Denitions. 2. Conguration space. We consider the regular 2?dimensional lattice 2 and denote by L := f; jj < g the set of nite non-empty subsets of 2. Elements of L are denoted by n ; ; M; N. The complement of a set 2 is c = 2 n. For two sites x; y 2 2 we dene jx? yj := 2 i= jx i? y i j: (2.) The state space is := f+;?g 2 and its elements (= congurations) are denoted by ; ; : : :. 2 is written as (x). For M 2 L, 2 we dene M (x) if x (x) = 2 M (2.2) + if x 62 M: The value of at a site x 2 For ; 2 we dene M M c to be the conguration (x) x 2 M M M c(x) = (x) x 62 M: The restriction of to a volume 2 is denoted by := f+;?g and we write 2 for the restriction of a conguration. On we have the natural action of translations y, y 2 2 dened by y (x) := (x?y); x 2 2. We can also let a translation work on a function f and on a probability measure on : y f := f y and y (f) := ( y f). A probability measure is called translation invariant if y = for all y 2 2. The -algebra generated by the evaluation maps (x); (x)(!) :=!(x); x 2 is written as F = f(x); x 2 g. When = 2, we set F := F 2. The tail eld -algebra F is dened as F := \ 2LF c: (2.4) The conguration space is a compact metric space in the product topology. A function f on is called local if it depends only on a nite number of coordinates, i.e. there is a M 2 L such that f() = f() whenever M = M. The minimal set M such that this holds is called the dependence set of the function and is denoted by D f. A continuous function is a uniform limit of local functions. Denition 2. A function f :! IR is called right-continuous in 2 if 2.2 Restrictions to a one-dimensional layer. (2.3) f() = lim " 2f( ): (2.5) We consider the one-dimensional layer := f(i; 0) : i 2 g 2. The restriction of a conguration 2 to will be identied with a conguration := f+;?g via 0 (i) = (i; 0). Denote by the mapping :! 0 :! 0. Finite subsets of will be denoted by V n ; V; A; B and are identied with subsets of. L 0 will denote the set of nite subsets of, -elds on 0 are denoted by F 0 A, A 2 L0 etc. The -elds F 0 A will also be identied (when necessary) with f(i; 0) : i 2 Ag. When is a probability measure on (; F), we can consider its restriction := which is a probability measure on ( 0 ; F 0 ). To avoid confusion we always use the symbol for a probability measure on (; F) and the symbol for a probability measure on ( 0 ; F 0 ). For a probability measure on (; F), f :! IR a local function, we put, for 2 0, (f) := IE [fjf 0 ](): (2.6) This object is of course only dened on a set of measure one. Finally we also abbreviate ( j) := IE [ = jf c](). 2

3 2.3 Potentials. Denition 2.2 A potential is a real-valued function on L such that (M; ) 2 F M for all M 2 L. A potential is translation invariant if 8M 2 L, a 2 : L! IR (2.7) 2, 2 (M; ) = (M + a; a ): (2.8) To avoid confusion we will denote potentials on L by and potentials on L 0 0 by U. To be useful, a potential has to obey certain summability properties. Denition 2.3. A potential is absolutely convergent at 2 if for all 2 L j(m; )j < : (2.9) M\6=; 2. A potential U is uniformly absolutely convergent if for all 2 L sup 2 j(m; )j < : (2.0) M\6=; Denition 2.4 A potential is of nite range if there exists a nite constant R such that (M; :) 0 whenever diam(m) > R, where diam(m) := sup x;y2m jx? yj. Let U be a potential on L 0 0 and suppose that there exists a tail eld set 0 U 0 of points of convergence of U (i.e., for all! 2 0 U and for all V 2 P L0 the sum A\V 6=; U(A;!) is well-dened). Then, for every V 2 L 0 and every! 2 0 U we can introduce the nite volume Gibbs measure!;u V () = where Z V (:) is a normalization factor if = V! V c Z V (!) expf? P A\V 6=; U(A; V! V c)g 0 otherwise, Z V (!) = V 2V expf? A\V 6=; (2.) U(A; V! V c)g: (2.2) ( We ask for 0 U to be in the tail eld to make sure that Z V (!) is well-dened.) Factors of temperature or a priori weights (reference measure) are supposed to be contained in the potential. The Dobrushin operator is then dened by taking expectations with respect to (2.): R U V (f)(!) := Z f()!;u V (d) (2.3) mapping bounded measurable functions f on 0 to functions R U V (f) on 0 U. 2.4 Weakly Gibbsian measures. Denition 2.5 A probability measure on ( 0 ; F 0 ) is weakly Gibbsian if there exists a potential U (on L 0 0 ) and a tail eld set 0 U of points of absolute convergence of U (cf. (2.9)) such that. ( 0 U ) = ; 2. for all V 2 L 0, for all B 2 F 0 V c and for every bounded measurable function f : 0! IR, ZB f d = Z B RU V (f)d: (2.4) 3

4 3 Kozlov potential for restrictions of Gibbs measures. We start from a one-parameter family of translation invariant Gibbs measures f ; > 0 g on. The parameter plays the role of inverse temperature and 0 is typically large. The potential of will be denoted by and is supposed to be of nite range, uniformly in. We denote by + [n;m] the measure conditioned on the event f 2 : (i; 0) = + 8i 2 [n; m] \ g. In what follows we abbreviate [i; j] \ by [i; j]. In order to formulate our basic assumptions concerning f ; > 0 g we need some more notation. We introduce a non-oriented graph G with vertex set 2 where an edge is put between two vertices x and y if there exists a set 2 such that x; y 2 and (; :) 6= 0 for some > 0. A path from x to y in G is a set of vertices fx ; : : : ; x n g with x = x, x n = y and (x i ; x i+ ) an edge for i = ; : : : ; n?. For ; 2 2 we dene the event E as E( ; 2 ) := f(; 0 ) 2 : 9 path from to 2 along vertices x 2 2 such that ((x); 0 (x)) 6= (+; +)g: (3.5) Finally, for ; dene d( ; 2 ) = inf x2 y2 2 jx? yj. 3. Basic assumptions ) Percolation estimate: There exists a constant > 0 such that uniformly in n; m 2 + [n;m] 2) Conditions on the potential : + [n;m] (E( ; 2 )) exp(?d( ; 2 )) (3.6). P M30 k (M; :)k for some 0 <. 2. is uniformly of nite range (see Denition 2.4) 3. There exists a non-decreasing function m : [; )! [0; ] such that lim! m () = and for every > 0, > 0 : n n i=0 with lim! c(; ) = for every > 0. (i; 0) m ()?! exp(?c(; )n) (3.7) Remark: The percolation estimate () is natural if a typical conguration under + [n;m] + [n;m] contains \a sea of plusses ((+; +))" and isolated islands (with exponentially small diameter) of other spin values. This holds typically for low temperature phases (small deformations of \all (+; +)") in the realm of Pirogov-Sinai theory. Assumption () implies the following estimate Proposition 3. Let f; g :! IR be local functions. Then we have Proof: see [5]. + [n;m] (f; g) 2kfk kgk exp([?d(d f ; D g )]): (3.8) 4

5 3.2 Right-continuity Suppose that f ; > 0 g is a one parameter family of translation-invariant Gibbs measures such that the basic assumptions of Section 3. are satised. Introduce K := f 2 0 : 8i 2 9 l i () < such that 8n l i () : n n j=0 (? (i + j)) < 8 and n n j=0 (? (i? j)) < g: (3.9) 8 This is a set in the tail eld and by the third point in assumption (2) and by a Borel-Cantelli argument (K) = for large enough. In the following lemma we denote by e the vector (; 0) 2 2. Lemma K 8f; g :! IR local, 8n l i (), 8i 2 where C depends on f and g only. and 8 large enough [i;i+n] [f; ne g] C exp(? n); (3.20) 2 Proof: Dene kfk := sup 2 jf()j, and let D f ; D g be the dependence set of f, resp. g. [i;i+n] [f; ne g] [i;i+n] 0 2kfk kgk C sup C exp C exp [i;i+n] d d [i;i+n] + [i;i+n] 2 n 4 2:2? (i + j)) j=0( 2 3 n 4 4? (i + j)) j=0(? E(Df ; D ne g) 2kfk kgk [i;i+n] + [i;i+n] 3(i+j;0) () A exp(?n) k (; :)k 3 5 exp[?n] 5 exp [?n]: (3.2) Where C 0 exp((diam(d f ) + diam(d g ))) (3.22) appears since d(d f ; D ne g) n? diam(d f )? diam(d g ). Looking now at the denition of the set K in (3.9) concludes the proof. Proposition 3.2 For large enough there exists a tail eld set K ~ with ( K) ~ = such that for all f :! IR, the mapping has a version which is right-continuous at every 2 ~ K. f; : 0! IR :! (f); (3.23) Proof: First remark that the set K in (3.9) has the property that 2 K implies that for all n 2 IN, [?n;n]+ [?n;n] c 2 K. Fix a local function f on. Consider j (fj [?n;n]+ [?n?l;n+l]n[?n;n])? (fj [?n;n])j l i= j (fj [?n;n]+ [?n?i;n+i]n[?n;n])? (fj [?n;n] + [?n?i+;n+i?]n[?n;n] j 5

6 C l i= [?n;n]+ [?n?i+;n+i?]n[?n;n] (f; (n+i)e g) C 0 exp(?n=2) (3.24) where in the second inequality (3.25) g() = ( + (?; 0))( + (; 0)) (3.26) 4 and in the third inequality we have used Lemma 3.. Since (3.24) holds uniformly in l, we can take the limit l! to obtain j [?n;n]+ [?n;n] c (f)? (fj [?n;n])j C 0 exp(?n=2): (3.27) Let K 0 0 be the set of those 2 0 such that lim n" (:j [?n;n]) = (:). Then (K 0 ) = and K 0 is a translation-invariant tail eld set. Taking the limit n! in (3.27), we get for 2 K \ K 0 : lim n! j [?n;n]+ [?n;n] c (f)? (f)j = 0: (3.28) which proves the desired right-continuity for 2 K ~ := K \ K 0. Remark that the object (f) is of course only dened on a set of -measure one, and at rst sight it may be unclear whether [?n;n]+ [?n;n] c is in that set. However, in the lines of (3.24) one easily veries that for 2 K, the sequence f (fj [?n;n]+ [?n?l;n+l]n[?n;n]) : l 2 INg is a Cauchy sequence. Identifying [?n;n]+ [?n;n] c with the limit of that sequence yields a version of the conditional probability. 3.3 Weakly Gibbsian character of restrictions. We start from f : > 0 g, a one parameter family of translation-invariant Gibbs measures on satisfying the assumptions of section (2.). f : > 0 g are their restrictions to, considered as probability measures on 0. Kozlov in [0] introduces a translation-invariant potential associated to a translation invariant specication, and shows that under certain conditions on the continuity of the specication this potential is absolutely convergent and \consistent" with the specication. Since we do not have a priori a nice specication consistent with the restricted measures, we shall work directly with the a.s. right-continuous version of the conditional probabilities (cf. proposition 3.2). In terms of the conditional probabilities, the Kozlov potential U (A; ) is non-vanishing on intervals A = [i; j] only and is then given by U ([i; j]; ) = log (+ i (i;j] j+) (+ j [i;j) j+) (+ j (i;j) + i j+) ( [i;j] j+) : (3.29) Since is the restriction of to, we can rewrite the potential U ([i; j]; :) for jj? ij > R in the following form (see [5]): where U ([i; j]; ) = (? i)(? j ) 4 i() := exp( 3(i;0) log [i;j]+ [i;j] c ( i j) [i;j]+ [i;j] c ( i) [i;j]+ [i;j] c ( j) ; (3.30) [ (; )? (; + f(i;0)g f(i;0)gc)]); (3.3) which is a local function since is a nite range potential. For jj? ij R we have the bound ku ([i; j]; :)k C () uniformly in i; j 2. An explicit expression for U ([i; j]; ) can also be written down for jj? ij R but for our purposes this is not needed (see [5] for the Ising model). 6

7 Theorem 3. There exists 0 < < such that for > the Kozlov potential U is absolutely convergent on a translation-invariant tail eld set 0 U, and is weakly Gibbsian for this potential. Proof: By Lemma 3. and the representation (3.30) of U, we have the following estimate for 2 K and j > l i () U ([i; j]; ) C exp(?jj? ij=2): (3.32) By assumption 2c we have for > 0 : (f : l i () ng) exp(?cn): (3.33) Therefore, we conclude that there exists a translation-invariant tail eld set K 00 such that 2 K 00 implies that for all i 2 N i () := jfj i : l j () jj? ijgj < : (3.34) Finally use that there exists C < such that U ([i; j]; ) C to arrive at the following estimate for 2 K \ K 00 : [i;j]30 N 0() ju ([i; j]; )j C + i0 i=0 j>0 l i () C exp(?jj? ij=2) < : (3.35) This proves absolute convergence on 0 U := K00 \ ~ K. The fact that is weakly Gibbsian (i.e. that the consistency relation 2.4 holds) follows from the right-continuity of! (:j) for 2 ~ K (cf. proposition 3.2), and the fact that U is a resummation of the vacuum potential (see [5]). 4 Variational Principle. The usefulness of the concept of weakly Gibbsian measures depends of course on how much of the concepts and results of classical equilibrium statistical mechanics have \a weak counterpart". Therefore we would like to characterize the set of weakly Gibbsian measures by means of a variational principle, i.e. as the set of measures that minimize the free energy. Unfortunately we cannot derive the full variational principle in the general setup of the previous section, but we do have existence of thermodynamic functions in this context, and thus \one direction" of the variational principle. For the \other direction" (i.e., characterising the minimizers of the free energy as the weakly Gibbsian measures) we have to restrict ourselves to the Ising model. Therefore, we focus in this section on one particular family of measures namely the low temperature plus phases f ; > 0 g of the two-dimensional Ising model with zero magnetic eld. They are dened via the potentials :?(x)(y) if A = fx; yg and jx? yj = (M; ) = 0 otherwise (4.) and is the weak limit of the nite volume measures, +; (cf. (2.)) along an increasing sequence of cubes, with plus boundary conditions. In this section we derive a variational principle for the restrictions to a layer of these Ising measures, and we indicate where generalisation is possible. For all proofs we refer to [5]. In this entire section denotes the restriction of to and U is the corresponding Kozlov potential (cf. Section 3). 7

8 4. Existence of thermodynamic functions. Before we can even think of a variational principle we need to prove the existence of the thermodynamic functions that appear in it, namely the energy density and the pressure (or free energy density). 4.. Energy density. For 2 0 U we dene f U() := A30 jaj U (A; ): (4.2) Given a probability measure on 0 such that ( 0 U ) = and f U 2 L (), we dene e U := We introduce the interaction energy in a nite volume V free boundary conditions: Z f U ()d(): (4.3) with H f V () := AV U (A; ) (4.4) boundary condition!: H! V () := A\V 6=; U (A; V! V c): (4.5) In [5] we proved that there exist classes of \good" boundary conditions,! and \good" measures such that the expectation of the energy density in the measure, i.e. lim V " jv j V (H V! ), exists and equals e U independent of!. A measure will be \good" for our purposes when it is translation invariant and in a certain sense \close" to + (the Dirac measure concentrating on the reference conguration of the Kozlov potential). Analogously a boundary condition! will be \good" if it is \close" to the all plus conguration (see (4.7) and (4.9) below). We use T 0 to denote the set of translation invariant probability measures on 0. The symbols C; K; c; will always be constants whose values can vary from place to place. We still need the following denitions:. For i 2, 2 0 dene (cf. (3.9)) l i () := minfk 2 IN : 8n k and n n? j=0 n? (? (i + j)) n 8 j=0 (? (i? j)) g: (4.6) 8 2. The set of \good" measures. Let M denote the set of probability measures 2 T 0 for which there exists a constant c > 0 such that for every n 2 IN Notice that if U satises the bound (3.32) [l +=? 0 (:) > n] e?cn : (4.7) ju ([i; j]; )j C I[l i () > j? i] + C 2 e?(j?i)=2 (4.8) then e U of (4.3) is wel-dened for every 2 M. 8

9 3. The set of \good" boundary conditions. Notice that 2 M ) ( 0 ) = and 0 0 U. 0 := f! > 0; 9N 2 IN : 8i 2 with jij N; l +=? i (!) jij 3+ g: (4.9) Proposition 4. (Energy density for free boundary conditions). Let 2 M then, for large enough e U = lim V " jv j (Hf V ): (4.0) Proposition 4.2 (Energy density for xed boundary condition). Let 2 M and! 2 0. Then, for large enough e U = lim V " jv j (H! V ): (4.) The proof of these propositions does not use any specic property of the Ising model. In fact, it relies entirely on the inequalities (4.7), (4.8) and (4.9). The result is therefore not restricted to the restriction of the Ising model but it is valid for every Kozlov potential satisfying (4.8) for some constants C and C 2, i.e. for the more general restrictions f : > 0 g which we considered in the previous section Pressure. Let the partition function in a nite volume V with free boundary conditions be dened by In [5] we proved that Proposition 4.3 The limit Z f V := 2 0 V exp "? # U (A; ) AV : (4.2) P (U ) := lim V " jv j logzf V (4.3) exists and is called the pressure (or free energy density) for the potential U. Proposition 4.4 Let P (U ) be as in Proposition 4.3 and dene Z! V := V exp 2 4? A\V 6=; 3 U (A; V! V c) 5 : (4.4) Then for every! 2 0 P (U ) = lim V " jv j logz! V : (4.5) The proof of Proposition 4.3 essentially uses that Z f V = (+ (+ V 0j+ V c0))?. The limit in (4.5) can therefore be calculated from the cluster expansion. Therefore, Proposition 4.3 also holds in the general context of the previous section in the region of where the cluster expansion applied to converges. The same holds for Proposition

10 4.2 Entropy Given two probability measures and on ( 0 ; F 0 ) and a nite volume V 2 L, the entropy S V () and the relative entropy S V (j) are given by S V () :=? V () log V () (4.6) and 2 0 V S V (j) :=? 2 0 V V () log V () V () (4.7) if is absolutely continuous with respect to and S V (j) := + otherwise (we take 0log0 = 0). Note that S V (j) is always positive. For every nite volume V, every! 2 0 U, every 2 T 0, the following holds S V (j!;u V ) =?S V () + (H! V ) + logz! V 0 (4.8) where!;u V is the nite volume measure corresponding to the potential U (cf. (2.), or equivalently jv j log Z! V jv j S V () + jv j (H! V ): (4.9) Analogous to (4.6) and (4.7) the entropy density and the relative entropy density are dened by s() := lim V " s(j) := lim V " if the latter limit exists. (The limit (4.20) exists for every 2 T 0 (cf. [7]).) 4.3 First part of the variational principle jv j S V () (4.20) jv j S V (j) (4.2) Theorem 4. For every measure 2 M and every boundary condition! 2 0 we have. The relative entropy density exists and is independent of the boundary condition. 2. Let P (U ) be as in Proposition 4.3. Then s(ju ) := lim V " jv j S V ( V j!;u V ) (4.22) P (U ) = sup 2M hs()? e U 3. The supremum in (4.23) is reached for =. i : (4.23) Claims and 2 follow immediately from (4.8) and Propositions The third point follows when is an element of M and the Kozlov potential satises (4.8). Summarizing, we can conclude that the rst part of the variational principle is valid for every family of weakly Gibbsian restrictions f ; > 0 g of Gibbs measures f : > 0 g and corresponding Kozlov-potentials fu g satisfying the basic assumptions of Section 3. 0

11 4.4 Second part of the variational principle The second part of the variational principle characterizes the maximizers of (4.23) as the \Gibbs" measures for the potential U. Note that any maximizer of (4.23) satises s(ju ) = 0. To conclude 2 G(U ) from s(ju ) = 0, we need an extra technical condition: Theorem 4.2 Suppose that 2 M such that and then 2 G(U ). s(ju ) := lim V " jv j S V (j!;u V ) = 0 (4.24)! n lim n! exp[2 (? (i))] 2 n exp[?n] = 0 (4.25) i=?n In [5] it is shown that satises the hypotheses of Theorem 4.2. Throughout the proof of Theorem 4.2 we used specic properties of the Ising model. In particular, condition (4.25) is of a technical nature and is not optimal. It comes from a comparison between and a Bernoulli measure. The fact that satises (4.25) relies on the contour representation of the two-dimensional Ising model. Up to now we do not know how to extend this second part of the variational principle to the more general restrictions considered in the previous section. References [] J. van den Berg, C. Maes, Disagreement percolation in the study of markov random elds, Ann. Prob. 22, (994). [2] J. Bricmont, A. Kupiainen, R. Lefevere: Renormalization group pathologies and the denition of Gibbs states, Comm. Math. Phys. 94, (998). [3] R.L. Dobrushin, A Gibbsian representation for non-gibbsian elds, talk presented in Renkum, September 995. [4] R.L. Dobrushin and S.B. Shlosman, Gibbsian description of `non-gibbsian' elds, Russian Math. Surveys 52, (997). [5] A.C.D. van Enter, R. Fernandez, A.D. Sokal, Regularity properties and pathologies of positionspace renormalization transformations: scope and limitations of Gibbsian theory, J. Stat. Phys. 72, 879 (993) [6] R. Fernandez, C. E. Pster, Global specications and nonquasilocality of projections of Gibbs measures, Ann. Prob. 3, (997). [7] H.-O. Georgii, Gibbs measures and phase transitions, de Gruyter, Berlin New York (988). [8] Griths, R.B. and Pearce, P.A., Position-space renormalization transformations: some proofs and some problems, Phys. Rev. Lett. 4, (978). [9] R. B. Griths, P.A. Pearce, Mathematical properties of position-space renormalization-group transformations, J. Stat. Phys. 20, (979). [0] O. K. Kozlov, Gibbs description of a system of random variables, Problems Inform. Transmission 0, (974).

12 [] J.L. Lebowitz, E. Presutti, Statistical mechanics of unbounded spins, Comm. Math. Phys. 50, (976). [2] J.L. Lebowitz, C. Maes, E.R. Speer, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59, 7-70 (990). [3] J. L}orinczi, C. Maes and K. Vande Velde, Transformations of Gibbs measures, Prob. Th. Rel. Fields 2, 2-47 (998). [4] C. Maes, F. Redig, A. Van Moaert, Almost Gibbsian versus weakly Gibbsian measures, preprint, to appear in Stoch. Proc. Appl., (998). [5] C. Maes, F.Redig, A Van Moaert, The restriction of the Ising model to a layer preprint KU Leuven, (998). [6] C. Maes, K. Vande Velde: The interaction potential of the stationary measure of a high-noise spinip process, J. Math. Phys. 34, (993). [7] C. Maes, K. Vande Velde, Relative energies for non-gibbsian states, Comm. Math. Phys. 89, (997). [8] R.H. Schonmann, Projections of Gibbs measures may be non-gibbsian, Commun. Math. Phys. 24, (989). [9] Ya. G. Sinai, Theory of Phase Transitions : Rigorous Results, Oxford Pergamon Press,