Probab. Theory Relat. Fields 112, 121 ± 147 (1998)
Transformations of Gibbs measures1 JoÂzsef Lo}rinczi2 , Christian Maes3 , Koen Vande Velde4 Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium Received: 11 November 1997 / Revised version: 20 February 1998
Abstract. We study local transformations of Gibbs measures. We establish sucient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies. General results are illustrated by simple examples. Mathematics Subject Classi®cation (1991): 60K35, 60G60, 82B20, 82B05 1. Introduction Central in equilibrium statistical mechanics is a function, called interaction potential, from which relative energies can be computed. They determine the conditional probabilities distribution for a random ®eld inside a ®nite region given the ®eld outside. In general there can be more than one random ®eld for the given local conditional
1
Research partially supported by EC grant CHRX-CT93-0411 and KUL-OT 92/09 e-mail:
[email protected] 3 Onderzoeksleider F.W.O. (Flanders); e-mail:
[email protected] 4 Present address: Laboratory for Medical Imaging Research, ESAT + Radiology, UZ Gasthuisberg, Herestraat 49, B-3000 Leuven, Belgium; e-mail:
[email protected] 2
Keywords: Gibbs measures, weakly Gibbsian measures, quasilocality, relative energies, disagreement percolation
122
J. L} orinczi et al.
distributions. These random ®elds are called Gibbs measures with respect to the potential. In many problems, however, we have to investigate the probability law on transformed variables. We have in mind in particular renormalization group transformations or transformations induced by a (stochastic) dynamics on the con®guration space. Over the last decade many interesting examples have been produced in which non-Gibbsian states resulted by transforming Gibbs states (for a recent review see [11]). In the context of renormalization group theory or probabilistic cellular automata Gibbs measures depend on parameters such as the temperature, magnetic ®eld, noise level etc. The discussion of the Gibbsian or non-Gibbsian character of transformed measures therefore has to be done with respect to the parameter values. Typically it seems that at very high temperatures or suciently strong magnetic ®elds the emerging state remains Gibbsian [20, 22, 23, 36, 15, 28, 29], whereas for low temperature regions and/or small external ®elds non-Gibbsian images commonly appear [19, 20, 22, 38, 14, 32, 33, 41, 12]. There are examples warning one, however, that this separation of behaviour in the parameter space cannot be taken for granted [13, 10]. One of the points of our paper is giving sucient conditions excluding such high temperature pathologies. More recently it was suggested, and for certain cases proven [8, 35, 30, 3], that even for non-Gibbsian measures one can make sense of the notion of relative energy. A potential can be de®ned which is absolutely summable only for `typical' con®gurations. In [30] we called the resulting probability measures weakly Gibbsian measures. One purpose for which weakly Gibbsian measures might be useful is of the extending of a meaningful thermodynamic description also to those parameter regions in which the transformed measures are nonquasilocal (not almost Markovian). This paper is organized as follows. In the preparatory section following this introduction we introduce notations, ®x the classes of transformations we will consider further on, and establish some basic results on these transformations. The main section is the last one, divided into four subsections. The ®rst subsection presents general results. There, by using disagreement percolation methods, we investigate sucient conditions for the quasilocality of conditional distributions for transformed Gibbs measures. This is continued in the third subsection where we concentrate on relative energies for weakly Gibbsian states. Subsections 2 and 4 are devoted to more speci®c examples illustrating various points.
Transformations of Gibbs measures
123
2. Preliminaries Lattices and graphs. For convenience and without serious restriction of generality we will mostly consider the regular d-dimensional cubic lattice Zd . The symbol E will denote the set of ®nite subsets of the lattice, and we write Kc Zd nK, for every K 2 E. The distance between two ~ 2 E is given by dist
K; K ~ minfjx ÿ yj : x 2 K; y 2 Kg. ~ The sets K; K label @K stands for the boundary of K, i.e. the set of sites in Kc at distance 1 from K. Whenever considering in®nite volume limits K ! Zd , we take K along an increasing sequence of cubes. Often in the following we will look at Zd as being a graph. We turn Zd into a graph by adding edges between nearest neighbours sites. The graph G then consists of the pair made up by the vertex set Zd and the set of edges Ed . Two vertices x and y are adjacent, denoted x y, if they are connected by an edge. A path of length n is a sequence of distinct vertices x1 ; x2 ; . . . ; xn in which the consecutive vertices are adjacent. For two disjoint sets K1 ; K2 2 E, a path connecting K1 with K2 is a path which has a vertex in K1 and another in K2 . Con®gurations. At each site of the lattice a `spin' is placed. These spins take their values from the same ®nite set S. The (spin) con®guration d space is thus X S Z , and its projection to a ®nite volume K is XK S K . If r; r0 2 X are such that rx r0 x , for every x 2 K, then we say that the two con®gurations agree on the set K and denote this event by rK r0 K . We think of two con®gurations as being close if they agree on a suciently large K. This notion of closeness is formalized by employing the product topology on X. We further consider the measurable space (X; F) by associating X with its Borel r-®eld F. We also consider the sub-r-®elds FK of events measurable with respect to spins sitting within K. Functions. A function f : X ! R is continuous if for every e > 0 there exists a K 2 E such that jf
r ÿ f
r0 j < e whenever rK r0 K . The set C
X of continuous functions is a Banach space with respect to the norm jjf jj1 supr2X jf
rj. Continuous functions are approximated by local functions. Denote by Df the dependence set of f : X ! R, i.e. the minimal set (in the sense of inclusion) K for which f is FK measurable. The function f is local if Df 2 E. A system of independent spins is described by a product measure. We can take such a measure (e.g., products of the normalized one-site counting measure) as a reference measure on which an interaction will be superimposed.
124
J. L} orinczi et al.
Markov ®elds. A probability measure l on (X; F) is a Markov ®eld if for each K 2 E and every g 2 XK we have l-almost surely l
rK gK j rx ; x 2 Kc l
rK gK j rx ; x 2 @K
2:1
Gibbs measures. More generally than in (2.1), whenever the local conditional probabilities l
rK gK j rKc gKc have a continuous version with respect to g for g and K 2 E ®xed, then we say that l is quasilocal (or almost Markovian, see [40]). When, moreover, these conditional probabilities are strictly positive, we say that l is a Gibbs measure. Equivalent de®nitions and more details can be found in [16, 14]. Typically in classical equilibrium statistical mechanics the conditional probabilities in the left hand side of (2.1) are speci®ed in terms of a potential and one is interested in characterizing the corresponding probability measures. A potential U fUB gB2E is a family of real valued functions on X such that each UB is FB -measurable, with U; 0. Relative energies. This function gives the energy dierence obtained under a change made in the con®guration. For our purposes it suces to compare the energy for a con®guration r with the energy for a reference con®guration, say 1, 1x 1, 8x 2 Zd . Then we consider for each r 2 X the modi®ed con®guration K r 2 X, where
K rx rx , if x 62 K, and
K rx 1, if x 2 K. We say that a potential U allows a relative Hamiltonian if the relative energies (for K 2 E) X HK
r UB
r ÿ UB
K r
2:2 B\K6;
converge uniformly in r. (Here and henceforth convergence means convergence along a net of ®nite subsets of the lattice, directed by inclusion.) A sucient condition for uniform convergence is that the potential is absolutely summable, i.e. for any x 2 Zd X jjUB jj1 < 1
2:3 B3x
It is not dicult to see [40, 16] that if a potential allows a relative Hamiltonian, then there exists a Gibbs measure l such that for each K 2 E and l-almost every g 2 X HK
g ÿ log
l
rK gK j rKc gKc l
rK 1 j rKc gKc
2:4
Transformations of Gibbs measures
125
Sullivan has proved that the converse is also true [40]: Local conditional probabilities of a Gibbs measure can be expressed in terms of a potential allowing a relative Hamiltonian. Denote by S
S the permutation group for the ®nite set S, and take p 2 S
S. Also, let us denote by X UB
r ÿ UB
px r
2:5 Dpx H
r B3x
the relative energy associated with the change r 7! px r, where
px ry ry if y 6 x and
px rx p
rx . This is well de®ned if (2.2) is. With some abuse of notation we also write px f
r f
px r for the action of the permutation group transposed to functions f : X ! R. If l is a Gibbs measure for the potential fUB gB2E , then, provided the reference measure is permutation invariant, for all p 2 S
S, x 2 Zd and all f 2 C
X we have p
Dx H f l
pÿ1 x f l
e
2:6
This means that l pÿ1 x is absolutely continuous with respect to l, for every x, with a positive and continuous version of the Radon-NikodyÂm derivative given by the Boltzmann-factor exp
Dpx H . Weakly Gibbsian measures. A weaker notion of Gibbsianness can be obtained by relaxing the condition (2.2) on the relative energy. In [30] we called such measures weakly Gibbsian states. They have appeared in [3] and [35], and their study was inspired by a talk of R.L. Dobrushin in 1995, see [8, 9]. A probability measure l on
X; F is weakly Gibbsian if there is a tail event K 2 T \K2E FKc with l
K 1, and a potential fUB g which is l-almost surely summable, namely for all r 2 K X ÿ UB
r ÿ UB K r
2:7 HK
r B\K6;
converges and is ®nite so that (2.6) holds and (2.4) holds l-almost surely. It should be added that other de®nitions of weak Gibbsianness are around, but their spirit is similar. For a weakly Gibbsian state the local conditional probabilities need not be continuous; all that matters is that relative energies (2.7) can be well de®ned on a measure one subset such that (2.4) holds. In each example we know of, the con®gurations r 2 K are characterized by the fact that in (2.7) c HK
V r ! HK
r, as V ! Zd . This means that for the left hand side in (2.1) there is a version g of the conditional probabilities which, c though not continuous, satis®es limV !Zd g
V r g
r, for all r 2 K.
126
J. L} orinczi et al.
Transformations. The last ingredient we need is a class of transformations of probability measures l on
X; F. The image measure will be a probability measure on
X0 ; F0 . We take here X0 S 0 L , where d jS 0 j jSjL and L LZd for some integer L 1. (The only exception from this choice will be discussed in the example in Subsection 3.3 where the transformation will be chosen as a projection to a hyperplane L Zdÿ1 .) The transformation T is given by a probability kernel Y tx
nx jr dnx
2:8 T
dnjr x2L
where tx
jr dnx is a probability measure on S 0 for each r 2 X, x 2 L. Here dnx is the counting measure on S 0 . In this paper we consider the following two cases: (a) Stochastic parallel updating: In this case we take L Zd (or L 1), S 0 S. The updating kernels tx are bounded, positive and local, that is we assume that there exist e1 ; e2 > 0 such that e1 tx
nx jr e2
2:9
and tx
nx j is Bx -measurable with a Bx 2 E, x 2 Bx , with uniformly bounded diameter, diam Bx < r, for some r > 0. These transformations correspond to a discrete time dynamics, often called a probabilistic cellular automaton (see [27]). (b) Deterministic updating with no overlap: In this case we take the dependence region fBx gx2L as a partition of the lattice Zd , i.e. for x 2 L LZd , Bx is a cube of size Ld on Zd centred around x. For updating we take here tx
nx jr Inx Tx
r
2:10
that is, the indicator function of the event nx Tx
r, where now Tx : S Bx ! S 0 is a copy of one and the same T0 . We assume, in order to avoid trivialities, that for every a 2 S 0 there is a r 2 S Bx such that a Tx
r. An example of deterministic updating with overlap (as in cellular automata) will be presented in Subsection 3.3 below. At any rate, case (b) can always be taken as some (zero noise) limit of case (a). Since our transformations are local, all local functions are mapped into local functions, i.e. if Df W , then we have Z Y Tf
r f
n tx
nx jr dnx
2:11 X0
x2W
Transformations of Gibbs measures
127
In this paper we study some aspects of the image m lT of (Gibbs) probability measures l. The image measure m is the marginal on X0 S 0 L of the product measure l T de®ned on the product space (X X0 ; F F0 ): Z Z l
dr T
dnjrf
r; n
2:12 l T
f X
X0
for all F F0 -measurable real valued functions f . We list below some useful properties the proof of which is immediate: Lemma 2.1. The following properties hold: 1. If f
r; n g
n, then l T
f m
g. 2. If f
r; n h
rg
n, then l T
f l T
h g l
hTg. 3. If g
n G1
nG2
n with G1 and G2 local functions on X0 having disjoint dependence, i.e. DG1 \ DG2 ;, then Tg TG1 TG2 . For a 2 X0 , K 2 E0 fV L; V 2 Eg, write IQ a;K
n for the indicator function of the event nK aK . Then TIa;K x2K tx
ax jr, and if l is non-null, then l
TIa;K > 0. This enables us to de®ne laK as the projection to X of the measure l T conditional on having n a on K. That is, for h 2 C
X, laK
h :
l T
h Ia;K l
hTIa;K l T
1 Ia;K
n m
Ia;K
2:13
Obviously, l mlnK , and in particular, for all f 2 C
X, l
f l T
f 1 m
ln
f
2:14
where la l T jn a; a 2 X0 is obtained by conditioning on the second marginal of l T . Notice that laK can be written as R Q l
drh
r x2K tx
ax jr aK Q
2:15 l
h R l
dr x2K tx
ax jr Therefore it is easy to prove (see also, e.g., [14], section 2.4.8.) Lemma 2.2. Suppose l is a Gibbs measure on
X; F for the potential fUB g. Under condition (2.9), laK is also a Gibbs measure on
X; F for the potential UB
r ÿ log tx
ax jr if B Bx for some x 2 K UBaK
r
2:16 UB
r otherwise
128
J. L} orinczi et al.
For deterministic updating (case (b), (2.10)) the measure laK is equal to l conditioned on having ax Tx
r, for x 2 K, i.e., laK
h l
hjax Tx
r; x 2 K
2:17
Since the conditioning is made on disjoint blocks Bx , for every x 2 K, it is convenient to think of laK as a measure on ~ K x2K S Bx S Zd n[x2K Bx , where the Bx are thought of as being a X super-site with new single spin space S Bx , x 2 K. The conditioning ax Tx
r then determines a reference measure on each S Bx , x 2 K, which in general need not be permutation invariant. Lemma 2.3. If l is a Markov ®eld on
X; F, then laK is a Markov ®eld ~ obtained by adding edges to
Zd ; Ed between as well for the graph G d those x; y 2 Z for which x; y 2 Bz for some z 2 K. Proof. The statement is obvious once one realizes that laK -a.s. ÿ
2:18 laK
rV jrV c gV c laK rV jr@ V~ g@ V~ with @ V~
[
Bz
\
Vc
z2K Bz \V 6;
2:19 (
This lemma becomes relevant for our purposes because of the following simple fact. Lemma 2.4. For all local functions f 2 C
X0 with Df \ K ; for K 2 E0 ; a 2 X0 m
f jnK aK laK
Tf
2:20
Proof. Notice that by Lemma 2.1 we have m
fIa;K l
Tf TIa;K l T
Tf Ia;K laK
Tf m
Ia;K l
TIa;K l T
1 Ia;K
2:21
as claimed. 3. Transformations of Gibbs measures 3.1. Quasilocality of image measures The most recent reference on the concept of quasilocality is [15]. We also refer to [16, 14] for more details. The central issue in our context is to understand how m
f jnK aK depends on a 2 X0 , for arbitrary
Transformations of Gibbs measures
129
2 E0 . For this purpose we ®x a K 2 E0 and F0 K -measurable f and K; K 0 an F K -measurable function f . We take K V W 2 E0 , and write V nK K1 and W nV K2 . Furthermore, we choose a; a 2 X0 such that aK1 aK1 . Lemma 2.4 allows us to estimate how much in the state m events at the site of observation depend on regions far away (the distance being measured by the length of a path connecting K1 with K2 ). This amounts to estimating a truncated pair correlation function for which we can follow the methods in [2, 17]. For example, suppose that there is just one site y 2 K2 for which ay 6 ay . Then, as it is straightforward to see from Lemma 2.4 (with K1 [ K2 replacing K there), say under condition (2.9), m
f jnK1 [K2 aK1 [K2 ÿ m
f jnK1 [K2 aK1 [K2
laK1 [K2
Tf ; pya;a laK1 [K2
pya;a
3:1
where pya;a
ty
ay jr ty
ay jr
3:2
and l
f ; g l
fg ÿ l
f l
g denotes the truncated pair correlation function for a probability measure l and functions f ; g. The conclusion is that it is necessary for the quasilocality of m that the measure ln has good decay of correlations, uniformly in n 2 X0 . Sucient conditions can be obtained in a variety of ways. For example, [21] use that the so called ``constrained measures'' ln obtained after a majority transformation from the two dimensional Ising model at the critical point fall into the regime of complete analyticity. We need then to investigate measures which are not translation invariant. In ln , n plays the role of disorder. We use a general disagreement percolation approach which has worked well in other disordered settings (where the n correspond to independent random variables), see [17]. In what follows we take l to be a Markov ®eld on
X; F with single vertex conditional probabilities lx
jg l
rx jry gy ; y x
3:3
with gy 2 S, y x, so that l
ry gy ; y x > 0. The maximal variation of this conditional probability de®nes the parameters g qx max varlx
jg; lx
j g; g
3:4
We distinguish between two types of transformations: transformations with or without overlapping. Transformations under case (a) can allow overlapping, while case (b) is always non-overlapping.
130
J. L} orinczi et al.
Proposition 3.1. Suppose T is a transformation with non-overlapping blocks. There is a q > 0 such that if qx < q for every x 2 Zd , then m lT is quasilocal. In general q depends on the transformation T , however if d 1, we can always take q 1. Corollary 3.1. Suppose T is a transformation with non-overlapping blocks of type covered by case (a). If l is a Gibbs measure on
X; F for a nearest neighbour pair potential fUB g, i.e. UB
r bU
rx ; ry , if B fx; yg, jx ÿ yj 1, and UB
r 0 otherwise, then there is a b > 0 such that m lT is a Gibbs measure for all b < b . If d 1, then m lT is a Gibbs measure for every b (i.e., b 1). Proposition 3.2. Suppose T is a transformation with overlapping blocks belonging to case (a). Suppose, moreover, that for every x kx max vartx
jr; tx
; r < k r; r
3:5
Then for k > 0 suciently small, Proposition 3.1 and Corollary 3.1 apply unchanged. If d 1, we can take k 1. We now turn to proving the above statements. The general strategy is as follows: We prove uniform convergence of the conditional probabilities m
f jnW nK aW nK as W ! Zd . For doing that we use (3.1) and the methods of [2, 17] for estimating truncated correlation functions and the in¯uence of boundary conditions in terms of disagreement percolation. Proof of Proposition 3.2. Fix x 2 Zd and consider the set Nx [z:Bz 3x Bz nfxg. The single site conditional probability of laK1 [K2 is given by aK1 [K2
lx
jg laK1 [K2
rx jrNx gNx
3:6
for those con®gurations g for which laK1 [K2
rNx gNx > 0. Its maximal variation parameters are aK1 [K2
qx
aK1 [K2
max varlx g; g
aK
[K
aK1 [K2
jg; lx
j g
3:7
Observe now that qx 1 2 can be made small uniformly in x and aK1 [K2 aK [K if we take both q and k small. This is clear because qx 1 2 0 if qx 0 and ky 0, for all y 2 Zd for which x 2 By . So we choose q aK [K and k so small that qx 1 2 q, for a ®xed q to be determined d below. Let kq be the Bernoulli measure on f0; 1gZ with density q. ~ with vertex set Zd where edges Furthermore consider the graph G
Transformations of Gibbs measures
131
connect any two sites x; y 2 Zd for which there is a z 2 Zd with the ~ be the property that x; y 2 Bz , or for which jx ÿ yj 1. Let pc
G ~ We threshold density for Bernoulli site percolation on the graph G. ~ have then that if q < pc
G, there exist C1 < 1 and m > 0 such that kq
A , B C1 exp
ÿm dist
A; B
3:8
Here A , B denotes the event that there is an open path (i.e., a path of occupied vertices) connecting the regions A; B 2 E. This is useful for applying the methods in [2, 17]. From there it follows that dierences in conditional expectations can be estimated as , K2 jm
f jnK [K aK [K ÿ m
f jnK [K aK [K j 2jjTf jj kq
K 1
1
2
2
1
2
1
1
2
3:9 K [z2K Bz . It remains to notice that pc
G ~ > 0 for every where K ~ ( (®xed) local transformation T , and that pc
G 1 if d 1. Proof of Proposition 3.1. It is useful here to take the approach outlined above Lemma 2.3. We glue the sites y 2 Bx , x 2 L, together into a single super-site, again denoted by x. Consider thus the random ®eld
sx ; x 2 L with sx 2 S Bx and joint distribution induced by laK1 [K2 . For example, in case (b) this means that sx
ry ; y 2 Bx such that ax Tx
r, if x 2 K1 [ K2 ; the rest of the dependence is between blocks and originates from correlations already present in l. We look again at the single-supersite conditional probabilities aK1 [K2
lx
aK1 [K2
je lx
sx jsy ey ; y x
3:10
where we put y x for those x; y 2 L for which dist
Bx ; By 1. The maximal variations are now given by aK1 [K2
Qx aK
aK1 [K2
max varlx e; e
aK1 [K2
; e; lx
; e
3:11
[K
Clearly, Qx 1 2 tends to zero as q goes to zero, for y 2 Bx . The rest of the proof is then similar to that of Proposition 3.2. ( Proof of Corollary 3.1. The statement follows immediately from Proposition 3.1 by noticing that qx ! 0 as b ! 0. Moreover, it is easily seen that m lT is non-null. Therefore we conclude that m is a Gibbs measure. ( Remark. The above deals with the so called `positive' results regarding the Gibbsian nature of the image measure. The results are overlapping with earlier work dealing with transformations of Gibbs measures in a well-understood region of the phase diagram. We believe that the
132
J. L} orinczi et al.
approach given above uni®es these results, at least for transformations of Gibbs measures for short range interactions. We refer to [20, 22, 23, 37, 7, 6, 14, 34, 31, 28, 4, 15] for the earlier results.
3.2. Counterexamples In this subsection we discuss by two examples why some of the conditions on the transformations appearing in Propositions 3.1 and 3.2 are necessary, and that the conclusions of these propositions cannot be taken for granted even in the regime of the so called complete analyticity for the original Gibbs measure l. First let us note that in Proposition 3.1 the temperature range where the statement is true depends on the transformation. It is important to keep account of this since it is easy to see that for arbitrarily high temperatures one can ®nd a transformation for which the image is non-Gibbsian. This has been ®rst discussed in [10]: Consider the standard nearest neighbour Ising model on Zd with d 2. (This is obtained from (3.2) below in the limit n ! 1.) The transformation is of type (b), see (2.10), where d L if ry ÿ1 for all y 2 Bx
3:12 Tx
r P otherwise y:y2Bx ry (Permutation invariance of the reference measure is not preserved here.) Then it is shown in [10] that for every temperature b there exists a side-length L
b such that for all L > L
b the measure obtained by transforming the Gibbs measure lb of the Ising model under T TL is ~ for any non-Gibbsian. What happens here is that in the new graph G vertex the number of edges incident to the vertex is increasing with L. Therefore we have for the corresponding percolation threshold ~ ! 0 as L ! 1, and hence we must take b b
L ! 0 as pc
G L ! 1 in order to ensure that lTL is Gibbsian. Secondly, we turn to discussing another interesting example which shows that for deterministic transformations we have to require in Proposition 3.2 that no overlap between the blocks Bx occurs. The following example shows why (originally due to J. van den Berg). Let us consider independent binary spins rx , x 2 Z, distributed identically by l
rx 1 p 6 1=2, l
rx ÿ1 1 ÿ p, 0 < p < 1. De®ne the transformation nx rx rx1 ;
x2Z
3:13
Transformations of Gibbs measures
133
This is a deterministic transformation with overlapping blocks. The
nx ; x 2 Z form a one-dependent random ®eld. In particular, the induced measure on the spins on the even (resp. on the odd) sites is a product measure. Still, the measure m on the
nx ; x 2 Z is in a sense strongly non-Gibbsian. Proposition 3.3. There is no version of conditional probabilities of the measure induced under the transformation (3.13) which is quasilocal for at least one con®guration. Proof. First notice that for m 1 n0 rÿm nÿm nÿ1 n1 nm rm1
3:14
Choose ai 2 fÿ1; 1g, for i 1; 2; . . . ; m, m > 1, and denote by la1m
f the expectation of a function f in the conditioned measure la1m
l
j r1 r2 a1 ; . . . ; rm rm1 am
3:15
Take f
r r1 , and look at how its expectation changes if we make the change a1m 7! a1m where the ai ai xi remain unchanged for i 1; . . . ; M < m. An easy and explicit computation shows that for every M < 1 and xi ; i 1; . . . ; M there are > 0, 0 < m 2M 1 and a1m ; a1m (equal to xi for i 1; . . . ; M) jla1m
r1 ÿ la1m
r1 j
3:16
In other words, since (3.16) involves a ®nite conditioning, whatever version of the local conditional probabilities we take, it is always nowhere quasilocal. ( The measure m is therefore non-Gibbsian in a very strong sense. The reason why this phenomenon occurs is that the severe constraint (3.13) makes it possible that spins exert an in¯uence on arbitrarily distant spins without this eect being shielded o by others in the intermediate region. There is no decay whatsoever in the corresponding measures (2.17). If, however, the constraint is taken to be less severe, the properties of the induced measure improve. Consider therefore a modi®ed transformation on the r-spins such that
rx ; x 2 Z are mapped into a pair of variables
nx ; fx ; x 2 Z, both taking the values 1: fx ÿ1 with probability w; 0 < w < 1; independently 8 if fx ÿ1 > < rx nx > : rx rx1 if fx 1
134
J. L} orinczi et al.
One can now repeat the computation in the proof of Proposition 3.3 but one easily ®nds that the dependence of the spin value at the origin on spins far away does not extend beyond the spins in the largest interval containing the origin on which f 1. Formally (and in the sense of (2.4)±(2.7)), the measure m on the variables
n; f is of the form m
dn; df exp
ÿHf
nk1=2
dnkx
df
3:17
where kw is the product measure with density x. The Hamiltonian Hf is not behaving well uniformly in f. Hf
nx ÿ Hf
n is a function of all such ny for which fz 1 for z between x and y, and it depends strongly on all these ny . For f 6 1, there is only a ®nite number of sites in the cluster of x, thus Hf
nx ÿ Hf
n is a strictly local function of n. However, although the range of the interaction is ®nite almost surely, it is not bounded uniformly in f. The situation is better than in the previous example: although m is still non-Gibbsian, it is weakly Gibbsian. Let us conclude this section by pointing out that a stochastic variant of this model is behaving even better. This is de®ned by putting (case (a)) exp
vrx rx1 nx
3:18 2 cosh v with v > 0. Then, by following the argument cf. Proposition 3.1 we get t
nx jrx ; rx1
qnx
p 1=2tanh
2v h tanh
2v ÿ h < 1
3:19
with tanh h 2p ÿ 1, and hence the image measure is Gibbsian. For v ! 1 the argument breaks down; then we have indeed the deterministic version and the resulting measure is not Gibbsian. The example shows how variations of the same transformation can yield genuinely non-Gibbsian, weakly Gibbsian or Gibbsian images. 3.3. Relative energies for image measures We restrict attention here to the case where the fBx g of (2.9)±(2.10) are non-overlapping boxes. Consider the permutation groups S
S respectively S
S 0 . Suppose there is an element p 2 S
S 0 for which a conjugated permutation ux 2 S
S Bx can be found such that the following invariance of updating kernels holds: tx
pnx jr tx
nx juÿ1 x r Dpx H
r
3:20
be the relative energy associated with the modi®cation Let r 7! px r (see (2.5)).
Transformations of Gibbs measures
135
Proposition 3.4. Suppose l is a weakly Gibbsian measure (see (2.6)± (2.7)) having then in particular Z u
3:21 l
drg
uÿ1 x r l
g exp
Dx H Then m pÿ1 x is absolutely continuous with respect to m, and the RadonNikodyÂm derivative equals d
m pÿ1 x
n ln
exp
Dux H dm
3:22
m-almost surely. Proof. We have m
pÿ1 x f
l
T pÿ1 x f
Z
Z l
dr
T
dnjrf
pÿ1 x n
3:23
On changing variables by pÿ1 x n 7! n and r 7! ux r, we use the hypothesis and the invariance property of the kernels (3.20) to ®nd that u m
pÿ1 x f l T
exp
Dx H f
3:24
We conclude the proof by applying the de®nition (2.13) and using that ( f is F0 -measurable. From Proposition 3.4 we learn that m will be a Gibbs measure if ln
exp
Dux H can be obtained as a continuous function in n. Indeed, this is what also appeared from Section 3.1 which was built on Lemma 2.4. We further illustrate that point via the following important example. Example. One of the problems in non-equilibrium statistical mechanics is to ®nd characterizations of time invariant measures going beyond the mere implicit stationarity condition mT m, see also [32]. The transformation T (as in case (a)) refers to the transfer operator of e.g. a PCA. In [27, 18] it was investigated how to characterize this stationary m as being the projection to a hyperplane of its Markovextension lm under T . lm is then a Gibbs measure on the space-time con®gurations. The problem is to see what can be said of measures m lP , resulting as restrictions of Gibbs measures to hyperplanes, i.e. Y P
dnjr I
nx r
x;0 dnx
3:25 x2Zdÿ1
with x 2 Zdÿ1 ,
x; 0 2 Zd , and dnx being the counting measure on S. Although this transformation P is not covered by any one of the cases discussed so far, Proposition 3.4 still applies. Suppose S fÿ1; 1g,
136
J. L} orinczi et al.
and take p corresponding to the spin ¯ip operation px f
r f
rx , de®ned by
rx y ry ; y 6 x; ÿrx ; y x. Then it is true that d
m px
n l
expDx H j r
;0 n dm
3:26
where Dx H
r is the relative Hamiltonian for the Gibbs measure l under a ¯ip at site x. For PCA this Hamiltonian is directly obtained from the transformation T . In general, Dx H
r can be decomposed into a sum H Dlower H Dx H Dupper x x
3:27
H is the contribution into the relative Hamiltonian conwhere Dupper x H taining interactions between spins in the upper half of Zd , and Dlower x is similarly the contribution of spins from the lower half. In the language of probabilistic cellular automata, ``upper'' refers to the ``past'' and ``lower'' refers to the ``future'', while the conditioning (3.26) is being made on the ``present''. If l has the global Markov property with respect to the hyperplane f
x; 0; x 2 Zd g (which is the case for PCA), then d
m px
n l
expDupper H j nl
expDxlower H j n x dm
3:28
In other words, in order to prove that m is Gibbsian, we have to understand the continuity properties of the transformation T (which is trivial for local PCA), and its time reversal with respect to m (which is far from trivial). In that sense, (2.14) gives the formula m
fTg m
ln
f f for the time reversal of the PCA T with respect to the stationary measure m. It is then clear that the continuity (Fellerproperty) of the time reversal is equivalent with the condition that m is Gibbsian. This intimate relation between a stationary measure and time reversal is also discussed in [25] and [26]. A proof that the projection m of the two dimensional Ising phase l at low temperatures is weakly Gibbsian can be found in [35]. In the above language, this Ising-projection is stationary (and reversible) under the so called transfer matrix (which is however highly non-local). 3.4. One-site Kadano transformation Here we consider the plus phase of the standard Ising model. More speci®cally, for n 1 we take the cubes Kn ÿn; nd \ Zd and cond sider on X fÿ1; 1gZ the probability measure
Transformations of Gibbs measures
ln
r
1 Zn
b
137
exp
ÿbHn
rI
r 1 on Kcn
for the Hamiltonian Hn
r ÿ
X
rx ry ÿ 1 ÿ
x;y2Kn xy
X x2Kn ;y2Kcn xy
rx ÿ 1
3:29
3:30
and with partition function Zn
b. As n ! 1, the weak limit ln ! l exists and is called the plus phase of the Ising model. We study the action of the Kadano transformation with block size equal to 1. This is a stochastic transformation de®ned by the probability kernel (see type (a) transformations above) tx
nx jr
exp
pnx rx 2 cosh p
3:31
The transformation is often used in image processing analysis (see [4,14]), and was already considered by Griths and Pearce in [20]. E.g. in [14], Section 4.3.3., it is proven that for a given 0 < p < 1 there exists a b0 > 0 such that for all b > b0 (small temperatures) the Gibbs measure l is transformed into a non-Gibbsian measure m l T under T . This holds for any dimension d 2 (but not for d 1 as we have seen before). Yet, by applying the methods of [3], one can show that m is weakly Gibbsian with respect to some set K, m
K 1, and a potential absolutely summable on K (recall Section 2 around (2.7)). A crucial issue in obtaining this result is to understand why and for which n the constrained measure l;n (see below) is clustering for n 2 K. We will show below that there is a set K X, m
K 1, for which the two-point correlation function l;n
f ; g l;n
fg ÿ l;n
f l;n
g decays exponentially fast with the distance between the dependence sets Df and Dg of local functions f and g as this distance becomes larger than a speci®c length determined by n 2 K. Here l;n is the measure obtained from l by conditioning on having n 2 X as image under the Kadano transformation; it is the Gibbs measure obtained as the in®nite volume limit of X 1 nx rx I
r 1 on Kcn ln;n
r exp
ÿbHn
r p Zn
b; n; p x2K n
3:32 (see Lemma 2). If l;n
f were continuous in n for every local function f on X, then m would be Gibbsian. Now suppose that 0 2 Df and consider the dierences l;n
f ÿ l;n
f where n n on Kn . As it is
138
J. L} orinczi et al.
easy to understand (and as we learn from (3.1)), this dierence is governed by correlations l;n
f ; g. Combining this with (3.22) we see that l;n
f ; g with f exp
Dux H measures how the relative energy is in¯uenced (with respect to a spin ¯ip at the site x) on changing the con®guration far away (monitored by g). The following proposition gives information on the control of these correlations. Suppose there is a constant m > 0 such that l
r0 rx ÿ l
r0 l
rx exp
ÿ2mjxj
3:33
This is always the case when b is suciently large or suciently small (and is expected to be true whenever b 6 bc , i.e. the system is away from the critical point). Proposition 3.5. There is a set K X, m
K 1, such that for every n 2 K there is l
n < 1 for which jl;n
r0 rx ÿ l;n
r0 l;n
rx j exp
ÿmjxj
3:34
whenever jxj > l
n. Proof. We notice the following facts: (1) By the FKG inequality l;n
r0 rx ÿ l;n
r0 l;n
rx 0
3:35
(2) Also, we have m
l;n
r0 rx l
r0 rx m
l;n
r0 m
l;n
rx l
r0 l
rx
3:36
by the de®nition of l;n (see (2.14)). (3) Since both l;n
r0 and l;n
rx are non-decreasing functions, we have again by the FKG inequality that m
l;n
r0 l;n
rx m
l;n
r0 m
l;n
rx
3:37
Hence, by combining observations (1±3) and (3.33), we get m
l;n
r0 rx ÿ l;n
r0 l;n
rx l
r0 rx ÿ l
r0 l
rx exp
ÿ2mjxj
3:38
The proof is then concluded by a standard application of the Borel-Cantelli lemma. ( At this stage one may start wondering which con®gurations n actually belong to the set K. Certainly the con®guration (i.e., nx 1, 8x) is an element of K, but how far Pcan we go with characterizations in terms of the magnetization 1=jV j y2V ny (with V 2 E large) ? In order
Transformations of Gibbs measures
139
to make this more speci®c and to link it more directly to the relative energies we consider E
x; g log
m
nx gx j ny gy ; y 6 x m
nx 1 j ny gy ; y 6 x
3:39
We want to estimate dierences like E
x; g ÿ E
x; g, where we assume that g g on a large set centred around x. For simplicity we restrict the discussion to d 2. Take a sequence of volumes Dk Z2 constructed iteratively by adding one of the sites xk nearest to the origin but not yet contained in Dkÿ1 , Dk Dkÿ1 [ xk , xk 2 Z2 , with the properties that jxk j > jxkÿ1 j; x1 0, D0 ;, D1 f0g. Also, we write Dk
x Dk x. For a connected set V we denote by V the union of V with the ®nite components of its complement. Proposition 3.6. Suppose b is suciently large. Fix x 2 Z2 , l < 1 and suppose that n 2 X is chosen in such a way that for all k > l and all connected sets V , V 3 x, jV j k 1 X 1 ny jV j y2V 2
3:40
holds. Then jE
x; nDk
x ÿ E
x; nj const
b; p exp
ÿm
p diam Dk
3:41 p whenever diam Dk > l, and m
p ! 1 as p ! 1 (diam Dk k is the c diameter of the set Dk , and nDk
x Dk
x n). The estimate (3.41) implies a sort of right continuity of the relative energies. The relative energy can be exponentially approximated by `good' con®gurations ending with all plus. It should be mentioned that although the approach in the proposition above based on the empirical magnetization performs well for the relative energies, the set of `good' con®gurations speci®ed by it is not really large. From the results of [3] it can be shown, however, that the resulting measure is actually weakly Gibbsian. Proof. Translation invariance allows us to restrict to x 0 and ®rst study the object h n
n En
0; n log
m n
n
0 m n
n
3:42
140
J. L} orinczi et al.
We have that h n
n log X
Zn;n :
Kn
Zn;n
exp
ÿHn;n
r
r2fÿ1;1g
Hn;n
r : ÿb
X
rx ry ÿ 1 ÿ b
x;y2Kn xy
ÿp
X
3:43
0
Zn; n
nx
rx ÿ 1 ÿ p
x2Kn
X x2Kn ;y2Kcn xy
X
3:44
rx ÿ 1
nx
3:45
x2Kn c
For a shorthand we put nk Dk
0 n (this is not to be confused with the value of n at a site). We de®ne for k 1 Unk
n : h n
nk ÿ hn
nkÿ1
3:46
More explicitly, for all k > 1 0
Unk
n log
Zn;nk
bZn; nkÿ1
b 0
Zn; nk
bZn;nkÿ1
b
3:47
Note that this expression is zero whenever n0 1 or nxk 1, and that the ratio of partition functions can be rewritten in terms of correlation functions to get Unk
n
1 ÿ n0 1 ÿ nxk l;nk
exp
2pr0 exp
2prxk log ;n n k 2 2
exp
2prxk ln k
exp
2pr0 l;n n
3:48
is the Gibbs state for Hn;n . where l;n n Now, by monotonicity (FKG inequality), limn Unk
n Uk
n exists for all n 2 X. Moreover, as we will show, for p suciently large, we have uniformly as n ! 1 and uniformly in b (large enough) that jUnk
nj C
b; p exp
ÿm
p diam Dk
3:49
for n and k satisfying the hypotheses of the proposition. From (3.49) we obtain that 2
2n1 X
k1
jUnk
nj c
n < 1
3:50
and since 2n1 X k0
Unk
n
1 X k0
Unk
n
3:51
(the extra terms on the right-hand side being identically zero), we can use dominated convergence to get
Transformations of Gibbs measures
lim n
2n1 X k0
Unk
n
1 X k0
141
Uk
n lim h n
n h
n n
3:52
We can now conclude that jE
0; nk ÿ E
0; nj jh
nk ÿ h
nj X 1 Ui
n ik1
1 X
jUi
nj
ik1
const
b; p exp
ÿm
p diam Dk as claimed.
3:53 (
For the rest it remains to prove (3.49). This is a consequence of the following Proposition 3.7. Take fx
r exp
2br
x. If p is large enough, then there exists m m
p > 0 (m
p ! 1 as p ! 1), such that if n satis®es (3.40) for x 0 and l diam Dk 2n 1, then ;n ;n jl;n n
f0 fxk ÿ ln
f0 ln
fxk j const
b; p exp
ÿm
p diam Dk
3:54 For proving Proposition 3.7 we use the strategy outlined in [5] and we need the following Lemma 3.1. Under the same conditions as before, there exists u u
p
u ! 1 as p ! 1, such that for all r l ln;n ln;n 0 , Kcr const
p exp
ÿu
pr
3:55
Here the notation A , B corresponds to the event of having a path of disagreement connecting the regions A and B. By a path of disagreement between A and B we mean a sequence x1 2 A, x2 ; . . . ; xn 2 B of consecutive nearest neighbour sites for which
rxj ; r0 xj 6
; , for all j 1; . . . ; n. Proof. Following [5] we call a ®nite set A Z2 enclosing if A and Ac are connected sets. Recall that for a connected set B we denote B for the union of B with the ®nite components of its complement. B is always enclosing. There is a one-to-one correspondence between enclosing sets and contours as they are usually de®ned for the Ising model (see e.g. [39]). Each contour borders an enclosing set, i.e. for each contour c there is an enclosing set A such that A Int c (the
142
J. L} orinczi et al.
interior of the contour). We use the notation fx 2 K : 9y 2 Kc ; x yg for the inner boundary of K.
@i K
Denote by C
r; r0 the maximal connected set containing 0 on which
r; r0 6
; . Denote by K
0 all enclosing sets containing the origin. Let C 2 K
0. 0 ln;n l;n n
C
r; r C X ln;n l;n n
r ÿon R; r on @C;
3:56
R@i C 0
r ÿon @i CnR; r0 on @C X ln;n
r ÿon R; r on @C
R@i C
ln;n
r0 ÿon @i CnR; r0 on @C We now need the following lemma that we will prove afterwards. Lemma 3.2. Let C 2 K
0 and R @i C. If n 2 X l and @i C contains at least r l sites, then there exists d d
p
d ! 1 as p ! 1, such that ;n 0 0 l;n n
r ÿon R; r on @C ln
r ÿon @i CnR; r on @C
const
p exp
ÿd
pj@i Cj
3:57
If we require that there is a path from 0 j@i C
r; r0 j > r. By Lemma 3.2 we have that for
to Kcr , n 2 X l
then indeed
;n 0 j@i Cj l;n const
p exp
ÿd
pj@i Cj n ln
C
r; r C 2
const
p exp
ÿm
pj@i Cj
3:58
and m
p ! 1 when p ! 1. We can now proceed as follows. Let Ws denote the number of enclosing sets bordered by a contour of length s and containing 0. By the isoperimetric inequality on the square lattice we have that Ws
s2 =164s . Hence c c ;n ;n ;n 0 l;n n ln 0 , Kr ln ln C
r; r \ Kr 6 ; 1 X X ;n 0 l;n n ln C
r; r C sr
C2K
0 j@i Cjs
const
p
1 X
exp
ÿm
ps
sr
const
p exp
ÿum
p
s2 s 4 16
3:59
Transformations of Gibbs measures
143
for some u > 0.
(
Proof of Lemma 3.2. For any enclosing set V Z2 we put X X X
rx ry ÿ 1 ÿ b
rx ÿ 1 ÿ p nx
rx ÿ 1 H~V ;n
r : ÿb x2V ;y2V c xy
x;y2V xy
H~V ÿ;n
r : ÿb
x2V
X X X
rx ry ÿ 1 ÿ b
ÿrx ÿ 1 ÿ p nx
rx 1 x;y2V xy
Z~;n
V :
X
x2V ;y2V c xy
x2V
exp
ÿH~V ;n
r
rx ;x2V
Z~ÿ;n
V :
X
exp
ÿH~V ÿ;n
r
3:60
rx ;x2V
and observe that ! X Z~ÿ;n
V exp
2bj@V j exp 2p nx Z~;n
V
3:61
x2V
We have that ;n 0 0 l;n n
r ÿ on R; r on @C ln
r ÿ on @i CnR; r on @C
ln;n
r ÿ on R j r on @C ln;n
r0 ÿ on @i CnR j r0 on @C 0 lC;n
r ÿ on R l;n C
r ÿ on @i CnR Y X Y wn
c wn
c0 c fcg;fc0 g inside C fcg compatible with fc0 g compatible with @i CnR
3:62
c0
The set fcg of contours is compatible with R if it is compatible with r ÿ on R, R [c Int c and R \ Int c 6 ; for each c. The weight wn
c of a contour c is given by ! X Z~ÿ;n
Intcn@i Intc nx wn
c : exp ÿ2bjcj ÿ 2p
3:63 Z~;n
Intcn@i Intc x2Intc
By using (3.61) we see that wn
c exp ÿ2p
X
! nx
3:64
x2@i Int c
Now regard the sets fcg; fc0 g in (3.62) as parts of a single set of contours f~cg fcg [ fc0 g. The contours ~c can overlap, but a site of @i C can be inside of at most two contours. Moreover, every site of @i C is in the interior of at least one contour of f~cg and j@i Cj > r.
144
J. L} orinczi et al.
Let V : [c~@i Int ~c. V is a connected set such that V 3 0 and jV j > r. Since n 2 X l , r l, X X 1X nx j@i Int ~cj
3:65 4 c~ ~c x2@ Int ~c i
This takes into account the worst situation where all ÿ spins are counted twice and all spins only once. Hence in (3.62) Y 1 Y Y 1 Y 0 wn
c wn
c0 eÿ2pj@i Int cj eÿ2pj@i Int c j
3:66 c
c
c0
c0
Now we can proceed as in Lemma 2.5 of [5]. Enumerate the set R as fx1 ; . . . ; xq g. Each xi , i 1; . . . ; q, is enclosed by a contour of the set fcg. The set fcg contains at most q contours. Suppose fcg has s contours; we denote them by ci1 ; . . . ; cis , where xij is the point of R with 1; . . . ; s, and the smallest index, enclosed by cij . Put Lij jcij j for j P Lk 0 if k 2 f1; . . . ; qg but k 6 Li1 ; . . . ; Lis . Let L c jcj. Then L1 ; . . . ; Lq gives a partition of L into q nonnegative numbers. It is well known that the number of such partitions is smaller than K4L for some constant K. On the other hand, the number of contours crossing a ®xed point x and having length l is bounded by l4l 8l . Therefore Y
X
c inside C fcg compatible with fc0 g compatible with @i CnR
wn
c
X
1
K32r eÿ2pr
rjRj
wn
c0
c0
fcg;fc0 g
Y
X
1
K32s eÿ2ps
sj@i CnRj
const
p exp
ÿd
pj@i Cj with a suitable d
p > 0.
3:67 (
We can now turn to prove Proposition 3.7. Proof of Proposition 3.7. For any con®guration n 2 X we have that ;n ;n jl;n n
f0 fxk ÿ ln
f0 ln
fxk j ;n;g ;n ;n jl;n n
ln
fxk j r gon@Dk lDk
f0 ÿ ln
f0 j ;n;g ;n e2b l;n n
jlDk
f0 ÿ ln
f0 j ;n;g e2b l;n l;n n
jlDk n
f0 1 ÿ 1 f0 j
3:68
The rest of the argument is similar to the ideas in [1, 2]. Pick a con l;n ®guration
r; r0 from the distribution l;n;g n . If there is no path Dk
Transformations of Gibbs measures
145
from 0 to Dck on which
r; r0 6
; , then there exists a -chain around 0 separating it from Dck and on which r r0 . It follows that there exists a maximal -chain D with this property inside Dk (maximal in the sense that it is contained in no other -chain). Therefore ln;n
f0 1 ÿ 1 f0 l;n;g Dk X ;n;g ;n;g lD l;n l;n n
f0 1 ÿ 1 f0 j DlDk n
D k
3:69
ÿchains D around 0
c ;n;g l;n ln;n 0 , Dck l;n;g Dk n
f0 1 ÿ 1 f0 j 0 , Dk lDk
Now, since r r0 on D ;n;g lD l;n n
f0 1 ÿ 1 f0 j D 0 k
3:70
for every D, we obtain ;n;g 2b ;n;g jlD l;n ln;n 0 , Dck n
f0 1 ÿ 1 f0 j 2e lDk k
3:71
Lemma 3.1 and (3.68), (3.71) then yield ;nk ;g k k k k k jl;n
f0 fxk ÿ l;n
f0 l;n
fxk j e2b l;n jlD l;n
f0 1 ÿ 1 f0 j n n n n n k k k l;n 0 , Kck 2e4b l;n n n
const
b; p exp
ÿm
p diam Dk
3:72
( References [1] van den Berg, J.: A uniqueness condition for Gibbs measures with application to the two-dimensional Ising ferromagnet, Commun. Math. Phys. 152, 161±166 (1993) [2] van den Berg, J., Maes, C.: Disagreement percolation in the study of Markov ®elds, Ann. Prob. 22, 749±763 (1994) [3] Bricmont, J., Kupiainen, A., Lefevere, R.: Renormalization group pathologies and the de®nition of Gibbs states, UC Louvain, preprint (1997) [4] Bruce, A.D., Pryce, J.M.: Statistical mechanics of image restoration, J. Phys. A 28, 511±532 (1995) [5] Burton R.M., Steif, J.E.: Quite weak Bernoulli with exponential rate and percolation for random ®elds, Stoch. Process. Appl. 58, 35 (1995) [6] Cammarota, C.: The large block spin interaction, Nuovo Cim. B 96, 1±16 (1986) [7] Cassandro, M., Olivieri, E.: Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem, Commun. Math. Phys. 80, 255±269 (1981)
146
J. L} orinczi et al.
[8] Dobrushin, R.L.: A Gibbsian representation for non-Gibbsian ®elds. Lecture given at the workshop `Probability and Physics', September 1995, Renkum (the Netherlands) [9] Dobrushin, R.L., Shlosman, S.B.: Gibbsian description of `non-Gibbsian' ®elds, Russian Math. Surveys 52, 285±297 (1997) [10] van Enter, A.C.D.: Ill-de®ned block-spin transformations at arbitrarily high temperatures, J. Stat. Phys. 83, 761±765 (1996) [11] van Enter, A.C.D.: On the possible failure of the Gibbs property for measures on lattice spin systems, Markov Proc. Rel. Fields 2, 209±225 (1996) [12] van Enter, A.C.D., LoÈrinczi, J.: Robustness of the non-Gibbsian property: some examples, J. Phys. A 29, 2465±2473 (1996) [13] van Enter, A.C.D., FernaÂndez, R., KoteckyÂ, R.: Pathological behaviour of renormalization-group maps at high ®elds and above the transition temperature, J. Stat. Phys. 79, 969±992 (1995) [14] van Enter, A.C.D., FernaÂndez, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization transformations: scope and limitations of Gibbsian theory, J. Stat. Phys. 72, 879±1167 (1993) [15] FernaÂndez, R., P®ster, Ch.-E.: Global speci®cations and non-quasilocality of projections of Gibbs measures, Ann. Prob. 25, 1284±1315 (1997) [16] Georgii, H.O.: Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin, New York (1988) [17] Gielis, G., Maes, C.: The uniqueness regime of Gibbs ®elds with unbounded disorder, J. Stat. Phys. 81, 829±835 (1995) [18] Goldstein, S., Kuik, R., Lebowitz, J.L., Maes, C.: From PCA's to equilibrium systems and back, Commun. Math. Phys. 125, 71±79 (1989) [19] Griths, R.B., Pearce, P.A.: Position-space renormalization transformations: some proofs and some problems, Phys. Rev. Lett. 41, 917±920 (1978) [20] Griths, R.B., Pearce, P.A.: Mathematical properties of position-space renormalization-group transformations, J. Stat. Phys. 20, 499±545 (1979) [21] Haller, K., Kennedy, T.: Absence of renormalization group pathologies near the critical temperature ± Two examples, J. Stat. Phys. 85, 607±638 (1996) [22] Israel, R.B.: Banach algebras and Kadano transformations, in: Random Fields, Proceedings, Esztergom 1979, J. Fritz, J.L. Lebowitz, D. SzaÂsz, eds., North Holland, Amsterdam, vol. 2., pp. 593±608 (1981) [23] Kashapov, I.A.: Justi®cation of the renormalization-group method, Theor. Math. Phys. 42, 184±186 (1980) [24] Kozlov, O.K.: Gibbs description of a system of random variables, Probl. Inform. Transmission 10, 258±265 (1974) [25] KuÈnsch, H.: Time reversal and stationary Gibbs measures, Stoch. Proc. Appl. 17, 159±166 (1984) [26] KuÈnsch, H.: Non-reversible stationary measures for in®nite interacting particle systems, Z. Wahrsch. verw. Geb. 66, 407±424 (1984) [27] Lebowitz, J.L., Maes, C., Speer, E.R.: Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59, 117±170 (1990) [28] LoÈrinczi, J.: On Limits of the Gibbsian Formalism in Thermodynamics, PhD Thesis, University of Groningen (1995) [29] LoÈrinczi, J.: Quasilocality of projected Gibbs measures through analyticity techniques, Helv. Phys. Acta 68, 605±626 (1995) [30] LoÈrinczi, J., Maes, C.: Weakly Gibbsian measures for lattice spin systems, J. Stat. Phys. 89, 561±579 (1997)
Transformations of Gibbs measures
147
[31] LoÈrinczi, J., Vande Velde, K.: A note on the projection of Gibbs measures, J. Stat. Phys. 77, 881±887 (1994) [32] Maes, C., Vande Velde, K.: The (non-) Gibbsian nature of states invariant under stochastic transformations, Physica A 206, 587±603 (1994) [33] Maes, C., Vande Velde, K.: The fuzzy Potts model, J. Phys. A 28, 4261±4271 (1995) [34] Maes, C., Vande Velde, K.: The interaction potential of a stationary measure of a high-noise spin¯ip process, J. Math. Phys. 34, 3030±3031 (1993) [35] Maes, C., Vande Velde, K.: Relative energies for non-Gibbsian states, Commun. Math. Phys. 189, 277±286 (1997) [36] Martinelli, F., Olivieri, E.: Some remarks on pathologies of renormalizationgroup transformations for the Ising model, J. Stat. Phys. 72, 1169±1177 (1993) [37] O'Brien, G.L.: Scaling transformations for f0; 1g-valued sequences, Z. Wahrsch. verw. Geb. 53, 35±49 (1980) [38] Schonmann, R.H.: Projections of Gibbs measures may be non-Gibbsian, Commun. Math. Phys. 124, 1±7 (1989) [39] Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results, Oxford Pergamon Press (1982) [40] Sullivan, W.G.: Potentials for almost Markovian random ®elds, Commun. Math. Phys. 33, 61±74 (1973) [41] Vande Velde, K.: On the Question of Quasilocality in Large Systems of Locally Interacting Components, PhD thesis, K.U. Leuven (1995)