The Restriction of the Ising Model to a Layer
Abstract
We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the twodimensional low temperature Ising phases for which we prove a variational principle.
Keywords : nonGibbsian states, variational principle, projections, weakly Gibbsian measures.
1 Introduction
In this paper we study the restriction of the twodimensional Ising model to a (onedimensional) layer. The restriction of the plus (or minus) phase is known to be nonGibbsian below the critical temperature, see [35], [10]. Following suggestions of Dobrushin it was recently shown that this restriction is in fact weakly Gibbsian, see [5], [6], [34], [30]. We state and discuss various recent results on these restrictions. Using elementary methods, we rederive results on weak Gibbsianness below the critical temperature and the results on Gibbsianness above the critical temperature or in a magnetic field. We also add the result that further random decimations of the weakly Gibbsian restriction are again Gibbsian. Finally, we prove the existence of thermodynamic functions (energy and free energy density) and we discuss a variational principle for the weakly Gibbsian measure.
The study of restrictions of Gibbs measures (on dimensional configurations) to a sublayer (of dimension ) can be motivated in various ways. First of all they are interesting test cases for an extended Gibbsian description of nonGibbsian states. Since about ten years various examples of nonGibbsian states have been produced. Some of these go back to the work of Griffiths, Pearce and Israel, [16], [19], and have become (in)famous as so called renormalization group pathologies, [8]. Dobrushin’s program tries to understand the nonGibbsianness as coming from a perhaps too strict requirement on the potential. If, like is the case for unbounded spins, one asks for a potential which is summable on what are typical configurations for the state, one can get at least some effective interaction or physically relevant parametrization of e.g. the images under transformations of states.
A second motivation comes from the theory of interacting particle systems. One of the questions is to see under what conditions the invariant measures of a dynamics are Gibbsian. The simplest scenario is found for so called probabilistic cellular automata (PCA). These are stochastic dynamics for lattice spin systems under which the spins are updated synchronously in discrete time. If one starts a PCA (with positive transition probabilities) from an invariant measure, then the distribution of spacetime configurations (configurations on the spacetime lattice) turns out to be a Gibbs measure see [24]. Therefore, the invariant measure itself is a restriction of a dimensional Gibbs measure to a dimensional hypersurface.
In fact, the question in [35] about the Ising restrictions came quite naturally after it was found in [24] that for high noise dynamics the unique stationary state is Gibbsian as follows from considering the restriction of a Gibbs state in the regime of complete analyticity. The question about the Gibbsian nature of stationary states has of course been considered before, see e.g. [22], [21], [32], [33] and references therein. Understanding the locality of the time reversal with respect to the stationay state plays a crucial role in these. While most of these problems are still open (for general PCA’s), we feel that the Dobrushin program gives new inspiration towards a (weakly) Gibbsian description of these invariant measures if considered as restrictions of Gibbs measures.
A third motivation can be found in the study of surfaces, models on halfplanes with random boundary conditions (wetting phenomena). The restriction of the two dimensional Ising model to a layer can of course be seen as a surface state with respect to the two dimensional Ising measure. The problem of finding an interaction for this restriction consists in finding the interaction between spins at the boundary of an Ising sample (as a function of the configuration at this boundary). The relation with wetting is of a more technical nature. It turns out that in the study of the restriction of Ising measures, one is quickly confronted with questions like how far the influence of a configuration on a sublayer is felt in the bulk of the system. The wettingcontext can be used to get a very useful intuitive picture of why, at low temperatures, the restrictions of the plus phase of the Ising model are not Gibbsian. This was made precise in [8], see also [10]. An interesting further question (related to the convergence properties of the potential for the restriction) is to see whether there is good decay of correlations close to the surface when on this surface we impose a ’typical surfaceconfiguration’ (i.e. a sample of the restriction). We will answer this question in Section 4 (Proposition 4.1).
The paper is organized as follows: in Section 2 we introduce basic notations and definitions. We introduce the ’telescoping’ potential (à la Kozlov, [20]) in Section 3 and discuss its summability properties. In Section 4 we give an overview of the results on the restrictions of the Ising model. In Section 5 we prove the results of Section 4 using the telescoping potential, and finally in Section 6 we discuss the variational principle.
2 Definitions and Notations
2.1 Configuration space
We consider the regular dimensional lattice and denote by the set of finite subsets of . The complement of a set is . For two sites we define
(2.1) 
The state space is and its elements (= configurations) are denoted by greek letters . The value of at a site is written as . For , we define
(2.2) 
For we define to be the configuration
(2.3) 
The restriction of to a volume is denoted by and we write for the restriction of a configuration .
On we have the natural action of translations , defined by . The algebra generated by the evaluation maps is written as . When , we set . The tail field algebra is defined as
(2.4) 
The configuration space is a compact metric space in the product topology. A function on is called local if it depends only on a finite number of coordinates, i.e. there is a such that whenever . The minimal set such that this holds is called the dependence set of the function.
Definition 2.1
A function is called rightcontinuous in if
(2.5) 
On we have the pointwise order if . A function is called monotone nondecreasing if for all , implies .
2.2 Potentials and specifications
Definition 2.2 (cf [10])
A local specification on is a family of probability kernels on , such that the following hold:

is a probability measure on for all ;

is measurable for all ;

if ;

if .
Definition 2.3
A probability measure is consistent with a specification (or vice versa), notation , if
(2.6) 
A specification is said to be translation invariant if , , and for all bounded measurable functions
(2.7) 
It is customary to slightly abuse the notation (and to circumvent property 3. above) by writing if one means to take configurations and identical on . One should think of , as the probability to find in given outside of .
Property 4 of Definition (2.2) is called selfconsistency and is most important in characterising equilibrium. It suggests constructing probability measures as weak limits of some (perhaps along a subsequence). Such weak limits automatically exist by compactness but their consistency with the specification is only immediate if is a continuous function for all continuous . One then deals with a socalled Feller (or quasilocal) specification. This is not the context of the Dobrushin program where more general specifications have to be considered and hence that is not obvious in general.
In a Gibbsian formalism one considers a special class of specifications, the so called Gibbsian specifications which are of the BoltzmannGibbs form:
(2.8) 
where is a normalization factor
(2.9) 
and is an “interaction potential”:
Definition 2.4
A potential is a realvalued function on
(2.10) 
such that for all (put ).
A potential is translation invariant if , ,
(2.11) 
Definition 2.5

A potential is convergent in if for all
(2.12) is welldefined. We always understand an infinite sum as welldefined when so that ,

A potential is absolutely convergent in if for all
(2.13) 
A potential is uniformly absolutely convergent if for all
(2.14)
Let be a potential and suppose that there exists a set in the tail field of points of convergence of ( and and the sum is welldefined). Then, for every and every we can introduce the finite volume Gibbs measure
(2.15) 
( We ask for to be in the tail field to make sure that is welldefined.) Factors of temperature or a priori weights (reference measure) are supposed to be contained in the potential. The Dobrushin operator is then defined by taking expectations with respect to (2.15):
(2.16) 
mapping bounded measurable functions on to functions on .
Definition 2.6
A probability measure on is weakly Gibbsian if there exists a potential and a tail field set of points of absolute convergence of (cf. (2.13)) such that

;

, and for every bounded measurable function ,
(2.17)
A somewhat less stringent definition of weak Gibbsianness is obtained by asking that there is a tail field set of points of convergence of such that 1. and 2. of Definition 2.6 hold.
If the potential in Definition (2.6) is uniformly absolutely convergent, then is a (bona fide) Gibbs measure.
3 Vacuum and Telescoping Potential.
In this section we shall introduce the socalled telescoping potential which will be a very useful tool in the study of the restrictions of the Ising model. A natural potential associated to a specification is the socalled vacuum potential (in our case the vacuum will always be the configuration of all plusses). To construct this potential, start from
(3.1) 
and write
(3.2) 
This last formula can be inverted (Møbius formula) and we get:
(3.3) 
The inversion is rapidly checked by remembering that where is the Kronecker delta. This potential is called vacuum because it has the property that whenever for some and . This follows easily from (3.3) using that if and . It is straightforward to check that the vacuum potential is the unique potential having this property and that it is translation invariant if is. In [30] it is proved that the vacuum potential is convergent and consistent with the specification , i.e.
(3.4) 
if and only if the specification is rightcontinuous.
A possible problem with this vacuum potential is that it may not be absolutely convergent (even when the specification is rightcontinuous). Therefore, in order to obtain absolute convergence Kozlov, in [20], introduces another kind of potential. We now present a simplified version of it which, for the occasion, we would like to call a telescoping potential. For this we turn again to (3.1) and we write
(3.5) 
where we have lexicographically ordered the sites in according to and
(3.6) 
We can now order the sites according to their distance from the ‘largest’ site . For this purpose we consider for every the sequence of increasing volumes with . We thus have the partition
(3.7) 
with and . Correspondingly, can be further telescoped as
(3.8) 
with
(3.9) 
for and
(3.10) 
(Observe that when is the ‘first’ site in ). We thus define the (telescoping) potential
(3.11) 
and otherwise.
To get more insight in the potential it is instructive to rewrite it for the onedimensional case. For the potential is nonvanishing iff is an interval. For such a we rewrite (3.9) as
(3.12) 
where we abbreviated e.g. for in .
Some properties of this potential are immediate. For example, whenever or when on the set . As a consequence and following (3.5)(3.10), the Hamiltonian (3.1) is telescoped as
(3.13) 
Moreover, the potential is explicitly translationinvariant if the specification is.
From now on we will assume that the specification is rightcontinuous. In the sequel this specification will always be the monotone rightcontinuous specification (introduced in [10]) consistent with the restriction of the Ising model.
In order to verify the consistency of this telescoping potential with the rightcontinuous specification, we can rewrite as a resummation of the vacuum potential (see also [20]). More precisely
(3.14) 
for , and for we have .
From this, one can check that
(3.15) 
if for every , is absolutely convergent in (see [20]). If there exists a set in the tail field such that is absolutely convergent in every point of then (3.15) together with the rightcontinuity of and the consistency of with (see 3.4) give that
(3.16) 
The representation (3.14) of the telescoping potential in terms of the vacuum potential is very useful, because one has a certain freedom in the choice of the sets . The only constraint on these sets is that the obtained potential (from (3.14)) is still consistent with the specification, i.e.
(3.17) 
Indeed, when the constraint (3.17) is satisfied we have
(3.18) 
and from this, together with the rightcontinuity of the specification, we conclude that the potential is consistent with the specification. The constraint (3.17) on the sets is of a geometric nature. In dimension we can choose e.g. , where is some strictly increasing function. The freedom in the choice of permits to ’tune’ a bit the convergence properties of the potential (for our study of the restrictions it will be okay to choose ). In the sets introduced before also satisfy this constraint, whereas e.g. do not satisfy the constraint.
In applications, the rightcontinuous specification is often constructed starting from a probability measure such that (cf. Definition 2.3). We then know that is weakly Gibbsian (cf. Definition 2.6) if there exists a tail field set of points of absolute convergence of such that . I.e. proving that is weakly Gibbsian boils down to showing that
(3.19) 
for a fullmeasure (tail)set of ’s.
Proposition 3.1
Let be a local rightcontinuous specification, and the telescoping potential defined by (3.9) and (3.10). Suppose that and such that ,
(3.20) 
with possibly unbounded.
Suppose further that a translation invariant tail field set so that , and
(3.21) 
Then the telescoping potential is absolutely convergent for and is weakly Gibbsian.
Of course, the left hand side of (3.20) is a local function for fixed while the right hand side can be highly nonlocal as a function of and deals with the dependence of the potential on as grows.
Remark 1 : Conditions (3.20) and (3.21) may seem weird or ad hoc. In Section 5 we show that they are satisfied for the onedimensional restriction of the plusphase of the twodimensional Ising model. In fact if is the restriction to a hyperplane of a measure then measures the correlations between spins at sites and in a measure where is a constrained measure obtained from and plays the role of a boundary condition. (This will become more clear in Section 4).
One should think of the as the radius of a
ball around site outside which the spins at sites are
only
weakly correlated to the spin at site in the constrained measure
of (4.7).
To satisfy (3.21) on a
set of full measure
for it suffices that for some
.
Remark 2 : For , it is convenient to use the left/right symmetry of the sets which are just lattice intervals . We then ask (3.20) (which looks to the left) together with the existence of finite for which
(3.22) 
(looking to the right). The assumption (3.21) in Proposition 3.1 can be replaced by the requirement that for each , there are finite so that for all , (3.20) and (3.22) hold. The idea is that looks in the configuration to the left of while looks to the right.
Proof of Proposition 3.1 : We have to check (3.19). Inserting (3.20) there are two sums to control. The sum for is easily taken care of using the exponential decay. The sum over has only a contribution if . We thus get that (3.19) is bounded by
(3.23) 
Using assumption (3.21) this is finite and the conclusion follows from the remarks above.
4 The restricted Ising model
4.1 The model
We consider the standard ferromagnetic nearest neighbor Ising model on the regular  dimensional lattice . The symbols will be reserved to indicate finite subsets of . Their complement is etc. The configuration space for the Ising model is Fix . For a finite box with free (or empty) boundary conditions, the Gibbs state for the Ising model assigns a probability
(4.1) 
to an Ising spin configuration . The normalization is the partition function (for free boundary conditions). The parameter is (proportional to) the inverse temperature and is called the magnetic field. The first sum in (4.1) is over the nearest neighbor pairs in . Each site has nearest neighbors and we write if the site is a nearest neighbor of . The infinite volume Ising state is obtained in the thermodynamic limit as along a sequence of sufficiently regular volumes. We can take the (weak) limit of and we write for this limit. In the same way, starting with and letting we can define . We refer to all of these as Ising states. When making no distinction between them, we denote these states by the common symbol and the corresponding random field is denoted . They are translation invariant probability measures on and they all satisfy the DobrushinLanfordRuelle equation
(4.2) 
almost surely. For and for sufficiently large there are other solutions to (4.2) (even nontranslation invariant ones if ) but we will restrict us in what follows to the Ising states introduced above. In particular, there is a critical value for which whenever . The standard methods, results and more details about the Ising model can be found in almost any textbook on statistical mechanics, see e.g. [37], [11] or [36].
We now fix a hyperplane or layer
(4.3) 
which we can identify with . The sites in are denoted by , which, though treated as elements of , ought to be identified with , when considering . Finite subsets of are written as and by we mean the complement of in . On we have a new configuration space with elements and we write . Of course, every gives rise to a unique via and much of the structure and notation of the spin system on is inherited quite straighforwardly from that on . For example, given , we put for and . We also write for the configuration which is equal to on and is equal to on when given .
This paper is about the restriction of the Ising states and to this layer . In other words, with the random field corresponding to the considered  dimensional Ising state, we want to study the  dimensional random field with
(4.4) 
Obviously, the distribution of is the one induced from that of . Writing and (or, in general) for this induced law from, respectively, the and we have for example
(4.5) 
for the expectation of any function which is  measurable (depends only on the ). In particular, for we have (the spontaneous magnetization). Similarly, the truncated correlations (or covariances) within the layer
(4.6) 
between any two functions and depending on a finite number of
coordinates in , decay exponentially fast in the distance between the
dependence sets of and , whenever this is the case in the
considered  dimensional Ising state (which is verified
away from
the critical point ).
The problem can therefore
not be
to evaluate the expectation value of specific observables in our restricted
state because this can be done starting from the wellknown Ising states.
Rather, we are interested in some global characterizations of
the restricted states . More specifically, we wish to
understand the as Gibbs measures for some interaction.
In the study of the convergence properties of this interaction potential it will turn out to be useful to know whether for the original Ising measure there is good decay of correlations close to the surface when on the surface we impose a typical configuration drawn from . This is the context of the following proposition. For the Ising measure on , define
(4.7) 
This is defined for almost every surface configuration .
One has the following result
Proposition 4.1
Suppose that there are constants so that for all . Then there is a set with such that for all there is a length for which
(4.8) 
whenever .
Proof : The crucial step is to observe that (by definition) . Moreover, by the FKGinequality (positive correlations), since is a bounded measurable function nondecreasing in . The constrained measure is itself an FKGmeasure so that . The conclusion (4.8) now follows from standard BorelCantelli arguments.
4.2 Results on Gibbsian characterizations
As announced in Section 1, we restrict ourselves to results concerning Gibbsian descriptions of restrictions to a layer of Ising states. We first give a summary of results describing the state of the art before Dobrushin’s 1995 talk, [5]. We then present the results of the Dobrushin program for these restrictions of the Ising model.
The beginning of the study of Ising restrictions was
Theorem 4.1 (Schonmann, [35])
Take . The projection of the twodimensional plus phase is nonGibbsian : there is no translation invariant absolutely and uniformly convergent potential for .
In the following, a further decimation of this was considered. This measure is obtained as the restriction of the twodimensional to or, alternatively, as the restriction of to the decimated integers .
Theorem 4.2 (Lőrinczi, Vande Velde, [29])
For sufficiently large , for all is a (bonafide) Gibbs measure.
Decimation of nonGibbsian measures can thus be (bona fide) Gibbsian measures (and the opposite is also true). We can extend this result to random decimations. In other words, we assign a Bernoulli variable to each site . The are independent and identically distributed with density . We consider the restriction of (or ) to the (random) set of occupied sites.
Theorem 4.3
There is so that for sufficiently large , for all is a (bonafide) Gibbs measure for almost all .
One can ask what happens when the temperature is large or when the magnetic field is nonzero. While it is rather easy to show that for sufficiently small or for sufficiently large , the Ising restrictions are Gibbsian, it is less trivial to show the following
Theorem 4.4 (Lőrinczi, [25])
For , the Ising restriction is Gibbsian whenever .
Theorem 4.1 was given a more intuitive proof in [8]. In fact, something more was obtained (so called absence of quasilocality).
Theorem 4.5 (van Enter, Fernandez, Sokal, [8])
Take sufficiently large. For any specification with is not a (bona fide) Gibbs measure.
This was extended to any dimension by
Theorem 4.6 (Fernandez, Pfister, [10])
Take sufficiently large. For any specification with , is not a (bona fide) Gibbs measure.
On the positive side and as we already mentioned following property 4. in Definition 2.2, from [10] it also follows there exists (for all ) an everywhere rightcontinuous local specification so that . We observed via (3.4) (see also [30] that this implies that the corresponding vacuum potential is always convergent (on )). As we have shown again around (3.4), this implies that (including ) is weakly Gibbsian for the (everywhere) convergent vacuum potential. The question about the absolute convergence of the vacuum potential (on a set of measure one) was also solved:
Theorem 4.7 (Dobrushin, Shlosman, [6])
For and sufficiently large, is weakly Gibbsian for the absolutely convergent vacuum potential.
For the telescoping potential we have the following
Theorem 4.8 (Maes, Vande Velde, [34])
For and sufficiently large, is weakly Gibbsian for the absolutely convergent telescoping potential.
The next Section will start with a more detailed presentation (and proof) of this last Theorem.
5 Proofs
We use here the telescoping potential constructed in Section 3 to prove the results of Section 4. The main thing to show is the exponential decay of this potential for large sets which will follow from the fact that it can be expressed as a correlation function in a two dimensional Ising model on a halfplane with a ”typical” surface configuration. The specification used in this section will always be the monotone rightcontinuous specification consistent with the restriction of the plus phase of the two dimensional Ising model to the line . The telescoping potential introduced in Section 3 is here for
(5.1) 
Using that is a specification consistent with the restriction of the plus phase of the twodimensional Ising model, we have
(5.2)  
where is the constrained measure of (4.7). More specifically, consider the events where is an increasing sequence of squares centered around the origin. For a continuous function on ,
(5.3) 
This, of course, is a function of the only, which act as extra boundary conditions. The limit (5.3) is over the finite volume Ising measures with plus boundary conditions outside the square and boundary conditions in the middle of the square (on , cutting the square in two equal parts). For , we have
(5.4) 
The extra term for and in (5.2) and (5.4) comes from the interaction inside the layer ) and corresponds to the onedimensional Ising model. One can check that, uniformly in .
Let be defined via
(5.5)  
(This notation suggests of course that coincides with the tail field set of points of absolute convergence of the potential introduced in Section 2. We show that this is indeed the case.)
Clearly, is a translation invariant set in the tail field. It is easy to see that whenever is sufficiently large. In fact, under this condition, for , the are exponential random variables. In the following proposition we show that the potential satisfies the bound (3.20) of section 3. For the sake of completeness we repeat the proof of [34]
Proposition 5.1
Proof: Looking at (5.2), we see that the crux of the matter consists in proving that uniformly in the size of the boxes
(5.8) 
whenever . This was done in [34].
We repeat the two main steps. They were inspired by the proof of some ergodic properties of the plus phase in [4]. The first step reformulates the required estimate in terms of a percolation event. Denote by the event that there is a path of consecutive nearest neighbor sites , , connecting with on which . Here and are two independent copies of the random field with law . Then,
(5.9) 
For the second step, we use that there is a finite constant so that
(5.10) 
for all sufficiently large , uniformly in the size (see [4]). The argument is now completed by noticing that
(5.11) 
The conclusion is that
(5.12) 
If , only one spin out of eight can be minus in for and hence the right hand side of (5.12) is then smaller than .
Remark 1: Notice that no use was made of a cluster expansion in the proof above. In fact, a naive application of this cluster expansion is quite impossible as it would yield too much; the attempt in [31] failed for that reason. This is similar to the analysis of Gibbs fields for a random interaction in the Griffiths’ regime, see e.g. [2], [7], [13]. We must only concentrate on a specific covariance and percolation techniques seem to be rather powerful in such cases.
Remark 2: An important ingredient in the previous proof is in the step (5.11). Therefore it seems that the proof is necessarily restricted to one dimension. This however is not the case. We can prove the decay of the covariance (as in (5.8)) also in higher dimensions. This we will deal with in a future publication.
So far we dealt with Theorem 4.8. We now prove the other theorems of Section 4. For the regular decimation we can only prove Theorem 4.2 for whereas Lőrinczi et al. included also .
a)The case or (Theorem 4.4).
In this case we don’t need the steps above. The covariance (the left hand side of (5.12)) is exponentially small uniformly in the boundary condition. That is,