The Schonmann projection as a g-measure-, how Gibbsian is it?
aa r X i v : . [ m a t h - ph ] F e b The Schonmann pro jectionas a g -measure–how Gibbsian is it? Aernout van Enter ‡ and Senya Shlosman ♮,♯,♭ ‡ Bernoulli Institute of Mathematics,Computer Science and Artificial Intelligence,University of Groningen, Groningen, the Netherlands; ♮ Skolkovo Institute of Science and Technology, Moscow, Russia; ♯ Aix Marseille Univ, Universite de Toulon,CNRS, CPT, Marseille, France; ♭ Inst. of the Information Transmission Problems,RAS, Moscow, [email protected], [email protected]@skoltech.ru, [email protected] 23, 2021
Dedicated to the memory of our dear friend and colleague Dima Ioffe.
Abstract
We study the one-dimensional projection of the extremal Gibbsmeasures of two-dimensional Ising model, the ”Schonmann projec-tion”. These measures are known to be non-Gibbsian, since theirconditional probabilities as a function of the two-sided boundary con-ditions are not continuous. We prove that they are g -measures, whichmeans that their conditional probabilities have a continuous depen-dence on one-sided boundary conditions. Introduction
In 1989 Roberto Schonmann published the paper: Projections of Gibbs mea-sures may be non-Gibbsian, [Sch]. In this paper he was considering theIsing model on Z , and he showed that the projection of the ( − )-phase onthe axis Z ⊂ Z is ‘too long-range’ to be called a Gibbs distribution, ac-cording to the usual definitions [Geo], as it violates a continuity propertyin the boundary conditions (quasilocality). Such continuity conditions havebeen long known to be equivalent to the property of being a Gibbs measure[Koz, Sul, BGMMT].On the other hand, in the paper [DS], the situation was mended: thedefinition of the Gibbs state was properly generalized there, and from thismore general point of view the Schonmann projection can still be seen as aGibbs field, although in a weaker sense.Different derivations, generalisations and clarifications of these resultswere produced later, see e.g. [BKL, EFS, FerPf, MMR1, MMR2, MavdV].The peculiar property of this 2D →
1D projection is the following. Con-sider a square box V M of size 2 M with ( − ) boundary condition, where addi-tionally we put the (+) boundary condition on two segments: I ′ = [( − N, , ( − n, , and I ′′ = [( n, , ( N, , with 1 ≪ n ≪ N ≪ M. The corresponding ground state configuration ¯ σ equals − I ′ , I ′′ . So ¯ σ has twocontours, γ ′ and γ ′′ , of unit thickness, which surround the segments I ′ , I ′′ , andin particular ¯ σ (0 , = − . However, at any positive temperature T the typicalconfiguration may look quite different! More precisely: for any n ≫ T one can find N ( n, T ) such that for any N > N ( n, T )the typical configuration σ looks as follows – it contains just one contour γ, surrounding both I ′ and I ′′ , and having (+)-phase inside. In particular, theprobability of the event σ (0 , = +1 is quite large, and in fact is close to 1.The underlying mechanism is due to an entropic repulsion in the directionperpendicular to the intervals, which causes a wetting phenomenon to occur.In this note we want to study the following question: can one similarlyinfluence the behavior of a typical configuration σ at the origin by condi-tioning the ( − )-phase to take the value +1 on segments I ′ , I ′′ , ... which areallowed to be placed only to the left of the origin? A negative answer tothis question would mean that the Schonmann projection is a g-measure, i.e.that the distribution is continuous with respect to the one-sided conditioning.2e will argue that this is indeed the case. Namely, let the (+)-segments I ′ , I ′′ , ... ⊂ Z − be listed from right to left, and let γ ′ , γ ′′ , ... be the shortest pos-sible contours surrounding them (i.e. ground state configuration contours).Let T be some low temperature, and let γ , γ ... be exterior contours of atypical configuration σ, which surround the union of the various segments I ′ ∪ I ′′ ∪ ... . Note that their number might be smaller than the number ofthe segments themselves, as we already saw earlier. As is well known, thecontours γ , γ ... can fluctuate quite far away from the union I ′ ∪ I ′′ ∪ ... ,and the corresponding distance can be unbounded. However, typically notall fluctuations can happen. For example, the rightmost tip of the rightmostcontour γ will look pretty much the same as the right tip of the ground statecontour γ ′ . Moreover, the size of fluctuations of the right tip of γ around thecontour γ ′ goes to zero with temperature. Therefore, the origin stays outsidethe contour γ , and in particular the probability of the event σ (0 , = +1 isvery small.The rest of the paper is the rigorous justification of the above picture.The picture sketched above shows that there is no entropic repulsion andwetting in the horizontal direction, as opposed to the vertical one.We notice that this result answers a question raised by Bethuelsen andConache, in a private communication to A.v.E and later in [BC], who de-rived a partial result in this direction. We also mention that more gen-erally the relationship between one-sided and two-sided continuity proper-ties (g-measures versus Gibbs measures) has been a topic of recent interest[BEEL, BFV, vE, FGM, Ober]. We consider the low- T
2D Ising model in a square box V M of size 2 M with ( − )boundary condition, where additionally we put the (+) boundary conditionon a segment I = [( − N, , (0 , , N < M, thereby forcing a contour γ around I . We are interested in the probability that the point ( n, ∈ Int( γ ) ;we want to show that it is of the order exp {− βn } , uniformly in N and M. This claim is not at all immediate, due to the entropic repulsion of γ from I. If the segment I is very long, then the contour γ tends to go away from I by a distance ∼ p | I | . Yet, this deviation of γ from I goes in the verticaldirection only, and the contour γ passes quite close to the edges ( − N, , (0 ,
0) of the segment, as we are going to show.3o let γ be our (exterior) contour. Its distribution is given by the weight w ( γ ) = exp ( − β | γ | + X Λ:Λ ∩ γ = ∅ Φ (Λ) ) , (1)where the summation over Λ goes over connected subsets Λ ⊂ V M \ I, andthe Φ (Λ)-s are exponentially small in diam (Λ) : | Φ (Λ) | ≤ exp {− β diam (Λ) } , (2)see, for example, [DKS], Sect. 4.3.Let L = { ( x, y ) : x = − N } , R = { ( x, y ) : x = 0 } be two vertical lines, S be the strip between them, and H ± be the upper and lower half-planes.Define the set of four cut-points u , u , v , v of γ by the properties: • u , u ∈ γ ∩ L, v , v ∈ γ ∩ R, • the arcs γ (piece of γ from u to v ) and γ (piece of γ from u to v )lie inside the strip S. We denote by γ u , γ v the remaining two arcs of γ. Let us define Z ( u , u , v , v ) = X γ : u ,u ,v ,v ∈ γ w ( γ ) , where the summation goes over γ ’s with cut-points u , u , v , v . We first pretend that w ( γ ) = w ( γ u ) w ( γ ) w ( γ v ) w ( γ ) , and moreover Z ( u , u , v , v ) ≈ exp {− β ( u − u + v − v ) } Z ( u → v ) Z ( u → v ) , (3)where Z ( u → v ) = P γ w ( γ ) , Z ( u → v ) = P γ w ( γ ) , and the weights w ( γ ) , w ( γ ) are taken from (1) , i.e. w ( γ ) = exp ( − β | γ | + X Λ:Λ ∩ γ = ∅ Φ (Λ) ) , (4)4here now both γ and all Λ-s stay in the upper semistrip I × { , , , ... } , and similarly for w ( γ ) (compare with (1)).In what follows we keep N fixed and we abuse notation by writing u as( − N, u ) , etc. We will choose N large enough, depending on β .While considering the partition functions Z ( u → v ), Z ( u → v ) wewill distinguish between regimes when the points u i , v i are of order √ N orsmaller, or are of higher orders.1. Suppose first that u > √ N , and v > u . Then, Z ( u → v ) < Z ( u → u )(meaning: Z ( u → u ) ≡ Z (( − N, u ) → (0 , u ))). Moreover, the func-tion Z ( u → v ) is decreasing in v in the regime v > u > √ N , sincethe v -dependence of this partition function is governed by the locallimit theorem, as is explained in [DKS], Section 4.10. Because of theextra factor of exp {− β ( v − u ) } , coming from the weight w ( γ v ) , wecan disregard the contribution of the configurations with u > √ N , and v > u to the partition function P w ( u , u , v , v ) .
2. The same monotonicity in v holds when u ≤ √ N , and v > √ N , with the same conclusion.3. Iterating 1, 2, we come to the remaining case, when all four variables u , u , v , v are ≤ √ N in absolute values. Here we would like to use thelocal limit theorem for the random point v in the ensemble Z ( u → v )and claim the existence of a constant C, which does not depend on β,u and v, such that Z ( u → ( v + 1)) Z ( u → v ) < C. (5)For β large, the factor exp {− β } – which is the price, due to the weight w ( γ v ) , for the point v to be one step higher above the segment – beats C. Overall, the probability of the point v to be at height h above the seg-ment decays exponentially in h . In what follows we will justify the abovearguments. 5 Splitting the partition function
Here we present the rigorous counterparts to the relations (3) − (5) . We will use the standard South-West convention to define the contoursas self-avoiding loops. Let γ be an exterior contour surrounding the segment[( − N, , (0 , . Then the cut-point u is defined as the point of the lastintersection of γ with the line L before getting to the line R, provided γ isoriented clockwise. The other three points u , v , v are defined similarly.As a result, γ is a concatenation, γ = γ ◦ γ v ◦ γ ◦ γ u , where • γ is any legal path in S, joining u and v and lying above [( − N, , (0 , , • γ is any legal path in S, joining u and v and lying below [( − N, , (0 , , • γ u ( γ v ) is any legal path joining u and u (joining v and v ), notintersecting [( − N, , (0 , γ ◦ γ v ◦ γ ◦ γ u is legal.As this definition suggests, we will treat the arcs γ , γ of γ as independentvariables, which put restrictions on the allowed realizations of γ u , γ v .Having in mind the definition (1) , we write Z ( u , u , v , v )= X γ : u → v γ : u → v exp − β | γ | + X Λ:Λ ∩ γ = ∅ , Λ ⊂ H + Φ (Λ) exp − β | γ | + X Λ:Λ ∩ γ = ∅ , Λ ⊂ H − Φ (Λ) (6) × X γ u : u → u ,γ v : v → v γ ◦ γ v ◦ γ ◦ γ u is legal exp − β | γ u | + X Λ:Λ ∩ γ u = ∅ , Λ ∩ ( γ ∪ γ )= ∅ Φ (Λ) exp − β | γ v | + X Λ:Λ ∩ γ v = ∅ , Λ ∩ ( γ ∪ γ ∪ γ u )= ∅ − Φ (Λ) .1 The vertical parts Let us first consider the partition function Z ( v , v ) = X γ : v → v exp ( − β | γ | + X Λ:Λ ∩ γ = ∅ , Φ (Λ) ) (7)(where we do not have the restriction that γ stays away from the segment I. ) It is well-known and easy to show that for any ε > | γ | > (1 + ε ) k v − v k goes to zero as β → ∞ , uniformly in v , v . Therefore, the probability p ( β ) in the ensemble (7) of the event thatthe ‘first’ edge of γ – i.e. the edge starting from v – goes down towards v , thus connecting v to ( v − e ) – has the property that p ( β ) → β → ∞ . Let Z ↓ ( v , v ) be part of the partition function (7) restricted tosuch configurations. Clearly, Z ↓ ( v , v ) Z ( v − e , v ) ∼ e − β , since the terms Φ (Λ) are of smaller order. Therefore, Z ( v , v ) Z ( v − e , v ) = 1 p ( β ) Z ↓ ( v , v ) Z ( v − e , v ) ≤ e − cβ , for some c → β → ∞ . The same argument, slightly modified, applies tothe third and the fourth partition functions in (6) . Here we will treat the partition function Z ( u → v ) = X γ : u → v exp − β | γ | + X Λ:Λ ∩ γ = ∅ , Λ ⊂ H + Φ (Λ) . The properties needed are obtained in [IOVW], which is based on the randomwalk approximation of the random line γ , worked out in [OV]. One coulduse instead the random walk description introduced in [DS], and used for asimilar goal in [IST]. 7n this subsection we will drop the subscript 1 and will write u, v, γ insteadof u , v , γ . The model we have to deal with is defined by assigning the weight w ( γ ) = exp − β | γ | + X Λ:Λ ∩ γ = ∅ , Λ ⊂ H + Φ (Λ) to any path γ ⊂ S. We can go to an enlarged ensemble, with more variables– ( γ, Λ ) – consisting of a path γ and a finite collection Λ of connected sets, Λ = { Λ i ⊂ Z } , each intersecting γ, and defined by the weight w ( γ, Λ ) = exp {− β ′ | γ |} Y Λ i ∈ Λ Ψ (Λ i ) . (8)The special case of Λ = ∅ is not excluded. Here the functional Ψ satisfies thesame estimate (2) , but in addition is positive , which makes w a legitimatestatistical weight. The functional Ψ and the new temperature β ′ can bechosen in such a way that P Λ w ( γ, Λ ) = w ( γ ) – so the partition functionsfor the weights w and w are the same – while | β − β ′ | → β → ∞ . Wewill call a pair ( γ, Λ ) a dressed path, or just a path. The idea of introducingthe hidden variables Λ and to consider the enlarged ensemble ( γ, Λ ) goesback to [DS], see [IST] for more details.Let x ∈ Z . We call the point x a splitting point of the dressed path( γ, Λ ) , if the intersection of the line l x = { ( x, y ) : x = x } with the curve γ is a single point, while all the intersections l x ∩ Λ i = ∅ , Λ i ∈ Λ . Let x , ..., x k be all the splitting points of the path ( γ, Λ ) . Then ( γ, Λ ) is split by the lines l x i into k + 1 irreducible pieces ( γ , Λ ) , ..., ( γ k , Λ k ) , and the dressed path( γ, Λ ) is their concatenation. We will call the irreducible pieces ( γ i , Λ i ) the animals. Note that w ( γ, Λ ) = k Y i =0 w ( γ i , Λ i ) , which paves the way to the definition of the random walk S – the effectiverandom walk representation.Let u = ( x, y ) , u ′ = ( x ′ , y ′ ) be two points in Z , x < x ′ . We define theweight s u,u ′ = ′ X ( γ, Λ ): γ : u → u ′ w ( γ, Λ ) , γ, Λ ) such that • the path γ goes from u to u ′ , and • the dressed path ( γ, Λ ) is its unique irreducible piece.These weights define the distribution of the random vector X = ( θ, ζ ) , whichdefines the steps of the walk S . Its starting point will be S = ( − N, u ) , while S i = ( − N, u ) + P ij =1 X j = u + ( T i , Z i ) , where T i = P j θ j , Z i = P j ζ j , (we follow here the notations of [IOVW], definition (45) ). The overalldistribution of S will be denoted by P ( − N,u ) . To study the ratio Z ( u → ( v +1)) Z ( u → v ) in (5) we can pass to the study of the prob-abilities in the ensemble P ( − N,u ) of the event { S : ( − N, u ) → (0 , v ) , S > } that the path S stays positive and arrives to the point (0 , v ) , resp. (0 , v + 1) . The very precise estimates of [IOVW] – see the relations (47-49) there – tellus that P ( − N,u ) { S : ( − N, u ) → (0 , v ) ; S > } ∼ C h + ( u ) h − ( v ) N / as N → ∞ , where • the function h + ( x ) = x − E x ( Z τ ) , where the random walk Z startsfrom the point x ∈ Z , x > , and the stopping moment τ is definedby τ = inf { n : Z n ≤ } ; • the function h − ( x ) is defined in the same way, but for the random walk( − Z ) ; • C = C ( θ, ζ ) > • the variables u, v ∈ (cid:2) , N / − δ (cid:3) , with any small δ > , which parameterwill be fixed from now on (say, δ = ).For the region u ∈ (cid:2) , N / − δ (cid:3) , v ∈ (cid:2) N / − δ , N / (cid:3) we have P ( − N,u ) { S : ( − N, u ) → (0 , v ) ; S > } ∼ C h + ( u ) v exp {− v / N } Var ( ζ ) N / , while in the region u, v ∈ (cid:2) N / − δ , N / (cid:3) we have P ( − N,u ) { S : ( − N, u ) → (0 , v ) ; S > } ∼ ψ (cid:0) u/N / , v/N / (cid:1) N / ψ on [0 , . Since in our case the random variable ζ is exponentially localized, i.e.Pr { ζ = k } ∼ exp {− c ( β ) | k |} with c ( β ) → ∞ as β → ∞ , we have that P ( − N,u ) { S : ( − N, u ) → (0 , v + 1) ; S > } P ( − N,u ) { S : ( − N, u ) → (0 , v ) ; S > } < C uniformly in N and u, v ∈ (cid:2) , N / (cid:3) . Combining the horizontal and vertical parts implies that with large proba-bility the cut points u , u , v and v all are at a not too large vertical distanceof the segment, which distance does not grow with N . Therefore the proba-bility that a point at horizontal distance n to the right of the segment will beinside the contour containing the segment decays exponentially in n , againuniformly in N (and M ). For the projected system this implies the continu-ity of the spin expectation in the origin on the left configuration (in the usualDynamic Systems interpretation where space is replaced by time, this meansa continuous dependence on the past configuration). We note that havingan arbitrarily large +-segment gives us the maximal (in FKG sense) value,which would maximally change the spin expectation at the origin from the(negative) infinite-volume Ising magnetization. Any other choice of configu-ration on the segment would have even less of an effect. Thus we concludethat the Schonmann projection produces a g -measure. We showed that the Schonmann projection applied on extremal low-temperatureGibbs measures of the 2-dimensional zero-field Ising model results in a mea-sure which, although not a Gibbs measure in the strict sense, for very lowtemperatures is a g -measure, and as such has a kind of one-dimensional Gibb-sian property. The property responsible for this is a lack of entropic repulsionwhen one inserts a long segment, in the direction of the segment.An earlier example of a non-Gibbsian g -measure was found in [FGM]. Onthe other hand, in [BEEL] a Gibbsian non- g -measure is displayed. The g -measure property thus cannot be seen as either weaker or stronger than theproperty of being a Gibbs measure.Presumably our result remains true for all subcritical temperatures, by ap-plying a coarse-graining argument as has been developed by Ioffe, Velenik10nd their collaborators on Ornstein-Zernike behavior, see e.g. [IOVW, OV].Although higher-dimensional versions of the non-Gibbsianness of the Schon-mann projected measures have been proved, there seems to be no naturalhigher-dimensional extension of the g -measure property. Acknowledgements:
Part of the work of S.S. has been carried out at Skoltech and at IITP RAS.The support of Russian Science Foundation (projects No. 14-50-00150 and20-41-09009) is gratefully acknowledged. Part of the work of S. S. has beencarried out in the framework of the Labex Archimede (ANR-11-LABX-0033)and of the A*MIDEX project (ANR-11- IDEX-0001-02), funded by the In-vestissements d’Avenir French Government program managed by the FrenchNational Research Agency (ANR).A.v.E. thanks the participants of the Oberwolfach miniworkshop [Ober], forvarious discussions as well as collaborations on related issues, and the Inter-national Emerging Action ”Long-Range” of the CNRS to make his partici-pation in the miniworkshop possible. He also thanks Rodrigo Bissacot, EricEndo and Roberto Fern´andez for earlier collaborations and various discus-sions on the issue of g -measures versus Gibbs measures. S.S. thanks SergePirogov, Yvan Velenik and Sebastien Ott for enlightening discussions. References [BGMMT] Barbieri,S., G´omez,R., Marcus,B., Meyerovitch,T. and Taati,S.,2021. Gibbsian representations of continuous specifications: the the-orems of Kozlov and Sullivan revisited. Communications in Mathe-matical Physics, to appear.[BFV] Berghout,S., Fern´andez,R. and Verbitskiy, E.A., 2019. On the rela-tion between Gibbs and g -measures. Ergodic Theory and DynamicalSystems, 38, pp. 3224-3249.[BC] Bethuelsen,S.A. and Conache,D., 2018. One-sided continuity prop-erties for the Schonmann projection. Journal of Statistical Physics172(4), pp. 1147-1163.[BEEL] Bissacot,R., Endo,E.O., van Enter,A.C.D. and Le Ny,A., 2018. En-tropic repulsion and lack of the g-measure property for Dyson mod-els. Communications in Mathematical Physics,363(3), 767-788.11BKL] Bricmont,J., Kupiainen,A. and Lefevere,R. , 1998. Renormalizationgroup pathologies and the definition of Gibbs states. Communica-tions in Mathematical Physics 194(2), pp. 359-388.[DKS] Dobrushin, R.L., Koteck´y, R. and Shlosman, S., 1992. Wulff con-struction: a global shape from local interaction (Vol. 104, pp. x+-204). Providence: American Mathematical Society.[DS] Dobrushin, R.L. and Shlosman, S.B., 1999. “Non-Gibbsian” statesand their Gibbs description. Communications in MathematicalPhysics, 200(1), pp.125-179.[vE] van Enter, A.C.D., 2018. One-sided versus two-sided stochastic de-scriptions. International conference on statistical mechanics of clas-sical and disordered systems, pp. 21-33, Springer.[EFS] van Enter, A.C.D., Fern´andez, R. and Sokal, A.D., 1993. Regularityproperties and pathologies of position-space renormalization-grouptransformations: Scope and limitations of Gibbsian theory. Journalof Statistical Physics 72(5), pp. 879-1169.[FGM] Fern´andez, R., Gallo, S. and Maillard, G., 2011. Regular gg