On the number of ground states of the Edwards-Anderson spin glass model
aa r X i v : . [ m a t h . P R ] O c t On the Number of Ground States of theEdwards-Anderson Spin Glass Model
Louis-Pierre Arguin ∗ Universit´e de Montr´ealMontr´eal, QC H3T 1J4 Canada
Michael Damron † Princeton UniversityPrinceton, NJ 08544, USA
October 30, 2011
Abstract
Ground states of the Edwards-Anderson (EA) spin glass model are studied oninfinite graphs with finite degree. Ground states are spin configurations that locallyminimize the EA Hamiltonian on each finite set of vertices. A problem with far-reachingconsequences in mathematics and physics is to determine the number of ground statesfor the model on Z d for any d . This problem can be seen as the spin glass version ofdetermining the number of infinite geodesics in first-passage percolation or the numberof ground states in the disordered ferromagnet. It was recently shown by Newman,Stein and the two authors that, on the half-plane Z × N , there is a unique groundstate (up to global flip) arising from the weak limit of finite-volume ground states fora particular choice of boundary conditions. In this paper, we study the entire set ofground states on the infinite graph, proving that the number of ground states on thehalf-plane must be two (related by a global flip) or infinity. This is the first result onthe entire set of ground states in a non-trivial dimension. In the first part of the paper,we develop tools of interest to prove the analogous result on Z d . Contents ∗ L.-P. Arguin was supported by the NSF grant DMS-0604869 during part of this work. † M. Damron is supported by an NSF postdoctoral fellowship. The uniform measure on the set of ground states 12 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.2 I ≤ I >
We study the Edwards-Anderson (EA) spin glass model on an infinite graph G = ( V, E ) offinite degree. We mostly take G = Z d (and further, d = 2), and G = Z × N , a half-plane of Z . For a finite set A ⊆ V , consider the set of spin configurations Σ A = {− , +1 } A and for σ ∈ Σ A , the Hamiltonian (with free boundary conditions) H J,A ( σ ) = − X { x,y }∈ Ex,y ∈ A J xy σ x σ y , (1)where the J xy ’s (the couplings ) are taken from an i.i.d. product measure ν . We assume thatthe distribution of each J xy is continuous with support equal to R . For inverse temperature β > A is G J,A ( σ ) = 1 Z J,A exp( − βH J,A ( σ )) , Z J,A = X σ ∈ Σ A exp( − βH J,A ( σ )) . As temperature approaches 0 ( β → ∞ ) the Gibbs measure converges weakly to a sum of twodelta masses, supported on the spin configurations with minimal value of H J,A . These spinconfigurations (related by global flip) can be seen to be characterized by the following localflip property: for each B ⊆ A , we have X { x,y }∈ ∂Bx,y ∈ A J xy σ x σ y ≥ . ∂B ⊂ E is defined as all edges { x, y } such that x ∈ B and y / ∈ B . The advantageis that this definition makes sense for infinite sets A . For this reason, we define the set ofground states on the infinite graph G at couplings J by G ( J ) = { σ ∈ {− , +1 } V : ∀ A ⊂ V finite, X { x,y }∈ ∂A J xy σ x σ y ≥ } . (2)In other words, elements of G ( J ) are the spin configurations minimizing the Hamiltonianlocally for the coupling realization J . Clearly, σ ∈ G ( J ) if and only if − σ ∈ G ( J ). Thegoal of this paper is not to determine precisely the cardinality of G ( J ) but rather to rule outpossibilities other than two or infinity. Our main result is to prove such a claim in the caseof the EA model on the two-dimensional half-plane. Theorem 1.1.
For the EA model on the half-plane G = Z × N , the number of ground states |G ( J ) | is either with ν -probability one or ∞ with ν -probability one. A main question in the theory of short-range spin glasses is to understand the structureof the set G ( J ), and in particular its cardinality. This problem is the zero-temperatureequivalent of understanding the structure and the cardinality of the set of pure states, theset of infinite-volume Gibbs measures of the EA model that are extremal. It is easy to checkthat for G = Z , G ( J ) has only two elements: the flip-related configurations σ defined bythe identity σ x σ y = sgn J xy . However, it is not known how many elements are in G ( J ) for G = Z d when d >
1. (We will see in the next section that the cardinality of G ( J ) must bea constant number ν -almost surely.) It is expected that |G ( J ) | = 2 for d = 2 [11, 15] (seealso [9] for a possible counterargument to this). There are competing predictions for higherdimensions. The Replica Symmetry Breaking (RSB) scenario would predict |G ( J ) | = ∞ for d high enough, and the droplet/scaling proposal would be consistent with |G ( J ) | = 2 inevery dimension. We refer to [12, 14] for a detailed discussion on ground states of disorderedsystems or pure states at positive temperature.There have been several works on ground states of the EA model in the physics andmathematics literature; a partial list includes [1, 6, 9, 11, 12, 13, 14, 15]. The present workappears to give the first rigorous result about the entire set of ground states G ( J ). Previousrigorous results have focused on the so-called metastates on ground states . A metastate is a J -dependent probability measure on {− , +1 } V supported on ground states. It is constructedusing a sequence of finite graphs G n converging to G . For a given realization J and n , theground state on G n is unique up to a global flip. We identify the flip-related configurationsand write σ ∗ n ( J ) for them. A metastate is obtained by considering a converging subsequenceof the measures (cid:0) ν ( dJ ) δ σ ∗ n ( J ) (cid:1) n , where δ σ ∗ n ( J ) is the delta measure on the ground state of G n for the coupling realization J . If κ denotes a subsequential limiting measure, then samplingfrom κ gives a pair ( J, σ ) ∈ R E × {− , +1 } V . A metastate is the conditional measure κ given J and is denoted κ J . It is not hard to verify that κ J is supported on G ( J ).3t was proved in [1] that the ground state of the EA model on the half-plane withhorizontal periodic boundary conditions and free boundary condition at the bottom is uniquein the metastate sense. Precisely, for a sequence of boxes G n that converges to the half-plane,the limit κ J produced by the metastate construction is unique and is given by a delta measureon two flip-related ground states. Though the metastate construction is very natural, it isimportant to stress that the measure thus obtained is not necessarily supported on the wholeset G ( J ). It may be that some elements of G ( J ) do not appear in the support of the metastate,due to the choice of boundary conditions on G n or to the fact that the subsequence in themetastate construction is chosen independently of J . Therefore uniqueness in the metastatesense does not answer the more general question of the number of ground states.It is natural from a statistical physics perspective to study the set G ( J ) by looking atprobability measures on it. One challenge is to construct probability measures on G ( J )that have a nice dependence on J , namely measurability and translation covariance. Themetastate (with suitably chosen boundary conditions) briefly described above is one suchmeasure. The main idea of the present paper is to consider another measure, the uniformmeasure on G ( J ) µ J := 1 |G ( J ) | X σ ∈G ( J ) δ σ . For µ J to be well-defined it is necessary to assume that |G ( J ) | is finite. Like the metastate, theuniform measure on ground states depends nicely on J : see Proposition 2.4 and Lemma 3.9.The strategy to prove a “two-or-infinity” result is to assume that |G ( J ) | < ∞ and to concludethat it implies that µ J is supported on two spin configurations related by a global flip (thatis, |G ( J ) | = 2). The approach is similar to the proof of uniqueness in [1] using the interfacebetween ground states, though new tools need to be developed. For spin configurations σ and σ ′ , define the interface σ ∆ σ ′ as σ ∆ σ ′ = {{ x, y } ∈ E : σ x σ y = σ ′ x σ ′ y } . It will be shown for the half-plane that Z ν ( dJ ) µ J × µ J { ( σ, σ ′ ) : σ ∆ σ ′ = ∅} = 1 . This implies that µ J is supported on two flip-related configurations for ν -almost all J since σ ∆ σ ′ = ∅ if and only if σ = σ ′ or σ = − σ ′ .Before going into the details of the proofs, we remark that the problem of determiningthe number of ground states for the EA model can be seen as a spin-glass version of afirst-passage percolation problem. Indeed, one question in two-dimensional first-passagepercolation is to determine whether there exist infinite geodesics. These are doubly-infinitecurves that locally minimize the sum of the random weights between vertices of the graph.This problem is equivalent to determining whether there exist more than two flip-relatedground states in the (ferromagnetic) Ising model with random couplings. The Hamiltonianof the ferromagnetic model is the same as in (1), but the distribution of J is restricted to4he positive half-line. The reader is referred to [17] for the details of the correspondence. Itwas proved by Wehr in [17] that the number of ground states for this model is either two orinfinity in dimensions greater or equal to two. On the half-plane, it was shown by Wehr andWoo [18] that the number of ground states is two. Contrary to the ferromagnetic case, thestudy of ground states of the EA spin glass model presents technical difficulties that stemfrom the presence of positive and negative couplings. This feature rules out monotonicity ofthe partial sums of couplings along an interface.The paper is organized into two main parts as follows. The first part develops generaltools to study ground states of the EA model. Precisely, in Section 2, elementary propertiesof the set G ( J ) are derived for general graphs. In particular, the dependence of G ( J ) on asingle coupling is studied. Properties of probability measures on G ( J ) are investigated inSection 3 with an emphasis on the uniform measure on G ( J ). The second part of the paperconsists of the proof of Theorem 1.1 and is contained in Section 4. Acknowledgements
Both authors are indebted to Charles Newman and Daniel Stein forhaving introduced them to the subject of short-range spin glasses and for numerous dis-cussions on related problems. L.-P. Arguin thanks also Janek Wehr for discussions on theproblem of the number of ground states in spin glasses and in disordered ferromagnets.
In this section, unless otherwise stated, we consider the EA model on a connected graph G = ( E, V ) of finite degree. We assume that there exists a sequence of subgraphs ( G n ) thatconverges locally to G . Throughout the paper, we will use the following notation: Ω = R E ,and F is the Borel sigma-algebra generated by its product topology; Ω = {− , +1 } V and F is the corresponding product sigma-algebra. We first note that the set of ground states is compact.
Lemma 2.1. G ( J ) is a non-empty compact subset of Ω (in the product topology) for all J .In particular, the set of probability measures on G ( J ) is compact in the weak-* topology onthe set of probability measures on Ω .Proof. The fact that G ( J ) is non-empty follows by a standard compactness argument, takinga subsequence of ground states for the Hamiltonian (1) with A = G n . The function σ P { x,y }∈ ∂A J xy σ x σ y is continuous in the product topology for a given finite A and J . Therefore,the set { σ ∈ {− , +1 } V : P { x,y }∈ ∂A J xy σ x σ y ≥ } is closed. Since G ( J ) is the intersection ofthese sets over all finite A by (2), it is closed. Being a closed subset of the compact spaceΩ , it is also compact. The second statement of the lemma follows from the first.5he next result is necessary for the uniform measure to be well-behaved and to laterapply the ergodic theorem to |G ( J ) | . Proposition 2.2.
The random variable J
7→ |G ( J ) | is F -measurable.Proof. Consider a sequence of finite graphs Λ n ⊂ G , a configuration σ n on Λ n and a config-uration ¯ σ n on the external boundary of Λ n (that is, all vertices that are not in Λ n but areadjacent to vertices in it). The condition that σ n is a ground state in Λ n with boundaryconditions ¯ σ n is a finite list of conditions of the form X { x,y }∈ S J xy ( σ n ) x ( σ n ) y ≥ X { x,y }∈ S J xy ( σ n ) x (¯ σ n ) y ≥ S of edges. For any given S , the set of J ∈ Ω such that condition(3) holds for fixed σ n and ¯ σ n is then measurable (that is, it is in F ). Intersecting over allrelevant sets S , we see that the following set is measurable: J ( σ n , ¯ σ n ) := { J : σ n is a ground state in Λ n for the boundary condition ¯ σ n } . Next take m < n and fixed configurations σ m on Λ m , σ m,n on Λ n \ Λ m and ¯ σ n on theboundary of Λ n . By a similar argument to the one given above, the set J ( σ m , σ m,n , ¯ σ n ) of J such that the concatenation of σ m and σ m,n is a ground state on Λ n with boundary condition¯ σ n is measurable. Taking the union over all σ m,n and ¯ σ n for a fixed σ m , we get that for m < n and σ m fixed, the following set is measurable: J ( σ m , n ) := { J : there is a ground state in Λ n (for some ¯ σ n ) that equals σ m on Λ m } . If there exists a sequence of (possibly J -dependent) configurations (¯ σ n ) such that thereare ground states ( σ n ) on Λ n with boundary condition ¯ σ n that converge to σ , then σ is in G ( J ). Conversely, if σ ∈ G ( J ), such a sequence ( σ n ) exists by taking ¯ σ n to be the restriction of σ to the boundary. It follows that ∩ n ≥ m J ( σ m , n ) is the event that there is an infinite-volumeground state σ for couplings J that equals σ m on Λ m . This event is thus measurable.For fixed m and a configuration σ m on Λ m , let F σ m ( J ) be the indicator of the event thatthere is an infinite-volume ground state σ for couplings J equal to σ m on Λ m . By the above,it is F -measurable. The proposition will then be proved once we show: |G ( J ) | = sup m X σ m F σ m ( J ) . (4)Here the sum is over all σ m on Λ m . For any m , the sum P σ m F σ m ( J ) equals the number ofdifferent σ m ’s that are equal to restrictions on Λ m of elements of G ( J ) . So for each m , X σ m F σ m ( J ) ≤ |G ( J ) | and the right side of (4) is at most |G ( J ) | . To show equality in (4), suppose first that |G ( J ) | is finite. We can choose n so that the restriction to Λ n of each element of G ( J ) is different.6or this n , P σ n F σ n ( J ) = |G ( J ) | and (4) is established. If |G ( J ) | = ∞ , then for any k ∈ N ,we can find n k such that P σ nk F σ nk ( J ) ≥ k . This is because we can take Λ n large enoughso that there are at least k elements of G ( J ) that are distinct on Λ n . Taking the supremumover k completes the proof of (4).In the case G = Z d , it is easy to see that for any translation T a by a vector a ∈ Z d , |G ( J ) | = |G ( T a J ) | where ( T a J ) xy = J T a ( x ) T a ( y ) . The ergodic theorem then implies that therandom variable |G ( J ) | is constant ν -almost surely. The same holds when G is the half-planeby considering only horizontal translations. Corollary 2.3.
For G = Z d or G = Z × N , the number of ground states |G ( J ) | is a constant ν -almost surely. The next result shows that if |G ( J ) | < ∞ then the uniform measure µ J is a randomvariable over F . Proposition 2.4.
Let B ∈ F and assume that |G ( J ) | < ∞ . The map J µ J ( B ) is F -measurable. Similarly, if B ′ is a Borel set in Ω × Ω , then the map J µ J × µ J ( B ′ ) is F -measurable.Proof. By a standard approximation, it is sufficient to prove the statement for B of the form B = { σ : σ = s A on A } for some finite set A and fixed configuration s A on A .Take a sequence of finite graphs Λ n converging to G . We define F s A ( J ) = number of σ ∈ G ( J )’s that equal s A on A .
Note that µ J ( B ) is simply F s A ( J ) divided by |G ( J ) | . The variable G ( J ) is F -measurable byProposition 2.2. Thus it remains to show that F s A ( J ) is also.Exactly as in the last proof, if n is so large that Λ n contains A and if s A,n is any fixedspin configuration on Λ n \ A , then the set J ( s A , s A,n ) of all J such that there is an elementof G ( J ) that (a) equals s A on A and (b) equals s A,n on Λ n \ A is measurable. Let F s A ,s A,n ( J )be the indicator of the event J ( s A , s A,n ) and consider the random variablesup n X s A,n F s A ,s A,n ( J ) . Here the supremum is over all n such that A ⊆ Λ n . The same reasoning to prove (4) showsthat F s A ( J ) is equal to the above and is thus measurable. This completes the proof of the firstclaim. The second assertion is implied by the first one since by a standard approximation,7ny measurable function on Ω × Ω can be approximated by linear combinations of indicatorfunctions of sets of the form B A := { ( σ, σ ′ ) : σ = s A on A , σ ′ = s A ′ on A ′ } for two finite sets A and A ′ of G . Since µ J × µ J ( B A ) is equal to the product of the µ J -probability of each coordinate, measurability follows from the first part of the proposition. In this section, we establish some elementary properties of the dependence of the set ofground states G ( J ) on a finite number of couplings.Fix an edge e = { x, y } . We will sometimes abuse notation and write for simplicity J e := J xy and σ e := σ x σ y . We are interested in studying how G ( J ) varies when J e is modified. For simplicity, we will fixall other couplings and write G ( J e ) for the set of ground states to stress the dependence on J e . From the definition (2), it is easy to see that if σ ∈ G ( J e ) and σ e = +1, then σ remainsa ground state for coupling values greater than J e . More generally: Lemma 2.5.
Fix an edge e = { x, y } . If J e ≤ J ′ e then G ( J e ) ∩ { σ : σ e = +1 } ⊆ G ( J ′ e ) ∩ { σ : σ e = +1 }G ( J e ) ∩ { σ : σ e = − } ⊇ G ( J ′ e ) ∩ { σ : σ e = − } In view of the above monotonicity of the set of ground states, it is natural to introducethe critical value of σ ∈ G ( J e ) at e . Namely, we define the critical value C e as C e ( J, σ ) := ( inf { J e : σ ∈ G ( J e ) } if σ e = +1;sup { J e : σ ∈ G ( J e ) } if σ e = − σ ∈ G ( J e ) and σ e = +1 = ⇒ J e ≥ C e ( J, σ ) σ ∈ G ( J e ) and σ e = − ⇒ J e ≤ C e ( J, σ ) . (5)An elementary correspondence exists between the critical values and the energy requiredto flip finite sets of spins. Lemma 2.6.
Let σ ∈ G ( J e ) . Then σ e C e ( J, σ ) = − inf A : e ∈ ∂A X { z,w }∈ ∂A { z,w }6 = e J zw σ z σ w , (6) In particular, for a given σ , C e ( J, σ ) does not depend on J e .
8n this section, we will often omit the dependence on J in the notation and write C e ( σ )for simplicity. From the above result, we see that this notation is consistent with the factthat all couplings other than J e are fixed in this section. Proof.
The independence assertion is straightforward from the expression. We prove theequation in the case of σ e = +1. The other case is similar. Let − e C e ( σ ) be the right side of(6). If C e ( σ ) + e C e ( σ ) >
0, there exists δ > C e ( σ ) − δ + inf A : e ∈ ∂A X { z,w }∈ ∂A { z,w }6 = e J zw σ z σ w > . In particular, σ ∈ G ( J ′ e ) for J ′ e = C e ( σ ) − δ , contradicting C e ( σ ) as the infimum of suchvalues. On the other hand if C e ( σ ) + e C e ( σ ) <
0, there must exist a finite set A such that C e ( σ ) + X { z,w }∈ ∂A { z,w }6 = e J zw σ z σ w < . In particular this would hold for C e ( σ ) replaced by some J e > C e ( σ ), contradicting thedefinition of C e ( σ ), because we should have σ ∈ G ( J e ) for all J e > C e ( σ ).The distance | J e − C e ( σ ) | from J e to the critical value is called the flexibility of e andis denoted F e ( σ ). (This quantity was first introduced in [13].) From above, it has a usefulrepresentation: F e ( σ ) := | J e − C e ( σ ) | = inf A : e ∈ ∂A X { z,w }∈ ∂A J zw σ z σ w . (7)In the same spirit as the critical values, for any edge e and σ ∈ G ( J e ), we define the setof critical droplets for e in σ . These are the limit sets of the infimizing sequences of finitesets in the expression (6) of the critical value C e ( σ ). Precisely, if (Λ n ) is a sequence of vertexsets, we say that Λ n → Λ if each vertex v ∈ V is in only finitely many of the sets Λ n ∆Λ(here ∆ denotes the symmetric difference of sets). We will say that Λ is a critical dropletfor e in σ if there exists a sequence of finite vertex sets (Λ n ) such that Λ n → Λ, e ∈ ∂ Λ n forall n and − X { x,y }∈ ∂ Λ n { x,y }6 = e J xy σ x σ y → σ e C e ( σ ) as n → ∞ . Write CD e ( σ ) for the set of critical droplets of e in σ . By compactness, this set is nonempty.Since the critical values are values of J e where there is a change in the set G ( J e ), it willbe useful to get bounds on them that are functions of the couplings only (not of σ ∈ G ( J e )).In this spirit, similarly to [13], we define the super-satisfied value for an edge e = { x, y } as S e := min X z = y { x,z }∈ E | J xz | , X z = x { y,z }∈ E | J yz | . (8)9e will say that an edge e is super-satisfied if | J e | > S e . The terminology is explained bythe following fact: by taking A = { x } and A = { y } in (2), one must have J e > S e = ⇒ σ e = +1 for all σ ∈ G ( J e ) J e < −S e = ⇒ σ e = − σ ∈ G ( J e ) . (9)Moreover, for the same choice of A , we get from Lemma 2.6 C e ( σ ) ≥ −S e if σ e = +1 and C e ( σ ) ≤ S e if σ e = −
1. (10)Our next goal is to prove that in fact | C e ( σ ) | ≤ S e (cf. Corollary 2.8). This is done byestablishing a correspondence between the two following sets: G + e = { σ ∈ Ω : σ e = +1 , ∀ A ⊂ V finite with e / ∈ ∂A , X { x,y }∈ ∂A J xy σ x σ y ≥ } ; G − e = { σ ∈ Ω : σ e = − , ∀ A ⊂ V finite with e / ∈ ∂A , X { x,y }∈ ∂A J xy σ x σ y ≥ } . In other words, G ± e are the sets of ground states on the graph G minus the edge e , where thespins of the vertices of e are restricted to have the same/opposite sign. Note that these setsdepend on the couplings but not on J e . Clearly, if σ ∈ G ( J e ) then either σ ∈ G + e or σ ∈ G − e depending on its sign at e . Moreover by (9), if J e > S e , then G ( J e ) ⊆ G + e and if J e < −S e ,then G ( J e ) ⊆ G − e . Equality is derived in Corollary 2.9 from the following correspondence. Proposition 2.7.
For σ ∈ G + e and Λ ∈ CD e ( σ ) , consider e σ where σ ∆ e σ = ∂ Λ , that is e σ v = ( σ v v / ∈ Λ − σ v v ∈ Λ . (11) Then e σ ∈ G − e and C e ( e σ ) ≥ C e ( σ ) . A similar statement holds for σ ∈ G − e with e σ ∈ G + e and C e ( e σ ) ≤ C e ( σ ) .Proof. Write D for the collection of sets of edges S such that S = ∂A for some finite set ofvertices A . We will use the following fact, which is verified by elementary arguments, andwhich was also noticed in [7]: if S , S ∈ D , then S ∆ S ∈ D .We will prove the proposition in the case σ ∈ G + e . The other case is similar. Choose asequence of finite vertex sets (Λ n ) such that e ∈ ∂ Λ n for all n , Λ n → Λ, and − X { x,y }∈ ∂ Λ n { x,y }6 = e J xy σ x σ y → C e ( σ ) as n → ∞ . Write S n = ∂ Λ n , S = ∂ Λ, let T ∈ D and take n so large that T ∩ S n = T ∩ S and T \ S n = T \ S . Let e J be the coupling configuration with value e J f = J f for f = e and10 J e = C e ( σ ) at e . X { x,y }∈ T e J xy e σ x e σ y = X { x,y }∈ T ∩ S n e J xy e σ x e σ y + X { x,y }∈ T \ S n e J xy e σ x e σ y (11)= − X { x,y }∈ T ∩ S n e J xy σ x σ y + X { x,y }∈ T \ S n e J xy σ x σ y = X { x,y }∈ T ∆ S n e J xy σ x σ y − X { x,y }∈ S n e J xy σ x σ y . Since T ∆ S n ∈ D and σ ∈ G ( e J ), we have P { x,y }∈ T ∆ S n e J xy σ x σ y ≥
0. Therefore, X { x,y }∈ T e J xy e σ x e σ y ≥ − X { x,y }∈ S n e J xy σ x σ y . The right side tends to 0 as n → ∞ by the definition of S and e J , so P { x,y }∈ T e J xy e σ x e σ y ≥ e σ ∈ G ( ˜ J ). Clearly, e σ ∈ G − e , and by (5), C e ( e σ ) ≥ e J e = C e ( σ ).We prove three corollaries of the proposition. The first is the claimed bounds on C e ( σ ). Corollary 2.8 (Super-satisfied bounds) . Let e be an edge. If σ ∈ G ( J e ) , then | C e ( σ ) | ≤ S e . Proof.
We prove the bound when σ ∈ G + e . The case σ ∈ G − e is similar. The lower boundwas noticed in (10). As for the upper bound, by Lemma 2.7, there exists e σ ∈ G − e such that C e ( σ ) ≤ C e ( e σ ). The claim then follows from C e ( e σ ) ≤ S e again by (10).A useful fact about Corollary 2.8 is that it replaces the critical value that a priori dependson an infinite number of couplings by a quantity that depends on finitely many. Anothercorollary is that for J e low enough or large enough, the set G ( J ) is independent of J e : Corollary 2.9. If J e > S e , then G ( J e ) = G + e . If J e < −S e , then G ( J e ) = G − e .Proof. Suppose first that J e > S e . Then, from (5) and Corollary 2.8, one has G ( J e ) ⊆ G + e .Conversely, if σ ∈ G + e , it suffices to show that for any finite set of vertices A with e ∈ ∂AJ e + X { z,w }∈ ∂A { z,w }6 = e J zw σ z σ w ≥ . By Corollary 2.8, we have S e − C e ( σ ) ≥ J e > S e . The proof for G − e is similar.11inally, we show that an infimizing sequence of sets for the critical values of an edge cannever contain certain super-satisfied edges. For this we need to introduce for e = { x, y }S xe = X { x,z }∈ E,z = y | J xz | . (12)Note that by definition, S e = min {S xe , S ye } . If d and e are two different edges, there exists avertex x which is an endpoint of d , but not of e . Having | J d | > S xd guarantees that the edge d is super-satisfied independently of the value of J e . Corollary 2.10.
Let d = { x, y } and e be edges such that x is not an endpoint of e and | J d | > S xd . If σ ∈ G ( J ) then no element Λ of CD e ( σ ) has d ∈ ∂ Λ .Proof. Let σ ∈ G ( J ) for some fixed J such that | J d | > S xd . Suppose d ∈ ∂ Λ for someΛ ∈ CD e ( σ ). Define e σ as in Proposition 2.7, so that σ ∆ e σ = ∂ Λ. For y ∈ R , let J ( e, y ) bethe coupling configuration that equals J f at f = e and y at e . On one hand, note that, byProposition 2.7, σ d = − e σ d and that e σ ∈ G ( J ( e, y )) for either small or large values of y . Onthe other hand, if | J d | > S xd for J , then | J d | > S xd in J ( e, y ) for all y ∈ R , because x is notshared by d and e . In particular, this implies by Corollary 2.9 that the sign at the edge d ofthe elements of G ( J ( e, y )) must be the same for all y ∈ R . This contradicts σ d = − e σ d . In this section we assume that |G ( J ) | is constant ν -a.s. and |G ( J ) | < ∞ . The first assertion holds for graphs with translation symmetry by the ergodic theorem asnoted in Corollary 2.3. We consider the family ( µ J ) consisting of the uniform measures on G ( J ) indexed by J ∈ Ω . Recall from Proposition 2.4 that this family has a measurabledependence on J . For concision, the following notation will be used throughout the paperfor the product measures on J and on one or two replicas of the spin configurations: M = ν ( dJ ) µ J or M = ν ( dJ ) µ J × µ J , (13)where the appropriate case will be clear from the context. In the first part, we use themonotonicity of the measure (defined below) to prove several facts, for example that thecritical droplet of any edge is unique. Second, we focus on the properties of the interfacesampled from M and prove that, if it exists, any given edge lies in it with positive probability. We first introduce the monotonicity property of the family ( µ J ). It is the analogue of themonotonicity of G ( J ) in Lemma 2.5 at the level of measures. To define it, we give the12ollowing notation. For any coupling configuration J = ( J f ) f ∈ E , fixed edge e and realnumber y , let J ( e, y ) be the coupling configuration given by( J ( e, y )) f = ( y if f = eJ f if f = e . (14)Consider any event A ⊆ Ω × { σ : σ e = +1 } . A simple consequence of Lemma 2.5, since |G ( J ) | is a.s. constant, is that for almost all J and for almost all y ≥ J e : µ J { σ : ( J, σ ) ∈ A } ≤ µ J ( e,y ) { σ : ( J, σ ) ∈ A } ; (15)on the other hand, if A ⊆ Ω × { σ : σ e = − } , then for almost all J and almost all y ≤ J e : µ J { σ : ( J, σ ) ∈ A } ≥ µ J ( e,y ) { σ : ( J, σ ) ∈ A } . (16)Similar statements hold for the product µ J × µ J . For example, the mixed case A ⊆ Ω × { σ : σ e = +1 } × { σ ′ : σ ′ e = − } yields for almost all J and almost all y ≥ J e and y ′ ≤ J e : µ J × µ J { ( σ, σ ′ ) : ( J, σ, σ ′ ) ∈ A } ≤ µ J ( e,y ) × µ J ( e,y ′ ) { ( σ, σ ′ ) : ( J, σ, σ ′ ) ∈ A } . (17)We refer to (15), (16) and (17) as the monotonicity of the family ( µ J ). It is a naturalproperty to expect from a family of measures on ground states. The results of this section,with the exception of Lemma 3.5, are derived solely from it and no other finer propertiesof the uniform measure. The main use of the monotonicity property is to decouple thedependence on J e in µ J from the dependence on J e in the considered event. This trick willappear frequently. The results of this section are stated for the measure M in (13) with onereplica of σ for concision. They also hold for the measure M on two replicas.A useful consequence of (15), (16), (17), and the continuity of ν is that ν -almost surelyno coupling value is equal to its critical value. This is a special case of the next proposition,taking B = { e } and h B c = C e . Proposition 3.1.
Let B ⊂ E be a finite set of edges and h B c : R E × {− , +1 } V → R be a function that does not depend on couplings of edges in B . Then for any given linearcombination P b ∈ B J b s b , provided that the coefficients s b ∈ R are not all zero, M { ( J, σ ) : h B c ( J, σ ) = X b ∈ B J b s b } = 0 . The same statement holds if h B c is a function of the couplings and two replicas ( J, σ, σ ′ ) h B c ( J, σ, σ ′ ) that does not depend on the couplings of edges in B .Proof. The event { σ : h B c ( J, σ ) = P b ∈ B J b s b } can be decomposed by taking the intersectionwith all possible spin configurations on B . Suppose first that σ b = +1 for all b ∈ B anddefine, for a given J ∈ R E , J ( B, y ) for y ∈ R B similarly to (14)( J ( B, y )) e = ( y e , e ∈ BJ e , e / ∈ B .
13y (15), µ J { σ : h B c ( J, σ ) = P b ∈ B J b s b , σ b = +1 ∀ b ∈ B } is smaller than the probabilityof the same event under the measure averaged over larger J b ’s. Writing { J ≥ B } for the eventthat y b ≥ J b for all b ∈ B , Z ν ( dJ B ) µ J { σ : h B c ( J, σ ) = X b ∈ B J b s b , σ b = +1 ∀ b ∈ B }≤ Z ν ( dJ B ) 1 ν { J ≥ B } Z { J ≥ B } ν ( dy ) µ J ( B,y ) { σ : h B c ( J, σ ) = X b ∈ B J b s b , σ b = +1 ∀ b ∈ B } . Integrating y over all of R B and dropping { σ b = +1 ∀ b ∈ B } gives the upper bound: Z ν ( dJ B ) 1 ν { J ≥ B } Z ν ( dy ) µ J ( B,y ) { σ : h B c ( J ( B, y ) , σ ) = X b ∈ B J b s b } . Note h B c ( J ( B, y ) , σ ) = h B c ( J, σ ) as h B c does not depend on couplings in B . Now use Fubini: Z ν ( dy ) Z dµ J ( B,y ) ( σ ) (cid:20)Z ν ( dJ B ) ν { J ≥ B } − { P b ∈ B J b s b = h Bc ( J ( B,y ) ,σ ) } ( J B ) (cid:21) , where 1 A ( J B ) denotes the indicator function of the event A . Because the linear combinationof J b ’s is non-trivial and h B c ( J ( B, y ) , σ ) does not depend on J B , the indicator function isequal to 1 on a set of J B ’s that is a hyperplane of dimension at most | B | −
1. Therefore itis ν -almost surely zero, and the inner integral equals zero. This completes the proof in thecase that σ b = +1 for all b ∈ B . To prove the other cases where σ b = − b ∈ B , itsuffices to average over { J ≤ b } (where this event is defined in the obvious way) for b and use(16). The proof of the second claim when h B c is a function of the couplings and two replicas( J, σ, σ ′ ) h B c ( J, σ, σ ′ ) is done the same way. In the case that σ b = +1 and σ ′ b = −
1, oneuses (17) and bounds µ J × µ J by the average of µ J ( b,y ) × µ J ( b,y ′ ) over { J ≥ b } × { J ≤ b } .One consequence of the above proposition is that the critical droplet CD e ( σ ) set cannotcontain two non-flip-related elements. In other words, infimizing sequences of finite sets ofedges entering in the definition (6) of the critical value converge to a unique set. This impliesin particular that the mapping of Lemma 2.7 is well-defined. Corollary 3.2.
For any edge e ∈ E , M { ( J, σ ) : ∃ T = T ∈ CD e ( σ ) with T = G \ T } = 0 .Proof. Suppose that CD e ( σ ) contains at least two critical droplets, T and T , not relatedby T = G \ T , with positive probability. Let S be the set of edges connecting T to T c (similarly for S ). Either S \ S or S \ S is non-empty. We may assume that S \ S isnon-empty. So there exists b such that M { ( J, σ ) : ∃ T , T ∈ CD e ( σ ) with b ∈ S \ S } > . (18)Assume that σ e = +1 and σ b = +1; the other cases are similar. Define C b,e ( J, σ ) = − inf A : b,e ∈ ∂AA finite X { x,y }∈ ∂A { x,y }6 = b,e J xy σ x σ y and C be ( J, σ ) = − inf A : e ∈ ∂Ab/ ∈ ∂AA finite X { x,y }∈ ∂A { x,y }6 = e J xy σ x σ y .
14n the event in (18), we have C b,e ( J, σ ) − J b = C e ( J, σ ) = C be ( J, σ ) because T and T are in CD e ( σ ). Thus (18) implies that M { ( J, σ ) : σ e = σ b = +1 , C b,e ( J, σ ) − C be ( J, σ ) = J b } > . This contradicts Proposition 3.1 using B = { e } and h B c ( J, σ ) = C b,e ( J, σ ) − C be ( J, σ ).We now state a lemma that will be used in Section 4.3.3. By Corollary 2.10, the criticaldroplet cannot go through certain super-satisfied edges. Therefore if there are such super-satisfied edges forcing the critical droplet of an edge f to go through some fixed edges e or e , then the flexibility (7) of f , by definition, cannot be smaller than both of those of e and e . The situation is depicted in Figure 4 where the super-satisfied edges appear in grey. Asin Corollary 2.10, the edges need to be super-satisfied independently of the value of J f . Forthis reason, we work with the value S xe defined in (12). Lemma 3.3.
Let e , e , f be edges. Let U be a set of edges with the property that all finitesets A with f ∈ ∂A and ∂A ∩ U = ∅ must have either e or e in ∂A . For each e ∈ U pick x ( e ) to be an endpoint of e that is not an endpoint of f . Then M { ( J, σ ) : F f ( J, σ ) ≥ min { F e ( J, σ ) , F e ( J, σ ) } , ∀ e ∈ U | J e | > S x ( e ) e } = 1 . We will now prove two lemmas about the measure M that will be useful later. Theyrequire an extra assumption on the type of events under consideration; see for example(19) and (21). The results show that an event of positive probability remains of positiveprobability after a certain coupling modification. They in fact provide explicit lower boundswhich will be needed when dealing with weak limits of the measure M in Section 4. Lemma 3.4.
Let A ⊆ Ω × { σ : σ e = +1 } be such thatIf ( J, σ ) ∈ A then ( J ( e, s ) , σ ) ∈ A for all s ≥ J e . (19) Then for each λ ∈ R , M ( A, J e ≥ λ ) ≥ (1 / ν ([ λ, ∞ )) M ( A ) . (20) If instead, we have A ⊆ Ω × { σ : σ e = − } and ( J ( e, s ) , σ ) ∈ A for all s ≤ J e then M ( A, J e ≤ λ ) ≥ (1 / ν (( −∞ , λ ]) M ( A ) . Proof.
We will prove the first statement; the second is similar. The left side of (20) equals Z ν ( dJ { e } c ) (cid:20)Z ∞ λ ν ( dJ e ) µ J { σ : ( J, σ ) ∈ A } (cid:21) , J b for b = e , and the second is over J e . This is Z ν ( dJ { e } c ) (cid:20)Z ∞ λ ν ( dJ e ) 1 ν (( −∞ , λ )) Z λ −∞ µ J { σ : ( J, σ ) ∈ A } ν ( dy ) (cid:21) (15) ≥ Z ν ( dJ { e } c ) (cid:20)Z ∞ λ ν ( dJ e ) 1 ν (( −∞ , λ )) Z λ −∞ µ J ( e,y ) { σ : ( J, σ ) ∈ A } ν ( dy ) (cid:21) (19) ≥ Z ν ( dJ { e } c ) (cid:20)Z ∞ λ ν ( dJ e ) 1 ν (( −∞ , λ )) Z λ −∞ µ J ( e,y ) { σ : ( J ( e, y ) , σ ) ∈ A } ν ( dy ) (cid:21) ≥ ν ([ λ, ∞ )) M ( A, J e < λ ) , where the third inequality comes from dropping ν (( −∞ , λ )) − . From this computation, M ( A, J e ≥ λ ) ≥ (1 / { ν ([ λ, ∞ )) M ( A, J e < λ ) + M ( A, J e ≥ λ ) }≥ (1 / ν ([ λ, ∞ )) M ( A ) . The next lemma does not use the monotonicity property, but its proof is similar in spiritto the previous one. Instead of considering coupling values that are far from the criticalvalue, we now consider values that are close. To show that an event of positive probabilityremains of positive probability after bringing the coupling closer to the critical value, weneed to use the fact that by definition, a ground state remains in the support of the uniformmeasure for all values of J e up to the critical value. Lemma 3.5.
Let c < d ∈ R and A ⊆ { ( J, σ ) : σ ∈ G ( J ) , σ e = +1 } ⊆ Ω × Ω be such thatIf ( J, σ ) ∈ A and J e ≥ c then ( J ( e, y ) , σ ) ∈ A for all y ≥ c . (21) Then for all d > c , M ( A, J e ∈ [ c, d ]) ≥ ν ([ c, d ]) M ( A, J e ≥ c ) . Proof.
From the second condition, for a fixed J with J e ≥ c , ♯ { σ : ( J, σ ) ∈ A } ≤ ♯ { σ : ( J ( e, y ) , σ ) ∈ A } for all y ≥ c . Since µ J is the uniform measure and A ⊆ { ( J, σ ) : σ ∈ G ( J ) } , this implies ν -almost surely µ J { σ : ( J, σ ) ∈ A } ≤ µ J ( e,y ) { σ : ( J ( e, y ) , σ ) ∈ A } for all y ≥ c . Therefore M ( A, J e ≥ c ) equals Z ν ( dJ { e } c ) Z ∞ c ν ( dJ e ) 1 ν ([ c, d ]) (cid:20)Z dc µ J { σ : ( J, σ ) ∈ A } ν ( dy ) (cid:21) (21) ≤ Z ν ( dJ { e } c ) Z ∞ c ν ( dJ e ) 1 ν ([ c, d ]) (cid:20)Z dc µ J ( e,y ) { ( σ, σ ′ ) : ( J ( e, y ) , σ, σ ′ ) ∈ A } ν ( dy ) (cid:21) = ν ([ c, ∞ )) ν ([ c, d ]) Z ν ( dJ { e } c ) Z dc µ J ( e,y ) { ( σ, σ ′ ) : ( J ( e, y ) , σ, σ ′ ) ∈ A } ν ( dy ) , which is smaller than M ( A, J e ∈ [ c,d ]) ν ([ c,d ]) . This implies the lemma.16 .2 Properties of the interface We now turn to properties of the interface σ ∆ σ ′ under the measure M = ν ( dJ ) µ J × µ J . The main result of this section is that if σ ∆ σ ′ is not empty, then it can be made tocontain any fixed edge of the graph with positive probability. A similar statement has beenproved in [1, Corollary 2.9] for the metastate measure on ground states. The conclusion isstraightforward by translation invariance in the case G = Z . A different approach is neededfor the half-plane G = Z × N . For the sake of simplicity, we prove the statement in the casethat the graph is planar and each face has four edges. The general statement for a graph G = ( V, E ) with finite degree can be proved the same way.
Proposition 3.6.
If there exists an edge e ∈ E such that M { ( J, σ, σ ′ ) : e ∈ σ ∆ σ ′ } > , thenfor any edge b ∈ E , M { ( J, σ, σ ′ ) : b ∈ σ ∆ σ ′ } > . Before turning to the proof, we record a fact: if σ and σ ′ are spin configurations then acycle (in particular, a face) of the graph cannot have an odd number of edges in σ ∆ σ ′ . Thisis a direct consequence of the following elementary lemma; see for example Theorem 1 in [2]. Lemma 3.7.
For any finite cycle C in the graph G , the parity of { e ∈ C : J e < } equalsthe parity of { e ∈ C : σ e = sgnJ e } . The following lemma interprets the event that an edge is in the interface in terms of thecritical values of e in the two ground states. Lemma 3.8.
For any edge e , M { ( J, σ, σ ′ ) : e ∈ σ ∆ σ ′ } > if and only if M { ( J, σ, σ ′ ) : C e ( J, σ ) = C e ( J, σ ′ ) } > .Proof. = ⇒ . By assumption, M { ( J, σ, σ ′ ) : σ e = +1 , σ ′ e = − } > . By (5), σ ∈ G ( J ) and σ e = +1 together imply that J e ≥ C e ( J, σ ). Similarly, σ ′ ∈ G ( J ) and σ ′ e = − J e ≤ C e ( J, σ ′ ). Therefore M { ( J, σ, σ ′ ) : σ e = +1 , σ ′ e = − , C e ( J, σ ) ≤ J e ≤ C e ( J, σ ′ ) } > . (22)To complete the proof, observe that Proposition 3.1 implies M { ( J, σ, σ ′ ) : C e ( J, σ ) = J e or C e ( J, σ ′ ) = J e } = 0 . ⇐ =. We may assume that with positive probability, on the event { C e ( J, σ ) = C e ( J, σ ′ ) } , σ and σ ′ have the same sign at e . Without loss of generality, taking σ e = σ ′ e = +1, M { ( J, σ, σ ′ ) : σ e = σ ′ e = +1 , C e ( J, σ ) = C e ( J, σ ′ ) } > .
17n particular, there exists a deterministic δ > M { ( J, σ, σ ′ ) : σ e = σ ′ e = +1 , C e ( J, σ ′ ) > C e ( J, σ ) + δ } > . Hence there is a subset of the couplings of positive ν -probability such that on this set µ J × µ J { ( σ, σ ′ ) : σ e = σ ′ e = +1 , C e ( J, σ ′ ) > C e ( J, σ ) + δ } > J e and take ( σ, σ ′ ) in the above event. By (5), we must have J e ≥ C e ( J, σ ) and J e ≥ C e ( J, σ ′ ). From Proposition 2.7, there exists σ ′′ ∈ G − e such that C e ( J, σ ′′ ) ≥ C e ( J, σ ′ ). In particular, by Corollary 2.9, σ ′′ ∈ G ( J ) for J e in the non-emptyinterval ( C e ( J, σ ) , C e ( J, σ ′′ )). Since µ J is supported on a finite number of spin configurations,this implies that on a subset of positive ν -probability µ J × µ J { ( σ, σ ′′ ) : σ e = +1 , σ ′′ e = − } > . Integrating over J completes the proof. Proof of Proposition 3.6.
By Lemma 3.8, it suffices to show that M { ( J, σ, σ ′ ) : C b ( J, σ ) = C b ( J, σ ′ ) } > . (23)Assume that M { ( J, σ, σ ′ ) : σ e = σ ′ e } > . (24)Without loss of generality, we can assume that b and e are edges of the same face. Otherwise,we simply apply the same argument successively on a path of neighboring faces from b to e .Let us denote the other edges of the square face by ˜ b and ˜ e . σ ∆ σ ′ contains e with positive probability. By the paragraph preceding the statement ofthe proposition, if it contains e it must also contain another edge of the face. If it contains b with positive probability we are done, so suppose it contains ˜ e with positive probability.Suppose also that with positive probability e b is not in the interface. The other case is provedthe same way and is simpler. We will indicate how to deal with it at the end of the proof.In our notation, e, ˜ e ∈ σ ∆ σ ′ and b, ˜ b / ∈ σ ∆ σ ′ . Therefore σ e = σ ′ e , σ ˜ e = σ ′ ˜ e , σ b = σ ′ b and σ ˜ b = σ ′ ˜ b on this event. The hypothesis (24) now reduces to M ( B ) > B = { ( σ, σ ′ ) : σ b = σ ′ b , σ ˜ b = σ ′ ˜ b , σ e = σ ′ e , σ ˜ e = σ ′ ˜ e } . By (15), for any J such that µ J × µ J ( B ∩ { σ : σ ˜ b = +1 } ) >
0, if J ′ is a configuration with J ′ ˜ b > J ˜ b , and J ′ a = J a for a = ˜ b , then µ J ′ × µ J ′ ( B ) >
0. Similarly, for any J such that µ J × µ J ( B ∩ { σ : σ ˜ b = − } ) >
0, if J ′ is a configuration with J ′ ˜ b < J ˜ b , and J ′ a = J a for a = ˜ b ,then µ J ′ × µ J ′ ( B ) >
0. In particular, this implies that if x is one of the endpoints of ˜ b thatis not also an endpoint of b , Z { J : | J ˜ b | > S x ˜ b } ν ( dJ ) µ J × µ J ( B ) > .
18e show that Z { J : | J ˜ b | > S x ˜ b } ν ( dJ ) µ J × µ J ( B ∩ { ( σ, σ ′ ) : C b ( J, σ ) = C b ( J, σ ′ ) } ) > , (25)thereby proving (23) and the proposition.The expression for the critical value C b ( J, σ ) can be written as follows. Let F = { b, ˜ b, e, ˜ e } .For I a non-empty subset of { ˜ b, ˜ e, e } , write I b,I for the collection of finite sets of vertices A whose boundary ∂A intersected with F equals the union of { b } with I . This collection mightbe empty for some choice of I . We restrict only to sets I for which I b,I is not empty. Let C b,I ( J, σ ) = sup A ∈I b,I − X { x,y }∈ ∂A { x,y } / ∈ F J xy σ x σ y . In this notation, the expression (6) becomes C b ( J, σ ) = max I ⊆ F \{ b } (X c ∈ I − J c σ c + C b,I ( J, σ ) ) . Let Λ ∈ CD b ( σ ), Λ ′ ∈ CD b ( σ ′ ) and note that both ∂ Λ and ∂ Λ ′ must contain at least oneedge of the face other than b . When | J ˜ b | > S x ˜ b , Corollary 2.10 gives that neither can contain˜ b , so they must both contain b and other edges in { e, ˜ e } . Therefore on this event, the abovedefinition of the critical values reduces to C b ( J, σ ) = max I ⊆{ e, ˜ e } (X c ∈ I − J c σ c + C b,I ( J, σ ) ) . Since the max is attained, it holds on the event { J : | J ˜ b | > S x ˜ b } that µ J × µ J (cid:16) B ∩ { C b ( J, σ ) = C b ( J, σ ′ ) } (cid:17) ≤ X I ⊆{ e, ˜ e } ,I ′ ⊆{ e, ˜ e } µ J × µ J (X c ∈ I − J c σ c + C b,I ( J, σ ) = X c ′ ∈ I ′ − J c ′ σ ′ c ′ + C b,I ′ ( J, σ ′ ) ) . The right-hand side is the same as X I ⊆{ e, ˜ e } ,I ′ ⊆{ e, ˜ e } µ J × µ J n C b,I ( J, σ ) − C b,I ′ ( J, σ ′ ) = X c ∈ I J c σ c − X c ′ ∈ I ′ J c ′ σ ′ c ′ o . The right-hand side of the equality in the event is a linear combination of the J c ’s, c ∈ I ∪ I ′ ,where the coefficients, which we call s c , can only take the values 0 , ± , ±
2. Most importantly,for each choice of
I, I ′ , the s c ’s cannot all be zero since I and I ′ are not empty, and σ c = − σ ′ c c ∈ { e, ˜ e } . Letting J I,I ′ be the set of non-zero { , ± , ± } -valued vectors s , with eachentry corresponding to an element in I ∪ I ′ , we see that the above is smaller than X I ⊆{ e, ˜ e } ,I ′ ⊆{ e, ˜ e } X s ∈J I,I ′ µ J × µ J n C b,I ( J, σ ) − C b,I ′ ( J, σ ′ ) = X c ∈ I ∪ I ′ J c s c o . To show (25), integrate over ν and use Proposition 3.1 with B = { e, ˜ e } and h B c = C b,I ( J, σ ) − C b,I ′ ( J, σ ′ ).This completes the proof in the case that ˜ b is not in the interface. If the probabilityof this is zero (that is, if (25) does not hold), then the proof is easier. We do not need tosupersatisfy J ˜ b ; we simply take I, I ′ to be subsets of { ˜ b, e, ˜ e } and complete the proof fromafter equation (25).Before turning to the proof of the main result, we mention that in the case that thegraph is invariant under a set of transformations (for example, translations), the uniformmeasure inherits a covariance property. Translation-covariant measures on ground statesare typically not easy to construct. The only other example known to the authors is themetastate on ground states constructed from suitable boundary conditions. An advantageof a translation-covariant measure is that the corresponding ν -averaged measure is preservedunder translations. Lemma 3.9.
Let G = Z d or G = Z × N and suppose |G ( J ) | < ∞ . The uniform measure µ J is translation-covariant. That is, if T is a translation of Z d or a horizontal translationof Z × N , then for any B ∈ F , µ T J ( B ) = µ J { σ : T σ ∈ B } for ν -almost all J .In particular, the measure M on Ω × Ω (or on Ω × Ω × Ω ) is translation-invariant.Proof. Using the fact that |G ( J ) | is constant ν -almost surely, one gets µ T J ( B ) = { σ ∈ G ( T J ) : σ ∈ B }|G ( T J ) | = { σ ∈ G ( J ) : T σ ∈ B }|G ( J ) | = µ J { σ : T σ ∈ B } . For the second assertion, let B ′ ⊂ R E × {− , +1 } V . Define T − B ′ = { ( J, σ ) : (
T J, T σ ) ∈ B ′ } . Then the first claim implies that the probability of T − B ′ is M ( T − B ′ ) = Z ν ( dJ ) µ J { σ : ( T J, T σ ) ∈ B ′ } = Z ν ( dJ ) µ T J { σ : ( T J, σ ) ∈ B ′ } . As ν is translation-invariant, we may replace ν ( dJ ) by ν ( dT J ) on the right side. The rightside then equals R ν ( dJ ) µ J ( σ : ( J, σ ) ∈ B ′ ) = M ( B ′ ) as claimed.20 The main result on the half-plane
In this section, we consider the EA model on the half-plane H = Z × N with free boundaryconditions at the bottom. Recall from Corollary 2.3 that the number of ground states |G ( J ) | is non-random. We continue to assume that |G ( J ) | < ∞ . Write M = ν ( dJ ) × ( µ J × µ J ) , where µ J is the uniform measure on G ( J ). We will use the notation that sampling from M amounts to obtaining a triple ( J, σ, σ ′ ) from the spaceΩ := R E H × {− , +1 } V H × {− , +1 } V H , where E H and V H denote the edges and vertices of the half-plane respectively. To showTheorem 1.1, it is sufficient to prove that M { ( J, σ, σ ′ ) : σ ∆ σ ′ = ∅} = 0. This implies that if |G ( J ) | < ∞ , then |G ( J ) | = 2. We will derive a contradiction from the following:assume that M { ( J, σ, σ ′ ) : σ ∆ σ ′ = ∅} > . (26)For this purpose, a representation of the interface σ ∆ σ ′ in the dual lattice will be used.Instead of thinking of an edge e as being in the interface, we think of the dual edge crossing e as being in it. We denote this dual edge by e ∗ . The interface represented this way is acollection of paths in the dual lattice. The reader is referred to Figure 1 for an illustrationof this representation. Note that these dual paths cannot contain loops; otherwise, σ or σ ′ would violate the ground state property (2). Moreover, it is elementary to see that theinterface cannot have dangling ends – dual vertices with degree one in the interface (forexample, using Lemma 3.7). A domain wall refers to a connected component of σ ∆ σ ′ ,viewed as edges in the dual lattice. In the case of the half-plane G = Z × N , we call anydomain wall that crosses the x -axis a tethered domain wall .The method used to derive a contradiction is similar in spirit to the one in [1]. From M we construct a measure on ground states in Z (denoted by f M ) with two contradictingproperties: on the one hand any interface sampled from f M must be disconnected; on the otherhand it must be connected. The construction of f M is outlined below and some propertiesare proved. The proof of non-connectivity is given in Section 4.2. The proof of connectivityfollows the method of Newman & Stein [13] and is in Section 4.3.The first step is to extend the measure M to include the critical values. This extensionis needed because the critical values are not continuous functions of ( J, σ ) in the producttopology; they depend on the couplings in a non-local manner, as can be seen from theformula (6). Therefore their distribution is not automatically preserved under weak limits.Enlarging the probability space to include them will bypass this obstacle. For illustration,consider the event that a fixed edge e has C e ( J, σ ) ∈ I , for some fixed open interval I . Theprobability of this event is not necessarily preserved under weak limits. However, after we21igure 1: An example of an interface between ground states on the half-plane. The edgesin σ ∆ σ ′ are the thick ones. The representation of the interface as dual paths is depicted bythe dotted lines. In this example, there are two domain walls and they are both tethered.include the variables C e ( J, σ ) in our space, this event becomes a cylinder event and thereforeits probability will behave nicely after taking limits.We remark that a different type of extension (but with the same spirit) was done in[1]. Namely, a measure called the excitation metastate (introduced first in [13]) was definedto include the critical values but also all information about local changes of the couplings.Implementing this type of construction turns out to be more delicate in the case of the uni-form measure. We therefore abandon it and turn to a simpler framework. The monotonicityproperty defined in Section 3.1 is the key tool for this approach.For a fixed J , edge e , and σ ∈ G ( J ), recall the definition of the critical value C e ( J, σ )from Lemma 2.6. Define the mapΦ by (
J, σ, σ ′ ) ( J, σ, σ ′ , { C e ( J, σ ) } e , { C e ( J, σ ′ ) } e ) , (27)where the last two coordinates are the collections of critical values of all edges. (This mapis only defined for σ, σ ′ ∈ G ( J ) but this does not create a problem because the support of µ J × µ J is equal to G ( J ) × G ( J ).) Let M ∗ be the push-forward of M by Φ on the spaceΩ ∗ := R E H × {− , +1 } V H × {− , +1 } V H × R E H × R E H . (28)Sampling from M ∗ amounts to obtaining a configuration ω = ( J, σ, σ ′ , { C e } e , { C ′ e } e ) ∈ Ω ∗ . We have not indicated the dependence of C e on σ and J , for example, because on Ω ∗ , it isno longer a function of the other variables. Note that the marginal of M ∗ on ( J, σ, σ ′ ) is M .22e now construct a translation-invariant measure f M on e Ω = R E Z × {− , +1 } Z × {− , +1 } Z × R E Z × R E Z . from the measure M ∗ using a standard procedure. An event in e Ω that only involves, in ameasurable way, a finite number of vertices of Z in σ and σ ′ , and a finite number of edgesthrough the couplings J e and the critical values C e and C ′ e will be called a cylinder event .Let T be the translation of Z that maps the origin to the point (0 , −
1) and for each n ≥ M ∗ n = 1 n + 1 n X k =0 T k M ∗ . (29)Note that the translated measure T k M ∗ is well-defined on cylinder events for k large enough.(If it is not defined, we can take it to be zero without affecting the limit below.) Moreover,the sequence of measures M ∗ n is tight. This is obvious for the marginal on ( J, σ, σ ′ ). The factthat it holds also when including the critical values is a direct consequence of Corollary 2.8.Therefore there exists a sequence ( n k ) such that M ∗ n k converges as k → ∞ , in the sense offinite-dimensional distributions, to a translation invariant measure on e Ω. Call this limitingmeasure f M . The weak convergence of the measures M ∗ n to f M implies that for any event B in e Ω lim inf n →∞ M ∗ n ( B ) ≥ f M ( B ) if B is open;lim sup n →∞ M ∗ n ( B ) ≤ f M ( B ) if B is closed;lim n →∞ M ∗ n ( B ) = f M ( B ) if f M ( ∂B ) = 0 . (30)(See, for example, Theorem 4.25 of [8].) Here we are using the fact that e Ω is metrizable,as these statements are true in general for probability measures on metric spaces. Theboundary ∂B is the closure of B minus its interior in e Ω (not to be confused with ∂A for A a finite set of vertices in the graph). Examples of open (resp. closed) cylinder sets are { h ( J, { C e } e , { C ′ e } e ) ∈ O } where h is a continuous function depending only on a finite numberof edges, and O is an open (resp. closed) set of R . Remark 1.
Note that if B only depends on the spins of a finite number of vertices andnot on the couplings and critical values, actual convergence of the probability holds, since B is open and closed thus ∂B = ∅ . This same conclusion is true if B is an event of theform { ( σ, σ ′ ) ∈ D, J ∈ I } for events D that depend on finitely many spins and sets I insome finite dimensional Euclidean space with boundary of zero Lebesgue measure. Indeed,it is a general fact that for any two events B and B ′ , ∂ ( B ∩ B ′ ) ⊆ ∂B ∪ ∂B ′ ; therefore ∂B ⊆ ∂ { ( σ, σ ′ ) ∈ D } ∪ ∂ { J ∈ I } . It follows that the set ∂B has f M -measure zero, since f M ( ∂ { J ∈ I } ) = ν ( ∂ { J ∈ I } ) = 0 (by the continuity of ν ) and f M ( ∂ { ( σ, σ ′ ) ∈ D } ) = f M ( ∅ ) = 0 . Since f M will be our object of study for the remainder of the paper, we will spend sometime explaining its basic properties. Suppose ω = ( J, σ, σ ′ , { C e } e , { C ′ e } e ) is sampled from f M .23irst, it follows directly from the construction that σ and σ ′ are almost-surely ground stateson Z . Also if we define F e = | J e − C e | and F ′ e = | J e − C ′ e | to be the flexibility of the edge e in σ and in σ ′ , then for any finite set A with e ∈ ∂A , F e ≤ X { x,y }∈ ∂A J xy σ x σ y f M -a.s. (31)and similarly for F ′ e . This is true because this relation holds with M ∗ -probability one on thespace Ω ∗ and for its translates by k (for k large enough that A ⊆ T k V H ) by (7). Moreover,both sides are continuous functions of ω . Thus the ω ’s satisfying the relation (31) form aclosed set. Equation (31) then follows from (30). It remains to take the infimum over all(countably many) finite sets A to conclude the following lemma. Lemma 4.1.
Let I e ( J, σ ) := inf A : e ∈ ∂AA finite P { x,y }∈ ∂A J xy σ x σ y . For any edge e , f M { F e ≤ I e ( J, σ ) } = 1 . The corresponding statement holds for σ ′ . In other words, flexibilities produced by the weak limit procedure from half-planes are nobigger than the ones computed directly from (7) in the full plane. This is to be expected sincethe former also take into account sets A that touch the boundaries of some translated half-planes. The last basic property we need is a result analogous to Proposition 3.1 (specificallythe consequence of that proposition that M ( C e = J e ) = 0) for the weak limit f M . Lemma 4.2.
For any edge e , f M { F e = 0 } = 0 . The corresponding statement holds for F ′ e .Proof. It suffices to prove the statement for F e . Because { F e = 0 } is not an open set, wecannot simply take limits in Proposition 3.1 to obtain the result. Consider the cylinder event {| J e − C e | < ε, | J e | < N } for ε > N >
0. Note that this set is open. (The cutoff in J e seems superfluous first but is useful in the estimate below.) The conclusion will follow from(30) once we show that for each fixed N , T k M {| J e − C e ( J, σ ) | < ε, | J e | < N } (32)can be made arbitrarily small uniformly in k (for k such that e ∈ T k E H ) by taking ε small.We prove the estimate for k = 0 only. It will be clear that the same proof holds for any k . Using the monotonicity (15) and the notation J ( e, s ) of (14), we have M ( | J e − C e ( J, σ ) | < ε, J e < N, σ e = +1)= Z ν ( dJ { e } c ) Z N −∞ ν ( dJ e ) 1 ν ( J e , ∞ ) Z ∞ J e ν ( ds ) µ J { σ : | J e − C e ( J, σ ) | < ε, σ e = +1 }≤ ν ( N, ∞ ) Z ν ( dJ { e } c ) Z ν ( dJ e ) Z ν ( ds ) µ J ( e,s ) { σ : | J e − C e ( J, σ ) | < ε, σ e = +1 }
24e now exchange integrals using Fubini and integrate over J e first to get the upper bound1 ν ( N, ∞ ) Z ν ( dJ { e } c ) Z ν ( ds ) Z µ J ( e,s ) ( dσ ) ν { J e : | J e − C e ( J, σ ) | < ε } . Recall that C e ( J, σ ) does not depend on J e . The interval { J e : | J e − C e ( J, σ ) | < ε } haslength 2 ε , hence given δ >
0, its ν -probability can be made smaller than δ , independently of C e ( J, σ ), by the continuity of ν . We have thus shown M ( | J e − C e ( J, σ ) | < ε, J e < N, σ e = +1) ≤ δν ( N, ∞ )Repeating the same proof, but using monotonicity in the other direction and taking J e > − N , M ( | J e − C e ( J, σ ) | < ε, J e > − N, σ e = − ≤ δν ( N, ∞ ) . This estimate holds for any k and (32) can be made uniformly small by taking ε small. In this section we show
Proposition 4.3. If (26) holds, then f M { σ ∆ σ ′ is not connected } > . The first key ingredient is to show that with positive M -probability, there are infinitelymany tethered domain walls in the interface on the half-plane. Lemma 4.4. If (26) holds, then with positive M -probability, σ ∆ σ ′ crosses the x -axis. More-over, with positive M -probability, σ ∆ σ ′ has infinitely many domain walls.Proof. The first claim is a direct application of Proposition 3.6. For the second, note thata connected component of σ ∆ σ ′ cannot cross the x -axis twice. If it did, it would containa dual path whose union with the x -axis encloses a finite set of vertices S . We must have P { x,y }∈ ∂S J xy σ x σ y ≥ σ ′ by (2). Since σ x σ y = − σ ′ x σ ′ y on ∂S , we conclude P { x,y }∈ ∂S J xy σ x σ y = 0, and this has probability zero by the continuity of ν . Therefore toeach dual edge crossing the x -axis contained in σ ∆ σ ′ , there corresponds a unique connectedcomponent of σ ∆ σ ′ . By horizontal translation-invariance of M (Lemma 3.9), if σ ∆ σ ′ containsone such dual edge, it must contain infinitely many. This gives the second claim.The next step is to prove that distinct connected components sampled from M do notdisappear after constructing f M . This is done by showing that the expected number ofcomponents intersecting a fixed box is uniformly bounded below in k . This is the content ofthe next lemma. We omit the proof; it is exactly the same as that of [1, Proposition 3.4].For any k ≥ n ≥
1, let I n,k = [ − n, n ] × { k } and let N n,k be the number of distinct tethered domain walls that cross the line segment I n,k . Write E M for the expectation with respect to M .25 emma 4.5. For fixed k ≥ , the sequence ( E M N n,k ) n is sub-additive. Therefore lim n →∞ (1 /n ) E M N n,k exists . Furthermore if (26) holds then there exists c > such that for all k ≥ and n ≥ , E M N n,k ≥ cn . This lemma yields Proposition 4.3. We omit the proof as it is identical to [1, Proposition 3.5].The proof there only deals with cylinder events involving only spins, and therefore limits gothrough using Remark 1.
In this section, we show
Proposition 4.6. f M ( σ ∆ σ ′ is not connected ) = 0 . This contradicts Proposition 4.3 and finishes the proof of Theorem 1.1. We will apply theNewman-Stein technique from [13]. The idea is to construct a random variable I (see below)that is defined on the event { σ ∆ σ ′ is not connected } . Proposition 4.6 will follow from both f M { I ≤ , σ ∆ σ ′ is not connected } = 0 (33)and f M { I > , σ ∆ σ ′ is not connected } = 0 . (34) I We first need information about the topology of interfaces σ ∆ σ ′ sampled from f M . This isthe content of the following proposition, which is analogous to Theorem 1 in [13]. The proofof part 1 relies on translation invariance and part 2 is a consequence of Lemma 3.7. Theproof of part 3 uses ideas of Burton & Keane [4]. Proposition 4.7.
With f M probability one, the following statements hold.1. If σ ∆ σ ′ is nonempty, then it has positive density.2. If σ ∆ σ ′ is nonempty, then it does not contain any dangling ends or three-branchingpoints.3. If σ ∆ σ ′ is nonempty, then it contains no four-branching points. In particular, eachdual vertex in the domain wall has degree two; thus each domain wall is a doublyinfinite dual path. Moreover, each component of the complement (in R ) of σ ∆ σ ′ isunbounded and has no more than two topological ends in the following sense. If C issuch a component then for all bounded subsets B of R , the set C \ B does not havemore than two unbounded components. Definition 4.8. A rung is a non-self intersecting finite dual path that starts at a dual vertexin a domain wall and ends at a dual vertex in a different domain wall. No other dual verticeson the path are in a domain wall. Let h = h ( ω ) be the first horizontal edge in the interface starting from the origin to theright. For almost every configuration ω such that the interface is nonempty, such an h existsbecause of translation and rotation invariance of f M . So we can define I = inf R E ( R ) , where the infimum is over all rungs R touching the domain wall of h ∗ and E ( R ) is the energy: E ( R ) = X { x,y } ∗ ∈ R J xy σ x σ y . See Figure 2 for a depiction of h and a rung under consideration. R h O Figure 2: An example of rung from the domain wall of h to another domain wall.Note that since no edge of a rung is in the interface σ ∆ σ ′ , we must have σ x σ y = σ ′ x σ ′ y forall edges { x, y } ∈ R . Therefore in the definition of E ( R ) it does not matter if we choose σ or σ ′ to perform the computation. 27 .3.2 I ≤ has zero probability We will now assume that f M { I ≤ , σ ∆ σ ′ is not connected } > , (35)and derive a contradiction. For a dual edge e ∗ , ε > K , let A e ( ε, K )be the event that (a) e ∗ is in a rung (between any two domain walls) with energy less than ε and (b) this rung has length (number of dual edges) at most K . Whenever I ≤ ε > h ∗ with energy less than ε . So A e ( ε, K ) occurs for some e and K , and under (35), there exists ε > K such that X e ∈ E f M ( A e ( ε, K )) ≥ f M ( ∪ e ∈ E A e ( ε, K )) > . By translation invariance, f M ( A e ( ε, K )) > e .Let us say that dual edges e ∗ and e ∗ are on the same side of a domain wall D if theyboth have a dual endpoint in the same connected component of the complement of D . Thefollowing lemma is the same as Lemma 1 in [13]. Lemma 4.9.
With f M -probability one, the following holds. If σ ∆ σ ′ is not connected, thenfor each domain wall D , either there are infinitely many dual edges e ∗ touching D such that A e ( ε, K ) occurs (in both directions along D and on each side of D ) or there are zero.Proof. For an edge e , let B e ( ε, K ) be the event that (a) A e ( ε, K ) occurs and (b) there existsa domain wall D such that e ∗ touches D and in at least one direction on D , there are noendpoints of dual edges h ∗ for which A h ( ε, K ) occurs for the same domain wall D on thesame side. For each e such that B e ( ε, K ) occurs we may associate e to a domain wall D .Note that in each realization ω in the support of f M , there are at most 4 edges associatedwith each domain wall (counting two directions and two sides of the domain wall).Let B ( n ) be the box of side length n centered at the origin, and let N n be the number ofdomain walls which have a dual vertex in B ( n ). Last, let us use the notation that e ∈ B ( n )if both of e ’s endpoints are in B ( n ). The above arguments imply that X e ∈ B ( n ) f M ( B e ( ε, K )) = E f M X e ∈ B ( n ) B e ( ε, K )) ≤ E f M N n . Here E f M stands for expectation with respect to f M . Distinct domain walls do not intersectso we can associate to each dual edge of the outer edge boundary ∂ e B ( n ) (that is, havingone endpoint in B ( n ) and one in B ( n ) c ) at most one domain wall that contains it. Thereforefor some suitable constants C , C > | B ( n ) | X e ∈ B ( n ) f M ( B e ( ε, K )) ≤ C | B ( n ) | | ∂ e B ( n ) | ≤ C | B ( n ) | − / → n → ∞ . By translation invariance, f M ( B e ( ε, K )) is the same for all e and thus equals 0,completing the proof. Remark 2.
Although the previous lemma was stated for the events A e ( ε, K ) , the same proofcan be used for a number of different events like A e ( ε, K ) . In [13], these events were called“geometrically defined.” Examples of such events are (a) the event that e ∗ is in a domainwall and is adjacent to a rung with a specified energy and (b) the event that e ∗ is in a domainwall and has a specified flexibility in σ or σ ′ . We will use these facts later in Section 4.3.3.Note that it is not enough to use only translation-invariance in the proof, as we would needto use (random) translations along a domain wall.Proof of (33) . For an edge e , ε > K , let A e ( ε, K ) be the event that A e ( ε, K ) occurs and one of the endpoints of e ∗ is in the domain wall of h ∗ . If I ≤
0, thenfor each ε there exists K such that A e ( ε, K ) occurs. By Lemma 4.9, we may find infinitelymany dual edges e ∗ n and f ∗ n (in both directions along the domain wall of h ∗ but on the sameside as e ∗ ) such that A e n ( ε, K ) and A f n ( ε, K ) occur. The e n ’s are chosen in one directionand the f n ’s in the other. Let R n be a rung corresponding to e n and let S n correspondingto f n . By relabeling the sequences ( e n ) and ( f n ) we may ensure that R n does not intersect S n for any n . (Here we are using the fact that the rungs have length at most K and so fora fixed n , there are finitely many n ’s such that R n intersects S n .) Calling D the domainwall containing h ∗ , both rungs S n and R n connect D to the same domain wall, say, D .Since R n and S n are disjoint, the dual path P consisting of R n , S n , the piece of D between e ∗ n and f ∗ n (call it P ) and the corresponding piece of D between the intersectionpoints of R n and S n with D (call it P ) is a circuit in the dual lattice. See Figure 4.3.2 fora depiction. The spin configurations σ and σ ′ sampled from f M are ground states, hence X { x,y } ∗ ∈ P J xy σ x σ y ≥ X { x,y } ∗ ∈ P J xy σ ′ x σ ′ y ≥ . For each edge { x, y } whose dual edge is in either R n or S n , we have σ x σ y = σ ′ x σ ′ y . For eachedge { x, y } whose dual edge is on either P or P we have σ x σ y = − σ ′ x σ ′ y . Using the factthat the energies of the rungs R n and S n are below ε , the above two inequalities reduce to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X { x,y } ∗ ∈ P J xy σ x σ y + X { x,y } ∗ ∈ P J xy σ x σ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε , and so P { x,y } ∗ ∈ P J xy σ x σ y < ε . As ε is arbitrary, the edge h has flexibility zero by Lemma 4.1(since I h = 0). By Lemma 4.2, this has zero probability, proving (33). I > has zero probability We now show (34) by assuming f M { I > , σ ∆ σ ′ is not connected } > . (36)29 RR n RS n P P n f n Figure 3: The rungs R n and S n form a circuit in the dual lattice together with the shadeddual paths P and P of the domain walls.and deriving a contradiction. The idea is that if I > I = I ( ω ) to emphasize the dependence of I on the configuration ω ∈ e Ω.For each edge e let e F e := min { F e , F ′ e } . By Lemma 4.2, f M ( e F e > e ) = 1 . (37)The rest of this subsection will serve to prove the following proposition. Fix ε > f be the edge connecting (1 ,
0) and (1 , g to be the edge connecting theorigin to (1 , X ε be the intersection of the following events:1. σ ∆ σ ′ is disconnected and I > g ∗ ∈ σ ∆ σ ′ ;3. f ∗ is in a rung R that satisfies E ( R ) < I ( ω ) + ε/ X ε , the edge h (used in the definition of I in the previous section) equals g . Proposition 4.10. If (36) holds, there exists ε such that for all but countably many <ε < ε , f M (cid:16) X ε , e F f > ε (cid:17) > . roof. We begin by finding deterministic replacements for many local quantities. Let E be the event that g ∗ ∈ σ ∆ σ ′ , f ∗ / ∈ σ ∆ σ ′ , σ ∆ σ ′ is disconnected and I >
0. By translationinvariance and by the assumption (36), we have f M ( E ) >
0. We denote the domain wall of g ∗ by D ( ω ) for ω ∈ E . By (37), we may choose ε > < ε < ε , f M ( E , there exists e ∗ ∈ D ( ω ) such that e F e > ε ) > . Furthermore, note that the distribution of e F e (for any edge e ) under the measure f M canonly have countably many atoms. We fix any such 0 < ε < ε in the complement of this setfor the rest of the proof, so that f M ( e F e = ε for some e ) = 0 . (38)Let E = E ∩ {∃ e ∗ ∈ D ( ω ) such that e F e > ε } .If ω ∈ E , we may find a rung R ( ω ) touching D ( ω ) such that E ( R ( ω )) < I ( ω ) + ε/ . (39)This is by the definition of I ( ω ). Let f ∗ ( ω ) be the dual edge in R ( ω ) that touches D ( ω ).There are countably many choices for f ∗ ( ω ), so we may find a deterministic e f ∗ such that f M ( E , f ∗ ( ω ) = e f ∗ ) > . In fact, by rotation and translation invariance we can take e f ∗ to be the fixed dual edge f ∗ : f M ( E , f ∗ ( ω ) = f ∗ ) > . By an argument identical to that given in Lemma 4.9, for f M -almost all ω ∈ E , there areinfinitely many dual edges e ∗ ∈ D ( ω ) (in both directions along D ( ω )) for which e F e > ε .(See Remark 2.) Therefore, for f M -almost every ω ∈ E , we may find dual edges e ∗ ( ω ) and e ∗ ( ω ) on D ( ω ) such that the piece of D ( ω ) from e ∗ ( ω ) to e ∗ ( ω ) contains g ∗ and such that e F e and e F e are bigger than ε . For any N , let B (0; N ) be the box of side length N centeredat the origin and for a spin configuration σ , let σ N be the restriction to B (0; N ). There areonly countably many choices, so we may find deterministic values of e , e , N, σ N , σ ′ N and R (whose first dual edge is f ∗ ) such that with positive f M -probability on E :1. B (0; N/
2) contains R , e ∗ , e ∗ and the piece of D ( ω ) between e ∗ and e ∗ ;2. σ ( ω ) (cid:12)(cid:12)(cid:12) B (0; N ) = σ N , σ ′ ( ω ) (cid:12)(cid:12)(cid:12) B (0; N ) = σ ′ N , e F e > ε, e F e > ε ;3. R is a rung with E ( R ) < I ( ω ) + ε/
2. 31all E the set of configurations satisfying the three above conditions. By construction, f M ( E ∩ { I > } ) >
0. Note that by the choice of σ N and σ ′ N , their interface contains g ∗ , e ∗ and e ∗ (and they are all connected through a single domain wall in B (0; N )), but theinterface does not contain f ∗ . In addition, if E occurs then R is a rung, and σ ∆ σ ′ must bedisconnected. Therefore E contains E ∩ { I > } . The same arguments also show that X ε ⊇ E ∩ { I > } . (40)Now, write D for the (deterministic) set of edges in B (0; N ) that are in σ N ∆ σ ′ N and canbe connected to g ∗ by a path of dual edges in σ N ∆ σ ′ N that stay in B (0; N ). This is just theconnected “piece” of D ( ω ) in B (0; N ) for configurations ω ∈ E . Let f ∗ , . . . , f ∗ n be the dualedges with both endpoints in B (0; N ) that are (a) incident to D , (b) not in σ N ∆ σ ′ N and (c)not equal to f . A depiction of these definitions is given in Figure 4. e e f R g O Figure 4: Depiction of definitions on the event E . The two domain walls are the dual dottedlines. The rung R is the thick path between the two domain walls. The edges f , f , . . . , f n along the domain wall are the grey edges.We claim that we can order the f ∗ i ’s so that for each i = 1 , . . . , n − f i has an endpoint x i that does not touch any edge from the set { f i +1 , . . . , f n , f } (note here we are consideringedges, not dual edges). To explain why this is true, we consider the graph whose edge setis equal to the union of the f i ’s (in the original lattice). Note that if C , . . . , C p are thecomponents of this graph then it suffices to give an ordering of each component and thenconcatenate these orderings together. So we may consider just one component, say, C . Wewill choose the edges g , . . . , g k of C in reverse order, so that our final ordering of C will32e g k , . . . , g . The desired condition on the f i ’s becomes the following for the g i ’s: for each i = 1 , . . . , k , g i has an endpoint that does not touch any edge from the set { f, g , . . . , g i − } .We now note that the graph whose edges are f, f , . . . , f n does not contain any cycles.If there were a cycle then it would force the interface σ ∆ σ ′ in the dual graph to have onetoo, which is impossible. Therefore the component C above can have at most one edge thattouches f . If there is such an edge, we let g be it; otherwise, we choose g arbitrarily in C .We now add edges in steps: at each step j ≥ G j be the current connected subgraphof C (that is, the graph whose edges are { g , . . . , g j − } ) and add g j to our collection ofedges so that it connects G j to its complement. This is always possible because C doesnot contain a cycle. We finish at step k with the desired ordering of the g j ’s, which, whenreversed, gives the desired ordering of C .We claim f M ( E , ∩ ni =1 {| J f i | > S x i f i } ) > , (41)where S x i f i is the super-satisfied value of the edge f i defined in (12). Essentially, the claimmeans that the event E is somewhat stable under modifications of couplings. Equation (41)will be proved in the lemma below. We first show how this implies the claim of the propositionusing Lemma 3.3. Let U be the set of all f i ’s. Note that by construction, for any finite set A such that f ∈ ∂A and ∂A ∩ U = ∅ , we must have e or e in ∂A . Let e G be the event that F f ≥ min { F e , F e } and | J f i | ≥ S x i f i for all f i ∈ U . The probability of e G under any translatesof M is equal to that of e G , which is 1 by Lemma 3.3. On the other hand, e G is a closed eventso f M ( e G ) is no smaller than lim sup k M ∗ k ( e G ) = 1. This implies from (41) f M ( E , e F f > ε ) ≥ f M ( E , e F f > ε, ∩ ni =1 {| J f i | ≥ S x i f i } ) = f M ( E , ∩ ni =1 {| J f i | ≥ S x i f i } ) > . Since X ε ⊇ E ∩ { I > } (and f M ( I = 0) = 0 by (33)), this concludes the proof of Proposi-tion 4.10. Lemma 4.11.
Let E be the event defined above (40) . Define f i and x i , i = 1 , ..., n as above (41) . If f M ( E ) > , then f M ( E , ∩ ni =1 {| J f i | > S x i f i } ) > .Proof. Write S for the set of dual edges in B (0; N ) that are not equal to any of the f ∗ i ’s orto f ∗ . Since f M ( E ) >
0, we can choose λ > f M ( E , | J e | ≤ λ for all e ∗ ∈ S ) > . Write E for this event. We will show that f M ( E , ∩ ni =1 {| J f i | > S x i f i } ) >
0. This will followif we find positive numbers a , . . . , a n , b , . . . , b n such that the following hold:1. a i < b i for all i ;2. a i +1 > b i for i = 0 , , . . . , n − b := λ .3. f M ( E , | J f i | ∈ [ a i , b i ] for all i } ) >
0; 33hese conditions imply that | J f i | > S x i f i for all i , as | J f i | ≥ a i > b i − and 4 b i − > S x i f i . Herewe are using the fact that x i does not touch the set { f, f i +1 , . . . , f n } . For q >
1, define E q := E ∩ {| J i | ∈ [ a i , b i ] for all i = 1 , . . . , q − } and for q = 1 define E q := E . We will proceed by induction to show that if f M ( E q ) > f M ( E q +14 ) > a q , b q , for q = 1 , ..., n −
1. Note that f M ( E ) >
0. Thecase q = n − f q are the same. The subsequent argument is similar in the other case.The idea is to use Lemma 3.4, which shows that the probability mass is somewhat conservedwhen one value of the coupling is increased for events satisfying (19). Two obstacles have tobe overcome. First, the properties of M (in particular, the monotonicity property) neededin Lemma 3.4 do not directly carry over under weak limits to f M . Therefore, we need togo back to M to apply the lemma. Second, weak convergence of the measures applies tocylinder events. Note that, from its definition, E q is an intersection of a finite number ofcylinder events except for { R is a rung with E ( R ) < I ( ω ) + ε/ } . To apply Lemma 3.4, wethus need to find a cylinder approximation for this condition.Let e B R ⊆ e Ω be the event { R is a rung with E ( R ) < I ( ω ) + ε/ } intersected with theevent { g ∗ ∈ σ ∆ σ ′ } . We will first define a double sequence of cylinder events ( e B Rj,l ) in e Ω withlim j →∞ lim sup l →∞ f M ( e B R ∆ e B Rj,l ) = 0 , (42)where ∆ represents the symmetric difference of events.Let B (0; j ) be the box of side-length j centered at 0 and let l ≥ j . For arbitrary spinconfigurations σ and σ ′ , the interface σ ∆ σ ′ splits into different connected components inthe following way. Two dual edges in B (0; j ) ∩ σ ∆ σ ′ (that is, they have both endpoints in B (0; j )) are said to be l -connected if they are connected by a path of dual edges in σ ∆ σ ′ , allof which remain in B (0; l ). Let D ( j, l ) , D ( j, l ) , . . . , D t ( j, l ) be the l -connected componentsof such edges in B (0; j ), where D ( j, l ) is the connected component containing g ∗ (if oneexists). Call these the ( j, l ) -domain walls (see Figure 5). We define a ( j, l ) -rung as a finitepath of dual edges in B (0; j ) which starts in a ( j, l )-domain wall and ends in a differentone, and no dual vertices on the path except for the starting and ending points are on a( j, l )-domain wall.On the event e B R , R is a ( j, l )-rung for all l ≥ j ≥ N . Let e B Rj,l ⊆ e Ω be the event that1. g ∗ ∈ σ ∆ σ ′ ;2. R is a ( j, l )-rung;3. no other ( j, l )-rung between D ( j, l ) and another ( j, l )-domain wall has energy less thanthe energy of R minus ε/ j →∞ lim sup l →∞ f M ( e B R \ e B Rj,l ) = 0 . (43)34 (0,j) B(0,l) Figure 5: In this figure, when we restrict the interface to B (0; j ), there are three components(connected inside this box). However, two of them are l -connected. Therefore, there are two( j, l )-domain walls in B (0; j ).Consider ω ∈ e B R . It suffices to prove that there exists J ( ω ) and for each j ≥ J ( ω ) there isa L ( j, ω ) such that j ≥ J ( ω ) and l ≥ L ( j, ω ) implies ω ∈ e B Rj,l . (44)This implies (43) because if e B R \ e B Rj,l occurs then either j ≤ J ( ω ) or both j ≥ J ( ω ) and l ≤ L ( j, ω ). Therefore the limit in (43) is bounded above bylim j →∞ lim l →∞ h f M ( j ≤ J ( ω )) + f M ( l ≤ L ( j, ω )) i = 0 . Take j ≥ N . Note that there are at most | B (0; j ) | number of ( j, l )-domain walls in B (0; j ).We claim that there exists L ( j ) such that all ( j, l )-rungs are rungs for l ≥ L ( j ). Indeed, if S is a rung then it is plainly a ( j, l )-rung. On the other hand, if S is a ( j, l )-rung, then eitherit connects distinct domain walls in σ ∆ σ ′ or simply two pieces of the same domain wall of σ ∆ σ ′ that are l -connected for l large enough. Now, since ω ∈ e B R , we must have g ∗ ∈ σ ∆ σ ′ .Moreover, R is a rung and so it is also a ( j, l )-rung for any l . By definition of e B R , no rungtouching D ( ω ) can have energy less than the energy of R minus ε/
2. Therefore for l ≥ L ( j ),no ( j, l )-rung can either, and we see that ω ∈ e B Rj,l for J ( ω ) = N and L ( j, ω ) = L ( j ) in (44).To show the other half of (42), it remains to prove thatlim j →∞ lim sup l →∞ f M ( e B Rj,l \ e B R ) = 0 . (45)We claim that if ω / ∈ e B R , there exists J ( ω ) such that for each l ≥ j ≥ J ( ω ), ω / ∈ e B Rj,l aswell. This implies (45) by the same argument as before. Since ω / ∈ e B R , at least one of three35efining conditions of e B R must fail. In each case, we will show that ω cannot be in e B Rj,l forall large j and l . First if g ∗ / ∈ σ ∆ σ ′ then we will never have ω ∈ B Rj,l , so we may assume thecontrary. If R is a ( j, l )-rung for some l ≥ j ≥ N then it connects two ( j, l )-domain walls.As in the previous paragraph, either these ( j, l )-domain walls are in fact distinct domainwalls or they are part of the same domain wall for j and l large enough. This argumentshows that if R is not a rung, there exists J ( ω ) such that it will also not be a ( j, l )-rung for l ≥ j ≥ J ( ω ). Finally, if g ∗ ∈ σ ∆ σ ′ and R is a rung, suppose that there is another rung S touching D ( ω ) with energy less than the energy of R minus ε/
2. Then the same argumentas above shows there exists J ′ ( ω ) such that for l ≥ j ≥ J ′ ( ω ), S will be a ( j, l )-rung withenergy less than E ( R ) − ε/ ω / ∈ e B Rj,l . This proves (45) and thus (42).Recall that the event E q is the intersection of the following:a. the three events that comprise E defined above (40) (the last one of which we canreplace by e B R );b. | J e | ≤ λ for all e ∗ ∈ S ;c. | J f i | ∈ [ a i , b i ] for all i = 1 , . . . , q − E qj,l be the cylinder approximation of E q that is, the event E q where e B R is replaced bythe cylinder event e B Rj,l . Note that E qj,l can be seen as an event in the translated space T k Ωfor k large enough such that the box B (0; l ) is contained in T k V H . Recall that in Ω as wellas in T k Ω, the flexibilities F e and F ′ e are functions of J and σ, σ ′ given by the formula (7).Note also by directly applying (42), we findlim l →∞ lim sup j →∞ f M ( E q ∆ E qj,l ) = 0 . (46)We claim that E qj,l (and T k E qj,l ) has the property (19):If ( J, σ, σ ′ ) ∈ E qj,l then ( J ( f q , s ) , σ, σ ′ ) ∈ E qj,l whenever s ≥ J f q . (47)To check this, we first remark that if | J f i | ∈ [ a i , b i ] for all i = 1 , . . . , q − | J e | ≤ λ forall e ∈ S for ( J, σ, σ ′ ) then this is plainly true for ( J ( f q , s ) , σ, σ ′ ) for any s . This handlesconditions (b) and (c) of E qj,l . To address condition (a), we first note that the event that g ∗ ∈ σ ∆ σ ′ (part of e B Rj,l in the third part of (a)) is unaffected by J f q , so it will continue to hold.In the other two parts of (a), no conditions involve the couplings except for F e > ε , F ′ e > ε .But since the spins at the endpoint of f q are the same, increasing J f q can only possiblyincrease F e and F ′ e as seen from (7). (Note here that F e and F ′ e are simply images under Φ of F e ( J, σ ) and F ′ e ( J, σ ) on Ω or T k Ω, so since this argument is valid on these spaces, it holds asstated on Ω ∗ or T k Ω ∗ .) Finally, to establish (47), it remains to show that if ( J, σ, σ ′ ) ∈ e B Rj,l ,then ( J ( f q , s ) , σ, σ ′ ) ∈ e B Rj,l for s ≥ J f q . Note that because l ≥ j ≥ N, the set D (definedbefore the statement of the present proposition) is contained in the ( j, l )-domain wall of g ∗ and since f ∗ q is adjacent to D , no ( j, l )-rung containing f ∗ can contain f ∗ q . So increasing the36alue of J f q to s can only increase the energies of ( j, l )-rungs that do not contain f ∗ . Thismeans that if no ( j, l )-rungs have energy less than the energy of R minus ε/ J, σ, σ ′ )then the same will be true in ( J ( f q , s ) , σ, σ ′ ) for s ≥ J f q . We have thus proved (47).We are now in a position to use Lemma 3.4. Since T k M is just a translate of M , the lemmaholds for the measure T k M as well, so we conclude that for all a ∈ R and k ≥ l ≥ j ≥ N , T k M ( E qj,l , J f q ≥ a ) ≥ (1 / ν ([ a, ∞ )) T k M ( E qj,l ) . (48)This holds trivially for M replaced by M ∗ , on the space Ω ∗ in (28), where the flexibilities areadded to the coordinates. We would like to take limits in this inequality. For this purpose,the reader may trace through the definition of E qj,l and see that this event is an intersectionof a cylinder event Y involving only spins and couplings and another event Z equal to { e F e > ε and e F e > ε } . The boundary ∂Y is included in the union of ∂ {| J e | ≤ λ ∀ e ∗ ∈ S } , ∂ {| J f i | ∈ [ a i , b i ] : ∀ i = 1 , . . . , q − } , ∂ { g ∗ ∈ σ ∆ σ ′ } , ∂ { R is a ( j, l )-rung } , and the boundaryof the event { no other ( j, l ) -rung between D ( j, l ) and another ( j, l ) -domain wall has energyless than the energy of R minus ε/ } . It is straightforward to see that the first four have f M -probability zero. As for the fifth one, notice that the energy of a ( j, l )-rung is a linearfunction of the couplings in the box B (0; j ) with coefficients +1 or −
1. There are only afinite number of such linear combinations. Therefore, the probability that the difference ofenergy between any two rungs is exactly ε/ ∂Z of f M -probability zero. Therefore by the discussion preceding Remark 1, we havelim k →∞ M ∗ k ( E qj,l ) = f M ( E qj,l ) . A similar argument holds for the left side of (48). Averaging over k and taking limits in thisinequality, we find f M ( E qj,l , J f q ≥ a ) ≥ (1 / ν ([ a, ∞ )) f M ( E qj,l ) , Now we take l → ∞ and j → ∞ , using (46) to obtain f M ( E q , J f q ≥ a ) ≥ (1 / ν ([ a, ∞ )) f M ( E q ) . By the induction hypothesis, f M ( E q ) >
0. To finish the proof of the lemma, it thus sufficesto take a = a q = 4 b q − + 1 and choose any b q > a q . In this subsection we use Proposition 4.10 to prove a final proposition about rung energies.This will allow us to reach a contradiction and establish (34).Recall that f refers to the fixed edge connecting (0 ,
1) to (1 ,
1) and g is the edge connectingthe origin to (1 , J f can be modifiedso that the energy of some rung that contains f ∗ decreases below the energies of all rungsthat do not contain f ∗ . To do this, we introduce two variants of I ( ω ), dealing with rungsthat contain f ∗ and rungs that do not. 37n the event X ε , we define the variable I ′ ( ω ) to be the infimum of energies of all rungsthat touch D ( ω ) (the domain wall that contains h ∗ = g ∗ ) and that do not contain f ∗ . Alsowe define e I ( ω ) to be the infimum of energies of all rungs that contain f ∗ . Later in theproof we will use a small technical fact: the distribution of e I ( ω ) − I ′ ( ω ) (under f M ) can haveonly countably many point masses. Therefore we may choose ε small enough so that theconclusion of Proposition 4.10 holds and so that f M ( ω : e I ( ω ) − I ′ ( ω ) = ε/ e I ( ω ) − I ′ ( ω ) = − ε/
4) = 0 . (49)This ε will be fixed for the rest of the paper.Let Y ε be the event that:1. σ ∆ σ ′ is disconnected and I > g ∗ ∈ σ ∆ σ ′ ;3. f ∗ is in a rung R that satisfies E ( R ) < I ′ ( ω ) − ε/ Y ε must have zero probability since by Lemma 4.9 andRemark 2 an event along the domain wall occurs infinitely often, whereas f ∗ must be uniquealong the domain wall by the definition of I ′ . On the other hand, we will use Proposition 4.10in Proposition 4.13 to show that the event Y ε must have positive probability. Proposition 4.12.
The following statement holds. f M ( Y ε ) = 0 . Proposition 4.13. If f M ( X ε ∩ { e F f > ε } ) > , then f M ( Y ε ) > . Proof of Proposition 4.12.
For a dual vertex b ∗ , let Y ε ( b ) ⊆ e Ω be the event that1. σ ∆ σ ′ is disconnected and I > b ∗ ∈ σ ∆ σ ′ ;3. there is a dual edge e ∗ , sharing a dual endpoint with b ∗ , that is the first edge of a rung R with E ( R ) < I ′ b,e ( ω ) − ε/
4. Here I ′ b,e is the infimum of energies of the rungs notcontaining e ∗ and touching the domain wall of b ∗ .In this notation, the Y ε corresponds to the case b = g and e = f . By definition of I ′ b,e , foreach domain wall D , there are at most two dual edges b ∗ such that Y ε ( b ) occurs (one foreach side of D ). By the same argument as in Lemma 4.9 (with B e ( ε, K ) replaced by Y ε ( b )),it follows that f M ( Y ε ( b )) = 0 for all dual edges b (see Remark 2), so f M ( Y ε ) = 0.38 roof of Proposition 4.13. On the event X ε ∩ { e F f > ε } , either the spins at the endpoints of f are the same or they are different (in both σ and σ ′ ). Let us suppose that: f M ( X ε , e F f > ε, σ f = σ ′ f = +1) > . The subsequent argument can easily be modified in the case σ f = σ ′ f = − e C f = max { C f , C ′ f } . We may choose a ∈ R such that f M ( X ε , e F f > ε, σ f = σ ′ f = +1 , e C f ∈ ( a, a + ε/ > e C f can have countably many point masses, we may furtherrestrict our choice of a so that f M ( e C f = a or a + ε/
8) = 0 . (51)By property (5), for each k , T k M (( J, σ, σ ′ ) : σ f = σ ′ f = +1 , J f < max { C f ( J, σ ) , C f ( J, σ ′ ) } ) = 0 . This is an open cylinder event in Ω ∗ , thus after averaging and taking liminf, f M ( σ f = σ ′ f = +1 , J f < e C f ) ≤ lim inf k →∞ M ∗ k ( σ f = σ ′ f = +1 , J f < e C f ) = 0 . (52)If J f ≥ e C f and e F f = max {| J f − C f | , | J f − C ′ f |} > ε , then J f > C f + ε and J ′ f > C ′ f + ε . Bycombining (50) and (52), we thus find f M ( X ε , e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) > . Recall that e I ( ω ) is the infimum of energies of all rungs that contain f ∗ . On the event X ε , we have e I ( ω ) < I ′ ( ω ) + ε/
2. Therefore if e B is the event that1. g ∗ ∈ σ ∆ σ ′ but f ∗ / ∈ σ ∆ σ ′ ;2. e I ( ω ) < I ′ ( ω ) + ε/ f M ( e B, e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) > . (53)Note that condition 2 of e B only makes sense if f ∗ is actually in a rung; however, in thesupport of f M , σ and σ ′ are ground states, so their interface does not contain loops. Thuswhen condition 1 of e B holds and ω is in the support of f M , f ∗ is in a rung.From this point on, the strategy is similar to the proof of Lemma 4.11. The idea is touse Lemma 3.5 to lower e I ( ω ) below I ′ ( ω ) − ε/
4. Let e P be the event e B with the condition e I ( ω ) < I ′ ( ω ) + ε/ e I ( ω ) < I ′ ( ω ) − ε/
4. We will show that f M ( e P ) > .
39 quick look at (53) can convince us that this is possible since J f could be lowered by 3 ε/ e I ( ω ) depends linearly on J f by definition, itwill be itself lowered by 3 ε/ I ′ ( ω ) by ε/
4. To make this reasoningrigorous, as in the proof of Lemma 4.11, we must bring the problem back to the half-planemeasure M and find a cylinder approximation for both e B and e P .Let B (0; j ) be the box of side-length j centered at 0 and let l ≥ j . Recall the definitionsof ( j, l )-domain walls and ( j, l )-rungs below (42). Let D ( j, l ) , D ( j, l ) , . . . , D t ( j, l ) be the( j, l )-domain walls in B (0; j ) and D ( j, l ) be the one containing g ∗ (if it exists). For l ≥ j and ω ∈ e Ω such that g ∗ ∈ σ ∆ σ ′ , write I ′ j,l ( ω ) (the cylinder approximation of I ′ ( ω )) as theinfimum of all energies of ( j, l )-rungs which touch D ( j, l ) but do not contain the dual edge f ∗ . Write e I j,l ( ω ) (the cylinder approximation of e I ( ω )) for the infimum of all energies of( j, l )-rungs which contain f ∗ . Let e B j,l ⊆ e Ω be the cylinder approximation of e B :1. g ∗ ∈ σ ∆ σ ′ but f ∗ / ∈ σ ∆ σ ′ .2. e I j,l ( ω ) < I ′ j,l ( ω ) + ε/ e P j,l of e P similarly with ε/ − ε/
4. Theremay be no ( j, l ) rungs, but their existence is implicit in condition 2 (in other words, it isimplied in condition 2 that the variables e I j,l ( ω ) and I ′ j,l ( ω ) are defined). We claim thatlim j →∞ lim sup l →∞ f M ( e B j,l ∆ e B ) = 0lim j →∞ lim sup l →∞ f M ( P j,l ∆ P ε ) = 0 . (54)We give the proof for e B . The proof for e P is identical with ε/ − ε/ ω be a configuration such that g ∗ ∈ σ ∆ σ ′ and f / ∈ σ ∆ σ ′ (this is truefor all configurations in e B or in e B j,l ). Note that for fixed j , e I j ( ω ) := lim l →∞ e I j,l ( ω ) existsand equals the infimum of energies of all rungs that stay in B (0; j ) and contain f ∗ . Clearly,lim j →∞ e I j ( ω ) = e I ( ω ) . The analogous statements are true for I ′ ( ω ) (defining I ′ j ( ω ) similarly). Therefore given δ > J ( ω ) such that j ≥ J ( ω ) implies that | e I j ( ω ) − e I ( ω ) | < δ/ | I ′ j ( ω ) − I ′ ( ω ) | < δ/ . For any such j we can find L ( j, ω ) such that for l ≥ L ( j, ω ), | e I j,l ( ω ) − e I j ( ω ) | < δ/ | I ′ j,l ( ω ) − I ′ j ( ω ) | < δ/ . j ≥ J ( ω ) and l ≥ L ( j, ω ), | e I j,l ( ω ) − e I ( ω ) | < δ and | I ′ j,l ( ω ) − I ′ ( ω ) | < δ . (55)We first show that lim j →∞ lim sup l →∞ f M ( e B \ e B j,l ) = 0 . (56)Suppose that ω ∈ e B . Then e I ( ω ) < I ′ ( ω ) + ε/ δ = δ ( ω ) so small that for j ≥ J ( ω ) and l ≥ L ( j, ω ), e I j,l ( ω ) < I ′ j,l ( ω ) + ε/ . Because ω ∈ e B , the first condition of e B j,l holds directly. Equation (56) follows from thisusing the same reasoning as for (43).We now prove that lim j →∞ lim sup l →∞ f M ( e B j,l \ e B ) = 0 . (57)As before, we need to show that if e I ( ω ) ≥ I ′ ( ω ) + ε/ J ( ω ) such that foreach j ≥ J ( ω ), there is an L ( j, ω ) such that if l ≥ L ( j, ω ) then e I j,l ( ω ) ≥ I ′ j,l ( ω ). If e I ( ω ) >I ′ ( ω ) + ε/ e U be the event that e I ( ω ) = I ′ ( ω ) + ε/
2. This event has f M -probability zero by (49) (forthe approximation of e P , one has e I ( ω ) = I ′ ( ω ) − ε/ j →∞ lim sup l →∞ f M (( e B j,l ∩ e U c ) \ ( e B ∩ e U c )) = 0 . However, e U has f M -probability zero, so this proves (57).Notice that e B j,l ∩ { e C f ∈ ( a, a + ε/ } ∩ { J f ≥ a + ε } is a cylinder event in e Ω. This event also makes sense under the measure T k M on the half-plane for e C f = max { C f ( J, σ ) , C f ( J, σ ′ ) } (where the critical values are functions as definedin (6)) and for k ≥ l ≥ j so that the boxes are contained in T k V H . We now analyze theprobability T k M ( e B j,l , C f ∈ ( a, a + ε/ , J f ≥ a + ε ). Let e K j,l ( ω ) := e I j,l ( ω ) − J f be theinfimum of the energies of ( j, l )-rungs where the contribution from the edge f is removed. If e I j,l ( ω ) < I ′ j,l ( ω ) + ε/ J f ≥ a + ε , then e K j,l ( ω ) = e I j,l ( ω ) − J f < I ′ j,l ( ω ) − a − ε/ . (58)Define A j,l ⊆ T k Ω as the intersection of the following events.1. f ∗ / ∈ σ ∆ σ ′ and σ f = σ ′ f = +1.2. g ∗ ∈ σ ∆ σ ′ and e K j,l ( ω ) < I ′ j,l ( ω ) − a − ε/ e C f ∈ ( a, a + ε/ σ, σ ′ are in G ( J ), the ground states in Z × N .Implicit in the second condition is that the variables e K j,l ( ω ) and I ′ j,l ( ω ) are actually defined;in particular, f ∗ must be in some ( j, l )-rung. Although the last condition does not give acylinder event, it will be used to apply Lemma 3.5. A j,l is an intermediary event between e B j,l and P j,l . On the set { σ, σ ′ ∈ G ( J ) } , e B j,l ∩ { e C f ∈ ( a, a + ε/ } ∩ { J f ≥ b } implies A j,l by(58), so T k M ( e B j,l , e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) ≤ T k M ( A j,l ) . (59)We claim that A j,l (and T k A j,l ) has the property (21) of Lemma 3.5:If ( J, σ, σ ′ ) ∈ A j,l and J f ≥ a + ε/
8, then ( J ( f, s ) , σ, σ ′ ) ∈ A j,l for all s ≥ a + ε/ . To verify this, note that the defining condition 1 of A j,l does not depend on J f , so if ( J, σ, σ ′ )satisfies it, so will ( J ( f, s ) , σ, σ ′ ) for all s . Next we argue that σ, σ ′ ∈ G ( J ( f, s )) for all s ≥ a + ε/
8. This holds because σ f = σ ′ f = +1, J f ≥ a + ε/ > e C f = max { C f ( J, σ ) , C f ( J, σ ′ ) } .Clearly condition 3 holds for ( J ( f, s ) , σ, σ ′ ) as the critical values do not depend on thecoupling at f . Last, because σ f = σ ′ f = +1, we see that e K j,l ( J ( f, s ) , σ, σ ′ ) does not dependon s since the contribution of J f to e I j,l is removed. Also the variable I ′ j,l does not depend on J f by construction. Therefore condition 4 holds for ( J ( f, s ) , σ, σ ′ ).We are now in the position to apply Lemma 3.5. Because A j,l satisfies the hypotheses ofthe lemma for c = a + ε/
8, we select d = a + ε/ T k M ( A j,l , J f ∈ [ a + ε/ , a + ε/ ≥ ν ([ a + ε/ , a + ε/ T k M ( A j,l ) . When A j,l occurs and J f ≤ a + ε/ e I j,l ( ω ) = e K j,l ( ω ) + J f < I ′ j,l ( ω ) − a − ε/ a + ε/ I ′ j,l ( ω ) − ε/ . Therefore, writing r = ν ([ a + ε/ , a + ε/ T k M ( A j,l , e I j,l ( ω ) ≤ I ′ j,l ( ω ) − ε/ ≥ r T k M ( A j,l ) , and by (59), T k M ( A j,l , e I j,l ( ω ) ≤ I ′ j,l ( ω ) − ε/ ≥ r T k M ( e B j,l , e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) . (60)Now A j,l ∩ { e I j,l ( ω ) ≤ I ′ j,l ( ω ) − ε/ } is contained in e P j,l . By (60), T k M ( e P j,l ) ≥ r T k M ( e B j,l , e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) . (61)We now want to average over k and take the limit in (61). First note that e P j,l is an eventthat only involves spins and couplings. Furthermore, the only non-trivial contribution to42he boundary ∂ e P j,l is ∂ { e I j,l ( ω ) ≤ I ′ j,l ( ω ) − ε/ } . This event is contained in the event thatthere are two distinct ( j, l )-rungs in the box B (0; N ) whose energies differ by exactly ε/ B (0; N ), this event has f M -probability zero by thecontinuity of ν . Thus by the discussion preceding Remark 1 we may take the limit on theleft to get lim k →∞ M ∗ k ( P j,l ) = f M ( P j,l ) . By (51), and reasoning similar to above, the boundary of the event on the right side of (61)also has f M -probability zero. Therefore we can average over k in (61) and take the limit tofinally get f M ( P j,l ) ≥ r f M ( e B j,l , e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) . (62)Finally, it suffices to take l → ∞ and j → ∞ . By (54), the right side converges to r f M ( e B, e C f ∈ ( a, a + ε/ , J f ≥ a + ε ) > . The probability is positive by (53). The left side of (62) converges to f M ( e P ) again by (54).Thus f M ( P ε ) >
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