Wilson loops in finite Abelian lattice gauge theories
aa r X i v : . [ m a t h . P R ] M a r WILSON LOOPS IN FINITE ABELIAN LATTICE GAUGE THEORIES
MALIN P. FORSSTRÖM, JONATAN LENELLS, AND FREDRIK VIKLUND
Abstract.
We consider lattice gauge theories on Z with Wilson action and finite Abelian gaugegroup. We compute the expectation of Wilson loop observables to leading order in the weak couplingregime, extending and refining a recent result of Chatterjee. Our proofs use neither duality relationsnor cluster expansion techniques. Introduction
Lattice gauge theories.
An important challenge in mathematical physics is to give precisemathematical meaning to quantum field theories (QFTs) that appear in the Standard Model. TheseQFTs describe fundamental forces in nature and elementary particle interactions. One way toproceed is to perform a Wick rotation, discretize space-time by a four-dimensional Euclidean lattice,and then try to approximate the continuum QFT by a well-defined probabilistic theory defined onthe lattice. Such lattice models are called lattice gauge theories and have been studied since at leastthe 1970s (see, e.g., [7]). The hope is that one can take a scaling limit of the lattice model andin this way somehow obtain a rigorously defined continuum field theory. This has proved difficultexcept in simpler cases of limited physical relevance. However, it is also natural to study latticegauge theories in their own right as statistical mechanics models, and this is the point of view wetake here. We refer to [5] for further background and references.1.2.
Wilson action.
In order to state our main results, we need to give a few definitions. Thegraph Z has a vertex at each point x P Z with integer coordinates and an edge between nearestneighbors, oriented in the positive direction, so that there are exactly four positively oriented edgesemerging from each vertex, denoted by e i , i “ , . . . , . We will let ´ e i denote the edge e i with theopposite orientation. Each pair e i and e j of directed edges defines an oriented plaquette e i ^ e j . If i ă j , we say that the plaquette e i ^ e j is positively oriented, and if i ą j , we say that the plaquette e i ^ e j “ ´ e j ^ e i is negatively oriented. Given N P N , let B N “ Z X r´
N, N s . Write E N forthe set of oriented edges contained in B N and E ` N Ă E N for the set of positively oriented edges.Write P N for the oriented plaquettes whose edges are all contained in B N and P ` N Ă P N for thepositively oriented plaquettes. Let p G, `q be finite Abelian group which throughout the paper willbe assumed to be non-trivial. The group elements will provide the “spins” for our model. Let ρ bea faithful and unitary matrix representation of G of dimension d . One example to keep in mind is G “ Z n . In this case an example of a faithful and unitary representation is given by j ÞÑ e πij { n ,implying that ρ p G q “ t e k ¨ πi { n u k Pt , ,...,n u consists of the n th roots of unity. Next, let Σ E N be theset of functions σ : E N Ñ G with the property that σ p e q “ ´ σ p´ e q . For e P E N , we will write σ e : “ σ p e q , and note that p σ e q e P E N is a 1-form on E N . Each element p σ e q P Σ E N induces a spinconfiguration on the oriented plaquettes in P N by assigning σ p : “ σ e ` σ e ` σ e ` σ e , p P P N (1) here e , e , e , e are the edges in the boundary B p of p (directed according to the orientation ofthe plaquette p ). We define the Wilson action with gauge group G by S p σ q “ S G,N p σ q “ ´ ÿ p P P ` N ℜ p tr p ρ p σ p qqq “ ´ ÿ p P P N ℜ p tr p ρ p σ p qqq , σ P Σ E N . Letting µ H denote the product Haar measure on Σ E N “ G E N , we obtain an associated probabilitymeasure µ β,N on Σ E N by weighting µ H by the Wilson action: dµ β,N p σ q “ Z ´ G,β,N e ´ βS p σ q dµ H p σ q . Note that dµ β,N induces a measure on spin configurations on plaquettes by (1). It will be useful todefine the function φ β : G Ñ R , by φ β p g q : “ e β ℜ p tr p ρ p g qqq , g P G. (2)While lattice gauge theories with finite Abelian gauge group are not of known direct physicalsignificance in the context of the Standard Model, they have been studied in the physics literature,see, e.g., [4, 1] and the references therein.1.3. Main results.
In a seminal paper [7], Wilson introduced a particular observable in order tostudy quark confinement in lattice gauge theories, see also, e.g., [5]. We say that a loop with edges t e j u Nj “ is simple if no edge is repeated. (In particular, we allow the corresponding walk on verticesto have double points.) A simple cycle is a finite collection of disjoint simple loops. The length | γ | of a cycle γ is the number of edges in γ . An edge e P γ is said to be a corner edge in γ if there isanother edge e P γ which shares a plaquette with e . Given a simple cycle γ Ă B N , the Wilson loopobservable W γ is defined on Σ E N by W γ : “ tr ´ ρ ´ÿ e P γ σ e ¯¯ . We will apply Ginibre’s inequality in Section 2.3 to see that, for a fixed γ , the limit x W γ y β : “ lim N Ñ8 E β,N r W γ s exists and is translation invariant. Here E β,N denotes expectation with respect to the measure µ β,N ,using free boundary conditions. We can now state our main result which extends Theorem 1.1 of [3] to the case when the gaugegroup is finite and Abelian and the representation is unitary, faithful, and d -dimensional (for someinteger ď d ă 8 ). In the statement, θ i p β q denotes the i th eigenvalue of the matrix Θ p β q “ ř g P G ρ p g q φ β p g q ř g P G φ β p g q (3)and λ p β q : “ max g P G r t u φ β p g q{ φ β p q . Since ρ is a faithful representation of G , λ p β q ă . See also Theorem 1.2.1 of [2]. We do not claim that there is a unique infinite volume limiting probability measure on spin configurations on Z , independent of the choice of boundary condition on B N . We may however consider subsequential limits, and theparticular observable we are interested in does not depend on the choice of limit. heorem 1.1. Consider lattice gauge theory with Wilson action corresponding to a finite andAbelian gauge group G and a unitary and faithful d -dimensional representation ρ of G . Then thereexists β ą such that the following holds when β ě β . Let γ be a simple oriented cycle in Z .There exist constants C and C , which are independent of β and γ , such that if ℓ is the numberof corner edges in γ , then ˇˇˇˇ x W γ y β ´ d ÿ i “ e ´ ℓ p ´ θ i p β qq ˇˇˇˇ ď C ”c ℓ ℓ ` λ p β q ı C . (4) Remark . In Section 3, where a proof of this result is given, we give exact expressions for theconstants C and C in (4). Using Remark 3.5, it follows from these equations that if ρ is one-dimensional, then C can be taken arbitrarily close to { by choosing β sufficiently large. Remark . Our proofs can easily be modified to work for closely related actions as well, such asthe Villain action. Also, with very small modifications, they work for any lattice Z m with m ě . Remark . In Section 2.2, we will show that there always exists a representation ˆ ρ which givesthe same theory as ρ and for which ˆ ρ p g q is diagonal for every g P G . If we for i P t , , . . . , d u and g P G let ρ i,i p g q denote the p i, i q -entry of the matrix ρ p g q , define λ i p β q : “ max g P G : ˆ ρ i,i p g q“ φ β p g q{ φ β p q and Λ i : “ ! g P G : ˆ ρ i,i p g q “ and φ β p g q{ φ β p q “ λ i p β q ) , then, as β Ñ 8 , we have that ´ θ i p β q „ λ i p β q ¨ ř g P Λ i ˆ ρ i,i ` g ˘ | Λ i | . This implies in particular that in order for the second term in (4) to be non-trivial in the limit, ℓλ i p β q needs to be bounded away from infinity for at least some i P t , , . . . , d u .1.4. Relation to other work.
This paper was inspired by, and builds upon, a recent paper ofChatterjee [3]. There, lattice gauge theory with Wilson action and gauge group G “ Z wasconsidered. Chatterjee obtained an expression for the first order term of the expected value of aWilson loop in the limit as β and | γ | tend to infinity simultaneously as in Theorem 1.1. This paperextends and refines Chatterjee’s result for the Wilson action to the case of a general finite Abeliangauge group and a faithful and unitary representation. In contrast to the argument in [3], ourargument does not use duality relations. While we were preparing this paper, Cao’s preprint [2]appeared. Cao obtains a more general result than our main theorem, which also allows for non-Abelian finite groups. However, we believe the present paper, while less general, is still of interestbecause our approach is quite different from Cao’s. Whereas the approach of [2] relies on a clusterexpansion and Stein’s method, ours is based on a rather elementary estimate of the probability fora plaquette to be part of a vortex of a given size. Acknowledgements.
MPF acknowledges support from the European Research Council, GrantAgreement No. 682537. JL is grateful for support from the Göran Gustafsson Foundation, theRuth and Nils-Erik Stenbäck Foundation, the Swedish Research Council, Grant No. 2015-05430,and the European Research Council, Grant Agreement No. 682537. FV acknowledges supportfrom the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the Ruth andNils-Erik Stenbäck Foundation. Thanks go to Juhan Aru for comments on an earlier version of ourpaper. . Preliminaries
This section collects some basic definitions and material needed for the proof of our main result.In some cases, if we could not find a clean reference, we have included derivations of results eventhough they could be considered well-known. We will freely use notions and results from discreteexterior calculus as discussed in Section 2 of [3].In the rest of the paper, we will assume that a finite Abelian gauge group, G , and a unitary andfaithful representation ρ of G have been fixed.2.1. Notation and terminology.
We will use the following notation and terminology. ‚ φ β will denote the function defined in (2) with parameter β . ‚ Σ E will denote the set of spin configurations on a set of edges E . ‚ Σ P will denote the set of spin configurations on a given set of plaquettes P which can arisefrom a spin configuration on the corresponding set of edges E . ‚ P p e q will denote the set of (oriented) plaquettes in Z which contain the (oriented) edge e . ‚ A surface is a set of plaquettes. ‚ d, δ, ˚ will denote the discrete exterior derivative operator, discrete coderivative operator,and discrete Hodge dual operator, respectively. See Section 2 of [3]. ‚ B c will denote the set of all oriented plaquettes that belong to the boundary of the oriented3-cell c . If we let f be the 3-form that is equal to on c and equal to otherwise, then δf is a 2-form which is a (signed) indicator function for the plaquettes in B c . Similarly, when p is a plaquette, B p will denote the set of edges in the boundary of p . ‚ Given a matrix A , A i,j will denote the element in row i and column j .2.2. Diagonal representations.
Recall that the trace of a matrix is invariant under a changeof basis. Moreover, note that the definitions of φ β , µ β,N and W γ depend on the representation ρ only through its trace. This implies in particular that if A is an invertible matrix, then therepresentations ρ and ˆ ρ : “ AρA ´ are both representations of G , and define the same theory. Inthis case, we say that the two representations ρ and ˆ ρ are equivalent. Since for any g, g P G , thematrices ρ p g q and ρ p g q commute and are invertible, the matrices p ρ p g qq g P G can be simultaneouslydiagonalized. In other words, there is a representation ˆ ρ which is equivalent to ρ and which issuch that ˆ ρ p g q is diagonal for each g P G . Since assuming that ρ p g q is diagonal for each g P G willsometimes simplify calculations, we can, and will, assume that ρ is already in this form. If ρ has thisform, then, since ρ is a unitary representation, all the diagonal elements of ρ p g q are roots of unityfor each g P G . Moreover, if ρ has this form, then since the eigenvalues of a diagonal matrix areequal to its diagonal elements, and eigenvalues are invariant under a change of basis, the eigenvaluesof the matrix Θ defined in (3) are given by θ i p β q “ ř g P G ρ i,i p g q φ β p g q ř g P G φ β p g q , i P t , , . . . , d u . Ginibre’s inequality.
The following result is proved in [6]. We will use it to prove existence ofthe infinite volume limit and translation invariance of the expectations of Wilson loop observables.This result is well-known and it is often stated in the literature that this fact follows from Ginibre’sinequalities, but since we have not found a clean reference, we give the short argument below.
Theorem 2.1 (Ginibre’s inequality) . Let p Γ , µ q be a finite probability space. For real-valued func-tions f and h on Γ , let x f y h “ ż f p x q e ´ h p x q dµ p x q N ż e ´ h p x q dµ p x q . et A be a set of real-valued functions on Γ such that for any f , f , . . . , f m P A and any choice ofsigns σ , σ , . . . , σ m , ij µ p x q µ p y q m ź i “ p f i p x q ` σ i f i p y qq ě . Then, for any f, g, ´ h in the convex cone generated by A , x f g y h ě x f y h x g y h . We want to apply this theorem with Γ “ G E N , where E N is the set of edges in the box B N , and µ is the corresponding (normalized) Haar measure. By Example 4 in [6], we can let A be the setof real-valued positive definite functions on Γ . It is well known that if ρ i,i p g q is a root of unity foreach g P G and i P t , , . . . , d u , then the functions ρ i,i `ř e PB p σ e ˘ , i.e., the functions of the type f p σ q “ ρ i,i `ś e PB p σ e ˘ , are positive definite. This implies in particular that for each i P t , , . . . , d u ,the function ´ h pp σ p qq “ β ř p P P ℜ p ρ i,i p σ p qq is in the cone Cone A generated by A for any set ofplaquettes P . Moreover, since any function which depends on only a finite number of plaquettescan be written as a finite weighted sum of the characters of G E N , any such function is in Cone A .(See also Example 4 in [6].)To get an infinite volume limit, one argues as follows. Fix M, N P N and assume that N ě M .For p σ p q P Σ P N , let h p σ q : “ e β M ř di “ ř p P PM ℜ ` tr p ρ p σ p qq ˘ ` β N ř p P PN r PM ℜ ` tr p ρ p σ p qq ˘ and h N p σ q : “ e β ř p P PN ℜ ` tr p ρ p σ p qq ˘ . Then for any function f P Cone A , we have ddβ N x f y h “ ddβ N Z B N ,h x f y h Z B N ,h “ A f p σ q ÿ p P P N r P M tr p ρ p σ p qq E h ´ x f y h A ÿ p P P N r P M tr p ρ p σ p qq E h . Since f P Cone A implies that f p σ q ř p P P N r P M tr ` ρ p σ p q ˘ P Cone A , applying Ginibre’s inequality itfollows that ddβ N x f y h ě and hence x f y h is increasing in β N . Since f depends only of a finite set of plaquettes P , } f } ă 8 ,and hence we can apply the monotone convergence theorem to conclude that lim N Ñ8 x f y h N exists.Moreover, this limit is translation invariant. To see this, let τ be a translation of Z , let f be afunction which depends only on the set of plaquettes in B Ď B M , and let N be so large that B M , τ B M , and τ ´ B M are all contained in B N . Then x f ˝ τ y h N “ x f y h N ˝ τ ´ ě x f y h M and x f y h N “ x f ˝ τ ˝ τ ´ y h N “ x f ˝ τ y h N ˝ τ ě x f ˝ τ y h M . Letting first N and then M tend to infinity, the translation invariance follows.2.4. Spins on edges and plaquettes.
While the probability measure µ β,N is defined by weightinga measure on edge spin configurations, we are primarily interested in the corresponding measure onplaquette spin configurations. To describe the latter measure, we first note that not all plaquettespin configurations arise from edge spin configurations. The exterior derivative operator d maps spinconfigurations p σ e q on edges to spin configurations p σ p q on plaquettes by d ` p σ e q e P E N ˘ “ p σ p q p P P N .Hence, if we let Σ P N denote the set of spin configurations on plaquettes in P N that can arise from ome spin configuration on the set of edges E N in a box B N , then by the discrete Poincaré lemma(see Lemma 2.2 of [3]), p σ p q P Σ P N if and only if d ` p σ p q p P P N ˘ “ .Suppose that f is a function which depends on σ P Σ E N only through the corresponding plaquetteconfiguration p σ p q p P P N P Σ P N . Then E β,N “ f p σ q ‰ “ Z ´ E N ,β ÿ σ P Σ E ` N f p σ q ź p P P N φ β ´ ÿ e PB p σ e ¯ “ Z ´ P N ,β ÿ σ P Σ P ` N f p σ q ź p P P N φ β ` σ p ˘ . The second equality above follows because, by the Poincaré lemma, each spin configuration in Σ E N corresponds to a spin configuration in Σ P N , and each spin configuration in Σ P N corresponds to thesame number of spin configurations in Σ E N .2.5. Simple loops and oriented surfaces.
We say that a collection of oriented plaquettes P Ď P N , is an oriented surface . A useful idea when estimating the expectation of the Wilson loopobservable is that the product of all spins along the loop can be written as a product of plaquettespins over an oriented surface whose boundary is equal to the loop. We next state and prove alemma which describes this connection between oriented surfaces P and oriented simple loops γ . Lemma 2.2.
Let γ be an oriented simple loop in B N . Then there exists an oriented surface P Ď P N such that ÿ e P γ σ e “ ÿ p P P σ p (5) Proof.
By Lemma 3.1 in [3], there is a set of plaquettes ˆ P Ď P N such that B ˆ P “ supp γ . We willshow that there is a unique way to assign orientations to the plaquettes in any such set ˆ P in sucha way that (5) holds. We will prove this by using induction on | ˆ P | . If | ˆ P | “ , then ˆ P “ t p u forsome plaquette p P P N , so we have either γ “ B p or γ “ ´B p , and hence (5) holds. Assume that thestatement of the lemma holds whenever the set ˆ P contains at most k elements, and assume that γ corresponds to a set ˆ P with | ˆ P | “ k ` . Then there is at least one plaquette ˆ p P ˆ P which is suchthat either(i) there are exactly one or three edges in E N which are in both B ˆ p and γ (in any orientation),or(ii) there are exactly two edges in E N which are in both B ˆ p and γ (in any orientation), and thesetwo edges are adjacent.Fix one such plaquette, and let γ be the closed loop which agrees with γ on all edges that are notboundary edges of ˆ p and which satisfies supp γ “ B ˆ P r t ˆ p u . Since | ˆ P r t ˆ p u| “ k , there is aunique way to assign orientations to the plaquettes in ˆ P r t ˆ p u to obtain a set of oriented plaquettes P for which (5) holds. By the definition of γ , there is now a unique way to assign an orientationto ˆ p , to obtain an oriented set of plaquettes P for which (5) holds. This concludes the proof. (cid:3) Vortices.
The following definition will play a crucial role in this paper. We mention that analternative definition will be given in Section 2.7.
Definition 2.3 (Vortex) . Let σ be a spin configuration on E N . A set of plaquettes V Ď P N is saidto be a vortex in σ if(i) ´ p P V whenever p P V ,(ii) the function ν : P N Ñ G defined by ν p p q : “ σ p if p P V otherwise (6)satisfies dν “ . vortex V in a spin configuration σ is said to be irreducible if there is no nonempty propersubset V Ă V such that V is also a vortex. The function ν given in (6) will be called the vortexfunction corresponding to V . If a vortex function ν corresponds to an irreducible vortex, then ν will be referred to as irreducible.When G “ Z , our definition of an irreducible vortex is equivalent to Chatterjee’s definition of avortex in [3].We now make a few observations about vortices. The next observation, although simple, will becrucial in a later proof. Lemma 2.4.
Let p σ e q be a spin configuration on E N , and let V be a vortex in σ . For each p P P N ,define σ Vp : “ if p P Vσ p otherwise. (7) Then p σ Vp q P Σ P N .Proof. Let ν be the function defined in (6). Then σ Vp “ σ p ´ ν p p q for any p P P N . Since p σ p q P Σ P N and dν “ by assumption, it follows that dσ V “ dσ ´ dν “ ´ “ and hence p σ Vp q P Σ P N as desired. (cid:3) The next lemma is a discrete version of the Bianchi identity. It will be used in Section 2.8 tobound the number of vortices of a given size.
Lemma 2.5. If p σ p q P Σ P N , then for any oriented 3-cell c in B N we have ÿ p PB c σ p “ . (8)The previous lemma follows immediately from the fact that the plaquettes in the product aboveare oriented so that each edge is counted exactly twice, once in each direction.We now state and prove the following lemma, which is similar to Lemma 3.2 in [3]. In this lemma,a plaquette p is said to be an internal plaquette of an oriented surface Q corresponding to a loop γ ,if no edge in B p Y ´B p is in γ . Lemma 2.6.
Let p σ e q be a spin configuration on the edges of Z , and let V be a vortex in σ , withcorresponding vortex function ν . Let γ be an oriented loop and let Q be a corresponding orientedsurface (in the sense of Lemma 2.2). If there is a cube B containing V which is such that ˚˚ B X Q consists of only internal plaquettes of Q , then ÿ p P Q ν p p q “ . (9) Proof.
Since ν is a vortex function, we have dν “ , and hence, by Lemma 2.3 in [3], δ ˚ ν “ . Usingthe assumptions of the lemma, it follows from Lemma 2.4 in [3] that ˚ ν is zero outside ˚ B , whichby Lemma 2.7 in [3] implies that ˚ ν “ ´ δg for some 3-form g that is zero outside ˚ B . Utilizing [3,Lemma 2.3] again, we conclude that ν “ ˚˚ ν “ d p˚ g q .Let S : “ ÿ p P Q ν p p q . Then S P G , and since Q is finite, the sum is finite. We need to show that S “ . To that end, notethat S “ ÿ p P Q d ˚ g p p q “ ÿ p P Q ÿ e PB p ˚ g p e q “ ÿ e P γ ˚ g p e q here γ is the boundary of Q , and the last equality follows from the fact that Q is an orientedsurface.Let us say that a 3-form is elementary if it is non-zero on exactly one 3-cell. Then the 3-form g is the sum of finitely many elementary 3-forms that are zero outside ˚ B . Since the Hodge-staroperator is additive, it will follow that S “ if we show that ÿ e P γ ˚ g p e q “ (10)for any elementary 3-form g that is zero outside ˚ B . Take any such g . Then ˚ g is an elementary1-form on the primal lattice. Let c be the cell where g is non-zero and let e “ ˚ c be the edgewhere ˚ g “ . Let P p e q be the set of all plaquettes containing e . Then P p e q is the set of dualsto the boundary plaquettes of c . In particular, the elements of P p e q are plaquettes in ˚ ˚ B . Let Q “ Q X P p e q . Then Q Ď ˚˚ B X Q , and hence by assumption, all elements of Q are internaledges of Q . In particular, e must be an internal edge of Q , i.e. e R γ . Clearly, this implies that (10)holds, and hence the desired conclusion follows. (cid:3) Our next lemma gives an alternative description of vortices as isolated sets of plaquettes withnon-zero spin.
Lemma 2.7.
Let σ P Σ E N be a configuration on the edges of E N . Let V be a bounded subset of P N which is such that(i) ´ p P V whenever p P V ,(ii) if p R V is a boundary plaquette for a 3-cell which intersects V , then σ p “ .Finally, define ν : P N Ñ G by ν p p q “ σ p if p P V or ´ p P V otherwise. (11) Then dν “ , and hence V is a vortex in σ .Proof. Since p σ p q P Σ P N , there is a 1-form h : E N Ñ G which is such that dh p p q “ σ p . Thisimplies in particular that dh p p q “ ν p p q for all plaquettes for which ν p p q “ σ p . By assumption, thisholds for any plaquette p which is a boundary plaquettes of at least one 3-cell which intersects V .This implies in particular that if c is a 3-cell which has a boundary plaquette which is in V then dν p c q “ d p dh qp c q “ d h p c q “ . On the other hand, if no boundary plaquette of c intersects V ,then, since ν is zero away from V , we also have that dν p c q “ . This shows that dν p c q “ for all3-cells c in Z , which is the desired conclusion. (cid:3) We now use the previous lemma to prove the following lemma, which generalizes Lemma 3.3in [3].
Lemma 2.8.
For any configuration p σ p q P Σ P N , the set V “ t p P P N | σ p ‰ u can be uniquelywritten as a disjoint union V “ Y N v j “ V j of irreducible vortices.Proof. Consider the graph with vertex set V and an edge between two plaquettes p , p P V if andonly if there is a 3-cell c such that p , p P B c . Let ˆ V , ˆ V , . . . , be the vertex sets for the connectedcomponents of this graph. By Lemma 2.7, the sets ˆ V i are vortices in σ . It remains to show thatthese vortices can be split into irreducible vortices. To see this, note first that since P N is finite, sois each vortex ˆ V i . Thus the desired conclusion follows if we can show that if W Ď W Ď ˆ V i are twovortices, then so is W r W . Given a subset W Ď ˆ V i , define ν W by ν W p p q “ σ p if p P W, otherwise. y definition, W Ď ˆ V i is a vortex in σ if and only if dν W “ . Since d is a linear operator, we have dν W r W “ d p ν W ´ ν W q “ dν W ´ dν W “ ´ “ , and hence the lemma follows. (cid:3) Subgroup vortices.
The assumptions on the representation ρ in Theorem 1.1, i.e. ρ is faithfuland unitary, will naturally define subgroups of G as follows.Recall from Section 2.2 that we can assume that ρ p g q is diagonal for each g P G . Next, note thatwhen this holds, then for each i P t , , . . . , d u , ρ i,i is a unitary and one-dimensional representationof G . Let G p i q : “ ker ρ i,i . Then G p i q is a normal subgroup of G , and ρ i,i naturally defines afaithful representation of G p i q : “ G { G p i q . Moreover, any element g P G can be uniquely writtenas p g p ,i q , g p i q q , where g p ,i q P G p i q and g p i q P G { G p i q . Using this decomposition for some fixed i Pt , , . . . , d u , if σ “ p σ p ,i q , σ p i q q P Σ E N is a spin configuration using the gauge group G , then σ p i q is a spin configuration on E N using the gauge group G p i q , and σ p ,i q is a spin configuration on E N using the gauge group G p i q . Conversely, if σ p i q is a spin configuration on E N using the gauge group G p i q and σ p ,i q is a spin configuration on E N using the gauge group G p i q , then p σ p ,i q , σ p i q q is a spinconfiguration on E N using the gauge group G .The main reason for introducing the decomposition above is that using it, all results in theprevious sections can be applied to σ p i q instead of σ . In particular, we will use the followingdefinition, which generalizes Definition 2.3. Definition 2.9 ( G p i q -vortex) . Let σ be a spin configuration on E N (using the gauge group G ). Let i P t , , . . . , d u . A set of plaquettes V Ď P N is said to be a G p i q - vortex in σ if(i) ´ p P V whenever p P V ,(ii) the function ν : P N Ñ G p i q defined by ν p p q : “ σ p i q p if p P V otherwise (12)satisfies dν “ .2.8. Distribution of vortices.
In this section we give upper bounds for a few probabilities relatedto the distribution of vortices. In particular, we give proofs of the natural analogues of Corollaries6.2 and 6.3 in [3]. Even when the gauge group is Z , which is the case studied in [3], our proofs aredifferent from the proofs in [3], and do not require that the model has a dual or that the rate ofconvergence of expected values of finitely supported functions is known.Recall from Section 2.2 that we can assume that ρ p g q is a diagonal matrix for every g P G . Giventhis assumption, we have λ i p β q “ max g P G : ρ i,i p g q“ φ β p g q{ φ β p q . It follows immediately from the definition of φ β that λ i p β q Ñ as β Ñ 8 . Proposition 2.10 (Compare with Theorem 6.1 of [3]) . Assume that G is a finite Abelian groupand let ρ be a faithful, unitary and diagonal d -dimensional representation of G . Consider latticegauge theory with gauge group G and representation ρ in a box B N , and let i P t , , . . . , d u . Let P Ď P ` N be any finite set of plaquettes and ν a G p i q -valued function with support exactly on P Y ´ P ,which is such that dν “ . Then the probability that σ p i q and ν agree on P is bounded from aboveby λ i p β q | P | . roof. Fix i P t , , . . . , d u . Let E P be the event that σ p i q p “ ν p p q for all p P P and, similarly, let E P be the event that σ p i q p “ for all p P P . For each σ P E P , define the plaquette configuration σ P by σ Pp : “ p σ p ,i q p , q if p P Pσ p otherwise.Since dν “ , Lemma 2.4 implies that p σ Pp q P Σ P N . Moreover, the mapping σ ÞÑ σ P is a bijectionfrom E P to E P . Define φ p i q β : “ exp ` β ℜ ` ρ i,i p σ p q ˘˘ and note that φ β “ ś di “ φ p i q β . Then µ β,N p E P q ď ř σ P E P ś p P P ` N φ β p σ p q ř p σ p qP Σ PN ś p P P ` N φ β p σ p q ď ř σ P E P ś p P P ` N φ β p σ p q ř σ P E P ś p P P ` N φ β p σ p q“ ř σ P E P ś p P P ` N φ β p σ p q ř σ P E P ś p P P ` N φ β p σ Pp q “ ř σ P E P ś p P P ` N r P φ β p σ p q ś p P P ` N X P φ β p σ p q ř σ P E P ś p P P ` N r P φ β p σ Pp q ś p P P ` N X P φ β p σ Pp q“ ř σ P E P ś p P P ` N r P φ β p σ p q ś j Pt , ,...,d u ´ś p P P ` N X P φ p j q β p σ p q ¯ř σ P E P ś p P P ` N r P φ β p σ Pp q ś j Pt , ,...,d u ´ś p P P ` N X P φ p j q β p σ Pp q ¯ ď ř σ P E P ś p P P ` N r P φ β p σ p q ˆś j Pt , ,...,d u r t i u ´ś p P P ` N X P φ p j q β p σ p q ¯˙ ś p P P ` N X P φ p i q β p σ p q ř σ P E P ś p P P ` N r P φ β p σ Pp q ˆś j Pt , ,...,d u r t i u ´ś p P P ` N X P φ p j q β p σ p q ¯˙ ś p P P ` N X P φ p i q β p σ Pp q“ ś p P P ` N X P φ p i q β p ν p p qq ś p P P ` N X P φ p i q β p q “ ź p P P ` φ p i q β p ν p p qq{ φ p i q β p q ˘ ď λ i p β q | P | . (cid:3) We now use the previous lemma to provide a proof of the following result, which extends Corollary6.2 in [3]. In contrast to the proof of the corresponding result in [3], we use neither any duality ofthe model, nor the rate of decay of correlations. Interestingly, this lemma is the only place in theproof of Theorem 1.1 which requires that β is sufficiently large. Proposition 2.11 (Compare with Corollary 6.2 in [3]) . Assume that G is a finite Abelian groupand let ρ be a faithful, unitary, and diagonal d -dimensional representation of G . Consider latticegauge theory with gauge group G and representation ρ in a box B N . Fix any β ą which is suchthat p| G | ´ q λ i p β q ă for all β ą β and all i P t , , . . . , d u . Assume that β ą β . Then theprobability that a plaquette p P P N belongs to an irreducible G p i q -vortex of size ě M is bounded fromabove by C p M,i q λ i p β q M , where C p M,i q : “ M ´ p| G p i q | ´ q M ´ ´ p| G p i q | ´ q λ i p β q . (13) Proof.
Let m be a positive integer and let p be a plaquette. Fix an arbitrary ordering of all the3-cells in Z at distance at most m from p . We will now describe an algorithm through which anyirreducible G p i q -vortex function, whose support contains p and has size at most m , can be found. et Q : “ t p u and choose any element g P G p i q r t u . Define a function ν : P N Ñ G p i q by ν p p q : “ $’&’% g if p “ p ´ g if p “ ´ p otherwise.Suppose sets of plaquettes Q , . . . , Q j and functions ν , . . . , ν j are given for some j P t , , . . . , m ´ u .Then define p Q j ` , ν j ` q as follows:(i) If dν j “ , set Q j ` : “ Q j and ν j ` : “ ν j .(ii) If dν j ı , there is at least one 3-cell c P B N for which dν j p c q “ . Let c j be the first 3-cell (with respect to the ordering of the 3-cells) for which dν j p c j q “ . There are then threepossibilities:(a) For all plaquettes p P B c j , either p or ´ p is in Q j . In this case there is no irreducible G p i q -vortex function ν which agrees with ν j on Q j . Set Q j ` : “ Q j and ν j ` “ ν j .(b) There are (at least) two (distinct) plaquettes p P B c j which are such that neither p nor ´ p is in Q j . Pick one such plaquette p j and any element g j P G p i q r t u . Define Q j ` : “ Q j Y t p j u and for each p P P N , let ν j ` p p q : “ $’&’% g j if p “ p j ´ g j if p “ ´ p j ν j p p q otherwise. (14)(c) There is exactly one plaquette p j P B c j such that neither p nor ´ p is in Q j . In this case,set Q j ` : “ Q j Y t p j u and let g j be the unique element in G p i q which is such that thefunction ν j ` defined in (14) satisfies dν j ` p c j q “ . Let ν j ` be this function.Given the ordering of the 3-cells, each irreducible G p i q -vortex function ν , with p P supp p ν q and | supp p ν q| ď m corresponds to at least one sequence ν , ν , . . . , ν m which can be found by thealgorithm above.At the first step of the algorithm, we are given a plaquette p and can let g be any elementin G p i q r t u . At the j :th step of the algorithm for j P t , . . . , m ´ u , we have at most fivedifferent choices for p j , and at most | G p i q | ´ different choices for g j . Finally, at the last step of thealgorithm, if we construct a G p i q -vortex of size m , there is exactly one way to choose p m and g m .It follows that the number of irreducible vortices of size exactly m that contain p can be at most m ´ p| G p i q | ´ q m ´ .By Proposition 2.10, if ν m is an irreducible G p i q -vortex function, then the probability that σ p i q agrees with ν m on P m : “ supp p ν m q for a random configuration σ is at most λ i p β q | P m | . If | supp p ν m q| “ m , then this probability is hence at most equal to λ i p β q m . This implies in particularthat µ β,N p p is in an irreducible vortex of size ě M q ď ÿ m “ M m ´ p| G p i q | ´ q m ´ λ i p β q m . The right-hand side in the previous equation is a geometric sum, which converges exactly if p| G p i q | ´ q λ i p β q ă . In this case, we obtain the upper bound M ´ p| G p i q | ´ q M ´ λ i p β q M ´ p| G p i q | ´ q λ i p β q . From this the desired conclusion follows. (cid:3) . Proof of the main result
The purpose of this section is to prove Theorem 1.1. To this end, fix a finite Abelian group G and a unitary and faithful d -dimensional representation of G , and consider lattice gauge theory on Z with gauge group G , representation ρ , and Wilson action. Our proof closely follows the proof ofTheorem 1.1 in [3].By Section 2.3, the limit x W γ y β “ lim N Ñ8 µ β,N p W γ q exists, where µ β,N is the probability measureon configurations on B N with free boundary data. By a compactness argument (see Section 6 of[3]), we may find a convergent subsequential limit of p µ β,N q N ě viewed as probability measures onspin configurations on the edges of Z (fixing the values to be say outside B N ). Fix such a limit µ β and note that x W γ y β “ µ β p W γ q .Recall that we have assumed that ρ p g q is a diagonal matrix for each g P G (see Section 2.2), andthat we have defined λ i p β q “ max g P G : ρ i,i p g q“ φ β p g q{ φ β p q . and θ i “ θ i p β q “ ř σ e P G ρ i,i p σ e q φ β p σ e q ř σ e P G φ β p σ e q . We now introduce the additional notation W γ p i q : “ ρ i,i ´ÿ e P γ σ e ¯ and note that W γ “ tr ´ ρ ´ÿ e P γ σ e ¯¯ “ d ÿ i “ ρ i,i ´ÿ e P γ σ e ¯ “ d ÿ i “ W γ p i q . The main part of the proof of Theorem 1.1 is divided into two lemmas, Lemma 3.1 and Lemma 3.3,the first of which we state now. This lemma is useful whenever ℓλ i p β q is bounded from above. Lemma 3.1 (Compare with Lemma 7.1 in [3]) . In the setting of Theorem 1.1, for each i Pt , , . . . , d u we have | µ β p W γ p i qq ´ e ´ ℓ p ´ θ i q | ď C p i q A e C ˚ i ℓλ i p β q „c ℓ ℓ ` λ i p β q , where C p i q A : “ C p ,i q ` C ˚ i C ˚ i ` C C p ,i q C ˚ i ` C ˚ i ` C p ,i q p C ˚ i q ` , (15) and C , C ˚ i , C p ,i q , and C p ,i q are defined by (17) , (23) , and (13) respectively.Proof of Lemma 3.1. Fix some i P t , , . . . , d u . Since γ has length ℓ , there is a cube B of width ℓ which contains γ . By Lemma 2.2, there is an oriented surface Q inside B such that γ is theboundary of Q . Let σ be a random configuration drawn from the Gibbs measure µ β . Let A be theevent that there is no G p i q -vortex of size ě in the configuration σ which intersects Q . Then, byProposition 2.11 and a union bound, µ β p A q ě ´ C p ,i q ℓ λ i p β q . Let V be the set of G p i q -vortices that intersect Q . By definition, W γ p i q “ ρ i,i ´ ÿ p P Q σ p ¯ “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . et V be the set of all members of V that have size ď and define W γ p i q : “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . If the event A occurs, then W γ p i q “ W γ p i q , and hence µ β ´ˇˇ W γ p i q ´ W γ p i q ˇˇ¯ ď p ´ µ β p A qq ď C p ,i q ℓ λ i p β q . Let b be the smallest number such that any G p i q -vortex of size ď is contained in a box of width b . By the definition of a G p i q -vortex, b is a finite universal constant. Let Q be the set of plaquettes p P Q that are so far away from γ that any cube of width b ` containing p does not intersect γ .Let V be the set of all P P V that intersect Q . Take any P P V . Then, by the definition of Q and V , it follows that any cube B of width b that contains P has the property that ˚˚ B X Q onlycontains internal plaquettes of Q . Therefore, by Lemma 2.6 ÿ p P P X Q σ p “ . This implies that if we set V “ V r V , then W γ p i q “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . (16)Let A be the event that no G p i q -vortex of size ě intersects Q r Q . Since each plaquette of Q r Q is contained in a cube of width b ` which intersects γ , it follows that | Q r Q | ď C ℓ, (17)where C is a universal constant which depends on b . Therefore, by Proposition 2.11 and a unionbound, µ β p A q ě ´ C C p ,i q ℓλ i p β q . (18)Recall that P p e q denotes the set of plaquettes containing an edge e . It is easy to see that anyvortex has size at least six, and that any vortex of size six is equal to P p e q for some edge e . Such avortex will be called a minimal vortex. Let V be the set of minimal G p i q -vortices that intersect Q but not Q , and define W γ p i q : “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . If the event A occurs, then V “ V , and hence by (16), W γ “ W γ . Consequently, by (18), µ β p| W γ ´ W γ |q ď p ´ µ β p A qq ď C C p ,i q ℓλ i p β q . Next, let V be the set of all G p i q -vortices in V that are equal to P p e q Y P p´ e q for some e P γ and define W γ p i q “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . Note that if e in an internal edge of Q and p P P p e q , then B p contains exactly one edge with withnon-zero spin in σ , namely e . Since Q is oriented and e is an internal edge, the contributions from e cancel, and hence W γ p i q “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ “ W γ p i q . inally, let V be the set of all members of V that are equal to P p e q Y P p´ e q for some non-corneredge e . Define W γ p i q “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ . Let A be the event that there is no corner edge e such that P p e q Y P p´ e q is a G p i q -vortex. ByProposition 2.11 and a union bound, µ β p A q ě ´ ℓ λ i p β q . If A occurs, then W γ p i q “ W γ p i q , and hence µ β p| W γ p i q ´ W γ p i q|q ď p ´ µ β p A qq ď ℓ λ i p β q . If P P V , then P “ P p e q Y P p´ e q for some edge e P γ . Since Q is oriented and e P γ , bydefinition, we have ÿ p P P p e qX Q σ p “ σ e , and hence if we let E be the set of non-corner edges which are such that P p e q Y P p´ e q is a minimal G p i q -vortex, then W γ p i q : “ ρ i,i ´ ÿ P P V ÿ p P P X Q σ p ¯ “ ρ i,i ´ ÿ e P E ÿ p P P p e qX Q σ p ¯ . (19)Now note that for edges e P γ which are not corner edges, the sets P p e q Y P p´ e q are disjoint.Moreover, note that if P p e q Y P p´ e q is a minimal G p i q -vortex, then ρ i,i p σ p q is, up to the orientationof e in p , the same, and non-zero, for all p P P p e q Y P p´ e q . In other words, if we for e P B p Y ´B p let I p p e q : “ I p e P B p q ´ I p´ e P B p q , then for any p, p P P p e q Y ´ P p e q , we have that ρ i,i p σ p q ¨ I p p e q “ ρ i,i p σ p q ¨ I p p e q . With this in mind,let E denote the set of all non-corner edges e P γ which are such that for all p, p P P p e q Y P p´ e q , ρ p i,i q p σ p ´ I p p e q σ e q ¨ I p p e q “ ρ p i,i q p σ p ´ I p p e q σ e q ¨ I p p e q . In other words, the set E consists of all non-corner edges e P γ which are such that the representa-tion ρ i,i of the sum of the spins around p , excluding e , is the same for each p P P p e q Y P p´ e q whenoriented such that e is traversed in the positive direction. If P p e q Y P p´ e q is a minimal G p i q -vortex,then ρ i,i p σ e q “ ρ i,i p σ p q for any p P P p e q which is oriented in this way. This implies in particularthat we can choose a plaquette p e P P p e q Y P p´ e q for each e P E in such a way that W γ p i q “ ρ i,i ´ ÿ e P E σ p e ¯ . On the event A , we have E “ E , and hence on this event W γ p i q : “ ρ i,i ´ ÿ e P E σ p e ¯ “ W γ . Again using (18), we have µ β p| W γ p i q ´ W γ p i q|q ď p ´ µ β p A qq ď C C p ,i q ℓλ i p β q . (20)Let γ be the set of all non-corner edges of γ . Further, let γ be the set of edges e P γ which aresuch that there are p, p P P p e q Y P p´ e q with ρ i,i p σ p ` σ p q “ if we orient p and p in such a waythat σ e cancels in the product. This implies in particular that if we condition on the spins of alledges which are not in γ (the non-corner edges of γ ), then we can recognize the set γ . Note that E “ γ r γ . Let µ β denote conditional probability and conditional expectation given p σ e q e R γ .Since no two non-corner edges belong to the same plaquette, it is not hard to verify that under his conditioning, p σ e q e P γ are independent spins. Moreover, by the argument above, conditioningon the spins outside γ determines γ . Next, for e P γ r γ , let σ e be the sum of the edges in p e ,excluding σ e (recall that p e P P p e q Y P p´ e q is oriented such that e is positive in p e ). Then µ β ` W γ p i q ˘ “ µ β ˆ ρ i,i ´ ÿ e P γ r γ σ p e ¯˙ “ µ β ˆ ź e P γ r γ ρ i,i ` σ p e ˘˙ “ ź e P γ r γ µ β ´ ρ i,i ` σ p e ˘¯ “ ź e P γ r γ ” ÿ σ e P G ρ i,i ` σ e ` σ e ˘ φ β p σ e ` σ e q ř σ e P G φ β p σ e ` σ e q ı “ ” ř σ e P G ρ i,i ` σ e ˘ φ β p σ e q ř σ e G φ β p σ e q ı | γ r γ | . (21)Recall that θ i “ ř σ e P G ρ i,i p σ e q φ β p σ e q ř σ e P G φ β p σ e q . Then µ β p W γ p i qq “ µ β p µ β p W γ p i qqq “ θ | γ | i µ β p θ ´| γ | i q . (22)Since ρ is a unitary representation of G , for any σ e P G we have that φ β p´ σ e q “ e β ℜ p tr p ρ p´ σ e qqq “ e β ℜ p tr p ρ p σ e q ´ qq “ e β ℜ p tr p ρ p σ e qqq “ e β ℜ p tr p ρ p σ e qqq “ e β ℜ p tr p ρ p σ e qqq “ φ β p σ e q . Also since ρ is unitary, we have that | ρ i,i p σ e q| ď for all σ e P G , and hence θ i ď . Moreover, if wedefine C ˚ i : “ sup β ą β „ ř σ e P G ` ´ ρ i,i p σ e q ˘ φ β p σ e q ř σ e P G φ β p σ e q ¨ λ i p β q ´ , (23)then θ i “ ´ ř σ e P G ` ´ ρ i,i p σ e q ˘ φ β p σ e q ř σ e P G φ β p σ e q ě ´ C ˚ i λ i p β q . This implies in particular that ´ C ˚ i λ i p β q ď θ i ď (24)and shows that, for any ď j ď ℓ , θ ´ ji ď θ ´ ℓi ď p ´ C ˚ i λ i p β q q ´ ℓ ď e C ˚ i ℓλ i p β q . (25)Consequently, for any ď j ď ℓ , θ ´ ji ´ ď jθ ´ ji p θ ´ i ´ q ď C ˚ i jλ i p β q e C ˚ i ℓλ i p β q . (26)By combining (22) and the triangle inequality, we get | µ β p W γ p i qq ´ θ ℓi | “ | θ | γ | i µ β p θ ´| γ | i q ´ θ ℓi | “ | θ | γ | i ` µ β p θ ´| γ | i q ´ ˘ ´ θ ℓ i ` θ ℓ i ´ ˘ |ď | θ | γ | i p µ β p θ ´| γ | i q ´ q| ` θ ℓ i p θ ´ ℓ i ´ q . By (26) and the fact that θ i ď , we have | θ | γ | i p µ β p θ ´| γ | i q ´ q| ` θ li p θ ´ ℓ i ´ q ď |p µ β p θ ´| γ | i q ´ q| ` C ˚ i ℓ λ i p β q e C ˚ i λ i p β q and hence | µ β p W γ p i qq ´ θ ℓi | ď |p µ β p θ ´| γ | i q ´ q| ` C ˚ i ℓ λ i p β q e C ˚ i λ i p β q . (27)Now recall that any plaquette p with ρ i,i p σ p q “ is contained in a G p i q -vortex, and that anyvortex has size at least six. Therefore, by Corollary 2.11, the probability that any given plaquette p has ρ i,i p σ p q “ is bounded from above by C p q λ i p β q . Consequently, µ β p| γ |q ď C p ,i q ℓλ i p β q . hus for any j ą , | µ β p θ ´| γ | i q ´ µ β p θ ´| γ | i t| γ |ď j u q| “ µ β p θ ´| γ | i t| γ |ą j u qď θ ´ ℓi µ β pt| γ | ą j uq ď θ ´ ℓi µ β p| γ |q j ď θ ´ ℓi C p ,i q ℓλ i p β q j (25) ď C p ,i q ℓλ i p β q e C ˚ i ℓλ i p β q j . (28)On the other hand, we have | µ β p θ ´| γ | i t| γ |ď j u q ´ | ď θ ´ ji ´ (26) ď C ˚ i jλ i p β q e C ˚ i λ i p β q . (29)Combining (28) and (29), using the triangle inequality and choosing j “ ? ℓ , we obtain | µ β p θ ´| γ | i q ´ | ď p C p ,i q ` C ˚ i q ? ℓλ i p β q e C ˚ i ℓλ i p β q . Combining this with (27), we infer that | µ β p W γ p i qq ´ θ ℓi | ď ” C p ,i q ` C ˚ i ? ℓ ` C ˚ i ℓ ℓ ı ¨ ℓλ i p β q e C ˚ i ℓλ i p β q . (30)Combining the equations above, we obtain | µ β p W γ p i qq ´ θ ℓi | ď | µ β p W γ p i qq ´ θ ℓ | ` µ β p| W γ p i q ´ W γ p i q|q ` µ β p| W γ p i q ´ W γ p i q|q` µ β p| W γ p i q ´ W γ p i q|q ` µ β p| W γ p i q ´ W γ p i q|q ` µ β p| W γ p i q ´ W γ p i q|qď ” C p ,i q ` C ˚ i ? ℓ ` C ˚ i ℓ ℓ ı ¨ ℓλ i p β q e C ˚ i ℓλ i p β q ` ” C C p ,i q ℓλ i p β q ı ` ” ℓ λ i p β q ı ` ” ı ` ” C C p ,i q ℓλ i p β q ı ` ” C p ,i q ℓ λ i p β q ı “ ” C p ,i q ` C ˚ i C ˚ i ? ℓ ` ℓ ℓ ı ¨ C ˚ i ℓλ i p β q e C ˚ i ℓλ i p β q ` ! C C p ,i q λ i p β q C ˚ i ` ℓ C ˚ i ℓ ) ¨ ” C ˚ i ℓλ i p β q ı ` C p ,i q λ i p β q p C ˚ i q ”` C ˚ i ℓλ i p β q ˘ ı . (31)Using that x ď e x and that x ď e x for x ą , it follows that | µ β p W γ p i qq ´ θ ℓi | ď " C p ,i q ` C ˚ i C ˚ i ? ℓ ` C C p ,i q λ i p β q C ˚ i ` ℓ C ˚ i ℓ ` ℓ ℓ ` C p ,i q λ i p β q p C ˚ i q * ¨ e C ˚ i ℓλ i p β q ď " C p ,i q ` C ˚ i C ˚ i ` C C p ,i q C ˚ i ` C ˚ i ` ` C p ,i q p C ˚ i q * ¨ „c ℓ ℓ ` λ i p β q ¨ e C ˚ i ℓλ i p β q . (32)Finally, combining the inequality | a ℓ ´ b ℓ | ď ℓ | a ´ b | with Taylor’s theorem, we find ˇˇ θ ℓi ´ e ´ ℓ p ´ θ i q ˇˇ ď ℓ | θ i ´ e ´ ℓ p ´ θ i q | ď ℓ p ´ θ i q ď ℓ p C ˚ i q λ i p β q ď e C ˚ i ℓλ i p β q ℓ . Combining this with (32), the desired conclusion follows. (cid:3) emark . The error term in the proof of Lemma 3.1 is made to work well together with theerror term in Lemma 3.3 below. However, some of the estimates above are far from optimal,especially when ℓλ i p β q ! . In particular, we several times use the estimate x ď e x , applied with x “ C ˚ ℓλ i p β q , which when ℓλ i p β q ! replaces something which is very small with somethingwhich is very close to one. An example of an upper bound on | µ β p W γ q ´ θ ℓ | which is much betterthan the bound given in (32) whenever ℓλ i p β q ă in fact immediately follows from (31): | µ β p W γ q ´ d ÿ i “ θ ℓi | ď d ÿ i “ ! C p ,i q e C ˚ i ` C ˚ i e C ˚ i ` ` e C ˚ i ` C p ,i q ` C C p ,i q )”c ℓ ℓ ` λ i p β q ı ℓλ i p β q . When ℓλ i p β q ! , the right-hand side of this inequality is much smaller than the right-hand sideof (4), enabling us describe the behaviour of µ β p W γ q in greater precision than what follows fromTheorem 1.1.Another alternative upper bound can be found by using the fact that C ˚ i ℓλ i p β q “ p C ˚ i q { ℓ { λ i p β q ` C ˚ i ℓλ i p β qq ˘ { ď p C ˚ i q { ℓ { λ i p β q e C ˚ i ℓλ i p β q to obtain (again using (31)) | µ β p W γ q ´ d ÿ i “ θ ℓi | ď ” C p ,i q ` C ˚ i p C ˚ i q { ℓ { ` C C p ,i q C ˚ i ` C p ,i q p C ˚ i q ` p C ˚ i q { ` ` C ˚ i ˘ ℓ C ˚ i ℓ { ı ¨ λ i p β q e C ˚ i ℓλ i p β q . If ℓ { ℓ { is bounded from above, then using this inequality instead of (32) would result in removingthe term a ℓ {| γ | from the inequality in Theorem 1.1 for a slightly different constant C . This mightbe useful if e.g. ℓ { ℓ { ă and λ i p β q ! b ℓ | γ | .Our next lemma, which generalizes Lemma 7.12 in [3], is useful when ℓ " λ i p β q . Lemma 3.3.
In the setting of Theorem 1.1, for each i P t , , . . . , d u , we have | µ β p W γ q| ď d ÿ i “ e ´ C ˚ p ℓ ´ ℓ q λ i p β q , where C ˚ : “ ´ cos ` π {| ker ρ i,i | ˘ | G | . Before we give a proof of Lemma 3.3, we will prove the following technical lemma, which will becrucial in the proof of Lemma 3.3. In this lemma, as in the rest of this section, we assume that ρ is a d -dimensional, faithful and unitary representation of G , which is such that ρ p g q is a diagonalmatrix for each g P G . Lemma 3.4.
Let K be a finite non-empty set and for each k P K , let g k P G . Finally, let i P t , , . . . , d u . Then, if β is sufficiently large, we have ˇˇˇˇ ř g P G ρ i,i p g q ś k P K φ β p g ` g k q ř g P G ś k P K φ β p g ` g k q ˇˇˇˇ ď ´ C ˚ λ i p β q . (33) Remark . Under stronger assumptions, the constant C ˚ on the right hand side can be mademuch better. For example, if the representation ρ is one-dimensional, the exact same proof givesthe same result but with C ˚ replaced by ˆ C p i q˚ “ inf β ą β max g P G : ρ i,i p g q“ ˇˇˇˇ ř g P G p ´ ρ i,i p g qq φ β p g q | K | ř g P G φ β p g q | K | ˇˇˇˇ . Proof.
Fix i P t , , . . . , d u and let S β p i, t g k uq : “ ř g P G ρ i,i p g q ś k P K φ β p g ` g k q ř g P G ś k P K φ β p g ` g k q . efine the subsets G and G of G by G : “ arg max g P G ź k P K φ β p g ` g k q , G : “ arg max g P G r G ź k P K φ β p g ` g k q . Since ź k P K φ β p g ` g k q “ exp ´ β ÿ k P K d ÿ j “ ℜ ` ρ j,j p g ` g k q ˘¯ , these sets are independent of the choice of β . Moreover, for any g P G , g P G and g P G r p G Y G q , we have ÿ k P K d ÿ j “ ℜ ` ρ j,j p g ` g k q ˘ ą ÿ k P K d ÿ j “ ℜ ` ρ j,j p g ` g k q ˘ ą ÿ k P K d ÿ j “ ℜ ` ρ j,j p g ` g k q ˘ and hence, as β Ñ 8 , ź k P K φ β p g ` g k q " ź k P K φ β p g ` g k q " ź k P K φ β p g ` g k q . (34)In particular, this implies that lim β Ñ8 ˇˇ S β p i, t g k uq ˇˇ “ | G | ¨ ˇˇ ÿ g P G ρ i,i p g q ˇˇ . (35)Since ρ is a unitary representation and ρ p g q is a diagonal matrix for each g P G we have that | ρ i,i p g q| “ for all g P G . If ρ i,i is not constant on G , it thus follows from (35) that lim β Ñ8 ˇˇ S β p i, t g k uq ˇˇ ă “ lim β Ñ8 ´ C ˚ λ i p β q | K | (36)and hence the desired conclusion holds in this case.Now assume that ρ i,i is constant on G . Note that this implies that | G | ď | ker ρ i,i | ă | G | . Also,note that in this case, both sides of (33) tend to as β Ñ 8 . Since ρ is unitary and faithful, forany fixed β ą , both sides of (33) are strictly smaller than , and hence the desired conclusion willfollow if we can show that the convergence of the right hand side is much faster than that of theleft hand side. To this end, note first that for any g P G , ˇˇ S β p i, t g k uq ˇˇ “ ˇˇˇˇˇ ř g P G ρ i,i p g q ś k P K φ β p g ` g k q ř g P G ś k P K φ β p g ` g k q ˇˇˇˇˇ “ ˇˇˇˇˇ ř g P G ρ i,i p g ` g q ś k P K φ β pp g ` g q ` g k q ř g P G ś k P K φ β pp g ` g q ` g k q ˇˇˇˇˇ “ ˇˇˇˇˇ ř g P G ρ i,i p g q ρ i,i p g q ś k P K φ β pp g ` g q ` g k q ř g P G ś k P K φ β pp g ` g q ` g k q ˇˇˇˇˇ “ ˇˇˇ ρ i,i p g q ˇˇˇ ¨ ˇˇˇˇˇ ř g P G ρ i,i p g q ś k P K φ β p g ` p g ` g k qq ř g P G ś k P K φ β p g ` p g ` g k qq ˇˇˇˇˇ “ ˇˇˇˇˇ ř g P G ρ i,i p g q ś k P K φ β p g ` p g ` g k qq ř g P G ś k P K φ β p g ` p g ` g k qq ˇˇˇˇˇ “ ˇˇ S β p i, t g k ` g uq ˇˇ , and hence we can assume that P G . Using the assumption that ρ i,i is constant on G , this impliesthat ρ i,i p g q “ for all g P G . Next, note that S β p i, t g k uq “ ´ ÿ g P G p ´ ρ i,i p g qq ź k P K φ β p g ` g k q φ β p ` g k q ¨ ś k P K φ β p ` g k q ř g P G ś k P K φ β p g ` g k q . (37)Using (34), it follows that ś k P K φ β p ` g k q ř g P G ś k P K φ β p g ` g k q Õ | G | ´ , as β Ñ 8 . efine G K : “ g P G : ` @ j P t , , . . . , d u : ρ j,j p g q “ or ρ j,j p g k q “ for all k P K ˘( . We will first show that for all ˆ g R G K , as β Ñ 8 , we have max g Pt ˆ g, ´ ˆ g u ź k P K φ β p g ` g k q φ β p ` g k q " max g Pt ˆ g, ´ ˆ g u φ β p g q | K | φ β p q | K | “ max g Pt ˆ g, ´ ˆ g u ź k P K φ β p g ` q φ β p ` q (38)or equivalently, that max g Pt ˆ g, ´ ˆ g u d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p g k q ¯ ą max g Pt ˆ g, ´ ˆ g u d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p q ¯ . We will show that this inequality holds by showing that for any g R G K , d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p g k q ¯ ` d ÿ j “ ℜ ´` ρ j,j p´ g q ´ ˘ ÿ k P K ρ j,j p g k q ¯ ą g P G : ρ i,i p g q“ d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p q ¯ . To this end, fix some g R G K and note that(i) cos p θ q ` cos p´ θ q ă cos p θ ` θ q ` cos p´ θ ` θ q whenever θ P p π { , π q and θ P p , π q , and(ii) ρ j,j p´ g q ´ “ ρ i,i p g q ´ .Combining these two observations, it follows that for any j P t , , . . . , d u such that ř k P K ρ j,j p g k q “ ř k P K ρ j,j p q “ | K | and ρ j,j p g q “ , ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p g k q ¯ ` ℜ ´` ρ j,j p´ g q ´ ˘ ÿ k P K ρ j,j p g k q ¯ ą ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ j,j p q ¯ ` ℜ ´` ρ j,j p´ g q ´ ˘ ÿ k P K ρ j,j p q ¯ . Since g R G K by assumption, there is at least one j P t , , . . . , d u such that this holds, and hence d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ i,i p g k q ¯ ` ℜ ´` ρ j,j p´ g q ´ ˘ ÿ k P K ρ i,i p g k q ¯ ě d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ i,i p q ¯ ` ℜ ´` ρ j,j p´ g q ´ ˘ ÿ k P K ρ i,i p q ¯ “ d ÿ j “ ℜ ´` ρ j,j p g q ´ ˘ ÿ k P K ρ i,i p q ¯ . This show that (38) holds. o compete the proof of the theorem, note that since lim β Ñ8 S β p i, t g k uq “ by assumption and ℜ ` ´ ρ j,j p g q ˘ ă . Using (38), this implies in particular that for any sufficiently large β , ˇˇ S β p i, t g k uq ˇˇ ď ˇˇˇˇˇ ´ ÿ g P G p ´ ρ i,i p g qq ¨ φ β p g q | K | φ β p q | K | ¨ ś k P K φ β p ` g k q ř g P G ś k P K φ β p g ` g k q ˇˇˇˇˇ “ ´ ÿ g P G p ´ ρ i,i p g qq ¨ φ β p g q | K | φ β p q | K | ¨ ś k P K φ β p ` g k q ř g P G ś k P K φ β p g ` g k q“ ´ ÿ g P G p ´ ℜ p ρ i,i p g qqq ¨ φ β p g q | K | φ β p q | K | ¨ ś k P K φ β p ` g k q ř g P G ś k P K φ β p g ` g k qď ´ ÿ g P G p ´ ℜ p ρ i,i p g qqq ¨ φ β p g q | K | φ β p q | K | ¨ | G |ď ´ ´ ´ cos ` π {| ker ρ i,i | ˘¯ ¨ λ i p β q | K | ¨ | G | . This concludes the proof. (cid:3)
Proof of Lemma 3.3.
Fix some i P t , , . . . , d u . As in the proof of Lemma 3.1, let γ be the setof all non-corner edges of γ , and let µ β denote conditional probability and conditional expectationgiven p σ e q e R γ . As observed earlier, p σ e q e P γ are independent spins under this conditioning.Take any e P γ . When p P P p e q , let σ ep : “ σ p ´ σ e . Then, for any ˆ σ e P G , µ β p σ e “ ˆ σ e q “ ś p P P p e q φ β p σ ep ` ˆ σ e q ř σ e P G ś p P P p e q φ β p σ ep ` σ e q . This implies that for any i P t , , . . . , d u , the expected value of ρ i,i p σ e q under µ β is given by | µ β p ρ i,i p σ e qq| “ ˇˇˇˇˇ ř ˆ σ e P G ρ i,i p ˆ σ e q ś p P P p e q φ β p σ ep ` ˆ σ e q ř σ e P G ś p P P p e q φ β p σ ep ` σ e q ˇˇˇˇˇ . Applying Lemma 3.4, with K “ P p e q , it follows that | µ β p ρ i,i p σ e qq| ď ´ C ˚ λ i p β q . This implies in particular that | µ β p ρ i,i p σ e qq| “ ´ C ˚ λ i p β q ď e ´ C ˚ λ i p β q . Since the edge spins p σ e q e P γ are independent given this conditioning, we obtain, for any i Pt , , . . . , d u , ˇˇ µ β ` ρ i,i p ÿ e P γ σ e q ˘ˇˇ “ ˇˇˇ ź e P γ µ β ` ρ i,i p σ e q ˘ˇˇˇ “ ź e P γ ˇˇ µ β ` ρ i,i p σ e q ˘ˇˇ ď e ´ C ˚ p ℓ ´ ℓ q λ i p β q . efine W γ p i q : “ ρ i,i p ř e P γ σ e q and let µ β be the measure we get when we condition on σ e for all σ e R γ . Then ˇˇ µ β p W γ p i qq ˇˇ “ ˇˇ µ β ` ρ i,i `ÿ e P γ σ e ˘˘ˇˇ “ ˇˇ µ β `ź e P γ ρ i,i p σ e q ˘ˇˇ “ ˇˇˇ µ β ´ ź e P γ ρ i,i p σ e q µ β ` ź e P γ r γ ρ i,i p σ e q ˘¯ˇˇˇ “ ˇˇˇ µ β ´ ź e P γ ρ i,i p σ e q µ β ` ź e P γ r γ ρ i,i p σ e q ˘¯ˇˇˇ ď µ β ´ˇˇˇ ź e P γ ρ i,i p σ e q ˇˇˇ ¨ ˇˇˇ µ β ` ź e P γ r γ ρ i,i p σ e q ˘ˇˇˇ¯ ď µ β ´ˇˇˇ µ β ` ź e P γ r γ ρ i,i p σ e q ˘ˇˇˇ¯ ď e ´ C ˚ p ℓ ´ ℓ q λ i p β q . This concludes the proof. (cid:3)
We are now ready to give a proof of Theorem 1.1.
Proof of Theorem 1.1.
We only need to prove the result for ℓ ă ℓ { , because the claim is automaticif ℓ ě ℓ { .From Lemmas 3.1 and 3.3, for each i P t , , . . . , d u we are given the two bounds | µ β p W γ p i qq ´ e ´ ℓ p ´ θ i q | ď C p i q A ¨ e C ˚ i ℓλ i p β q ¨ „c ℓ ℓ ` λ i p β q . (39)and | µ β p W γ p i qq| ď e ´ C ˚ p ℓ ´ ℓ q λ i p β q . From the second of these, using the triangle inequality, it follows that | µ β p W γ p i qq ´ e ´ ℓ p ´ θ i q | ď e ´ ℓ p ´ θ i p β qq ` e ´ C p i q˚ p ℓ ´ ℓ q λ i p β q ď e ´ C ˚ ℓλ i p β q ` e ´ C ˚ ℓλ i p β q ď e ´ C ˚ ℓλ i p β q . Combining this with (39), we obtain | µ β ` W γ p i q ˘ ´ e ´ ℓ p ´ θ i q | ` C ˚ i { C p i q˚ ď C p i q A e C ˚ ℓλ i p β q ”c ℓ ℓ ` λ i p β q ı ¨ ” e ´ C p i q˚ ℓλ i p β q ı C ˚ i { C p i q˚ “ C p i q A C ˚ i { C p i q˚ ”c ℓ ℓ ` λ i p β q ı . To obtain the conclusion of the theorem, note that by the triangle inequality | µ β ` W γ ˘ ´ d ÿ i “ e ´ ℓ p ´ θ i q | ď d ÿ i “ | µ β ` W γ p i q ˘ ´ e ´ ℓ p ´ θ i q | and hence by the previous equation, if we let C ˚ : “ min i Pt , ,...,d u {p ` C ˚ i { C ˚ q it follows that | µ β ` W γ ˘ ´ d ÿ i “ e ´ ℓ p ´ θ i q | ď d ÿ i “ ! p C p i q A C ˚ i { C ˚ q {p ` C ˚ i { C ˚ q ) ¨ ”c ℓ ℓ ` λ i p β q ı {p ` C ˚ i { C ˚ q . Finally, to obtain (4), note that since ρ p g q is a diagonal matrix, the matrix Θ defined in (3)is also a diagonal matrix, and hence the diagonal elements of Θ are exactly its eigenvalues, i.e. θ i “ Θ i,i . Since the eigenvalues of a matrix are invariant under a change of basis, the desiredconclusion follows. (cid:3) eferences [1] Borgs, C., Translation symmetry breaking in four dimensional lattice gauge theories , Commun. Math. Phys. 96,251–284 (1984).[2] Cao, S.,
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Department of Mathematics, KTH Royal Institute of Technology, 100 44Stockholm, Sweden.
E-mail address : [email protected] (Jonatan Lenells) Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stock-holm, Sweden.
E-mail address : [email protected] (Fredrik Viklund) Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stock-holm, Sweden.
E-mail address : [email protected]@math.kth.se