Pfaffian Correlation Functions of Planar Dimer Covers
PPfaffian Correlation Functionsof Planar Dimer Covers
Michael Aizenman ∗ Manuel La´ınz Valc´azar † Simone Warzel ‡ Version of November 14, 2016
Abstract
The Pfaffian structure of the boundary monomer correlation functionsin the dimer-covering planar graph models is rederived through a combi-natorial / topological argument. These functions are then extended into alarger family of order-disorder correlation functions which are shown to ex-hibit Pfaffian structure throughout the bulk. Key tools involve combinatorialswitching symmetries which are identified through the loop-gas representa-tion of the double dimer model, and topological implications of planarity.
The combinatorial problem of enumeration of dimer covers of graphs (aka dominotilings) has attracted interest from a diverse range of perspectives. These includestatistical mechanics, combinatorics, and algorithm complexity studies. In theirgroundbreaking papers, P. W. Kasteleyn, M. E. Fisher and H. V. N. Temperley [19,9, 26, 20], showed that for planar graphs the pure dimer problem admits a simplesolution in terms of a Pfaffian of what is now known as the Kasteleyn matrix. Thepure dimer partition functions is different in this sense from its monomer-dimer ∗ Departments of Mathematics and Physics, Princeton University, USA. † Princeton visiting student from Univ. Cantabria, Spain. ‡ Zentrum Mathematik, TU M¨unchen, Germany. a r X i v : . [ m a t h - ph ] N ov extension, for which its evaluation is computationally hard and thus not of simplePfaffian form [17].Extensive research has been devoted to various facets of dimer coverings, spe-cially in the case of planar and bipartite graphs. Examples include the close rela-tion between the partition functions of the dimer cover and of the Ising model [20,10, 24], non-existence of phase transitions [16], structure of the model’s correla-tion functions, the arctic circle phenomenon [7], continuum limits and their de-scription in terms of (conformal) field theory. More on this may be found in theoverviews [21, 8, 4] and references therein.Our main aim here is to present a simple geometric explanation of the Pfaffiannature of some of the model’s correlation function, through which the model’s n correlation functions can be determined from just the corresponding two pointfunction. The proofs given here bear close similarity to the methods which haverecently been developed for planar Ising spin models [2]. In analogy to the latter,the method relies on a combinatorial relation, which is valid for general graphs,combined with topological properties of planar graphs.It was already noted that for planar graphs the boundary monomer correlationfunctions, whose explicit definition is restated below, are given by Pfaffians of thecorresponding -point functions [25, 12]. The relation is less simple for the bulkmonomer correlation functions, but it was pointed out that these can be written asproducts of two Pfaffians [3].We start by giving an elementary geometric proof of the Pfaffian structure ofthe boundary monomer functions. The derivation also explains why these func-tions do not have the Pfaffian structure in the bulk. Furthermore, we formulatemore explicitly than was done in the literature the model’s disorder operators,and show that the expectation values of products of order-disorder operators yieldcorrelation functions which are simultaneously Pfaffian throughout the bulk andreduce to the simper monomer correlation functions for sites along a boundaryline.The disorder operators can be viewed as incomplete implementations of thedimer model’s Z gauge symmetry. From this perspective, their construction andbasic properties are similar to those of the corresponding concept for the Isingmodel, as discussed by L. P. Kadanoff and H. Ceva [18].The combinatorial and topological arguments presented here parallel the anal-ogous discussion of planar Ising model in the introductory sections of [2]. Anessential tool is a path integral representation of a duplicated system, which isreferred to as the double dimer model. The latter has been studied by R. Kenyonand D. Wilson (cf. [22, 23] and references therein) and is related to the monopole-dimer model recently studied in [5]. Given a finite graph G = ( V , E ) of vertex set V , a perfect matching or dimer coveris a subset of the edge set, ω ⊂ E , such that every vertex is covered by exactlyone edge. The set of perfect matchings is denoted Ω G . The dimer-cover partitionfunction counts the number of the graph’s perfect matchings.Perfect matchings can also be weighted through a complex-valued edge func-tion K : E (cid:55)→ C . Given such an edge weight, the weighted dimer-cover partitionfunction is Z G ,K := (cid:88) ω ∈ Ω G χ K ( ω ) (2.1)with χ K ( ω ) := (cid:89) b ∈ ω K b . Of particular interest is the effect on the dimer-cover partition function of theremoval of a collection of sites, M ⊂ V , which are regarded as covered by sepa-rate monomers. The collection of perfect matchings of the remaining vertices isdenoted by Ω G ( M ) and Z G ,K ( M ) := (cid:88) ω ∈ Ω G ( M ) χ ( ω ) (2.2)stands for the weighted partition function of the monomer-depleted graph. Itshould be noted that not all graphs admit a perfect matching. In particular if M is of odd cardinality, then at least one of the factors in Z G ,K × Z G ,K ( M ) van-ishes. For simplicity, we shall concentrate in this paper on the case Z G ,K (cid:54) = 0 forwhich the monomer correlation function for an even collection of disjoint sites { x , ..., x n } ⊂ V is well-defined as S n ( x , ..., x n ) := (cid:104) n (cid:89) j =1 η x j (cid:105) G ,K := Z G ,K ( { x , ..., x n } ) Z G ,K (2.3)The variables η x j should be thought of an operator in the functional integral rep-resenting the average (cid:104)·(cid:105) G ,K corresponding to the dimer partition function Z G ,K .These variables take a similar role as the spin variables in the related Ising model.In the planar set-up, monomer correlations have been studied early on by M. E.Fisher and J. Stephenson [11], who determined the fall-off of S ( x , x ) on thesquare lattice Z for K ≡ and two monomers in the bulk to behave asymptoti-cally as | x − x | − / for large separation (making the similarity to the Ising modeleven more apparent [24]). The values of other special placements of momomerpairs on a square lattice are also known (cf. [11, 15, 4]). In case of the infinitehalf-lattice Z × Z + , the monomer boundary correlations in case K ≡ have beencomputed not long ago by V. B. Priezzhev and P. Ruelle [25]. They turned out tobe Pfaffians with two-point function given by S (( ξ, , ( η, (cid:40) − π | ξ − η | if | ξ − η | is odd otherwise. (2.4) The removal of a site in a finite graph, or equivalently its cover by a monomer,has a drastic effect on the graph’s dimer covers: if Z Λ ,K (cid:54) = 0 then for parityreasons the modified graph has no dimer cover. The removal of an even numberof sites does not automatically invalidate the existence of a cover. Its effect on thedistribution of the dimer covers may be localized to a collection of random pathslinking pairwise the affected sites. A convenient way to arrive at such a stochasticgeometric picture of correlations is to consider the overlay of two sets of dimercovers, one of the original graph and the other of its depleted version resulting inthe double dimer model. This technique is reminiscent of the duplication whichis an effective tool in the study of the Ising model’s correlation functions in itsrandom current representation [13, 1].The configurations of doubled dimer covers of a graph G = ( V , E ) , depletedby corresponding sets of monomers M , M ⊂ V will be denoted here as: ω (2) = ( ω , ω ) ∈ Ω G ( M ) × Ω G ( M ) =: Ω (2) G ( M , M ) . (3.1)Clearly, each such configuration ω (2) is in one-to-one correspondence with a 2-multigraph with vertex set V and the collection of edges in ω (2) . The following x x x x x x Figure 1: A double dimer configuration ω (2) = ( ω , ω ) on a finite graph (whoseedges are indicated in grey) with disjoint sets of monomers covering selected sitesin one or the other copy of the graph. The edges of ω and ω , are marked insolid and dashed lines, correspondingly. The overlay results in a configurationof alternatingly marked paths connecting the monomers and alternatingly markedloops, as described in Lemma 3.1.deterministic statement concerning such pairs of matchings relates the duplicateddimer cover model with a system of loops and paths with prescribed boundariesgiven by the monomers, cf. Figure 1. Lemma 3.1 (Double matching as a loop / path system) . For any finite graph G =( V , E ) , let ω (2) ∈ Ω (2) G ( M , M ) be a pair of dimer covers of G depleted by adisjoint pair of monomers, M , M ⊂ G . Then the multiplicity with which theedges are covered by ω (2) coincides with that of a collection Γ = Γ( ω (2) ) of edge-disjoint loops and paths where each γ ∈ Γ is eitheri. a double loop covering a single edge,ii. a simple loop of an even number of non-repeated edges,iii. a simple path with boundary set ∂γ ⊂ M (cid:116) M .In case iii., the numbers of edges of γ is odd if and only if its two boundary sitesare in the same monomer set (i.e. either both in M or both in M ).The loop-path characterisation of double covers ω (2) in terms of Γ partitionstheir collection Ω (2) G ( M , M ) into equivalence classes, each of n s (Γ) elements,where n s (Γ) is the number of simple loops in Γ . Proof.
In the case of disjoint monomer sets, the degree of each site x ∈ V in themultigraph formed from the edge set of ω (2) is either or , and given by deg ω (2) ( x ) = deg ω ( x ) + deg ω ( x ) = (cid:40) , if x ∈ V\ [ M (cid:116) M ]1 , if x ∈ M (cid:116) M . (3.2)It follows that the collection of edges with multiplicity is the disjoint union ofloops (of no boundary) and paths with end points in M (cid:116) M , each made of simpleedges in ω j , at alternating values of j = 1 , . The stated constraints on the parityof the number of edges in the loops and paths readily follow from the constraintthat the path’s edges alternate between the two dimer covers. In case of the openpaths, the identity of the cover to which an edge of γ belongs can be determinedsuccessively starting from the end points. There is no such constraint for the n s (Γ) simple closed loops, and hence for each of these there are exactly two choices(independent among the loops) for the alternating values of j ∈ { , } .The above representation of Ω (2) G ( M , M ) in terms of loops and paths maybe extended by allowing the two sets of monomers to overlap, or coincide. Thecorresponding pure loop gas was recently studied in [5].The loop gas picture of the double-dimer partition functions Z (2) G ,K ( M , M ) := Z G ,K ( M ) Z G ,K ( M ) = (cid:88) ω (2) ∈ Ω (2) ( M ,M ) χ K ( ω ) χ K ( ω ) , (3.3)is particularly convenient in revealing switching symmetries of the double dimermodel’s connection amplitudes. Similar symmetries have been noted for the cor-relation functions of the Ising model, revealed there through its random currentrepresentation.The connection amplitudes are defined as restricted sums such as Z (2) G ,K ( M , M ; x j ↔ y j for j = 1 , . . . , N ) := (cid:88) ω (2) ∈ Ω (2) ( M ,M ) χ K ( ω ) χ K ( ω ) N (cid:89) j =1 (cid:20) x j ω (2) ←−→ y j (cid:21) . (3.4)where { x j , y j } j =1 ,...,N are pairs of sites in M (cid:116) M , and (cid:2) x j ω (2) ←−→ y j (cid:3) is an indica-tor function corresponding to the condition that the monomers x j , y j are connectedby a path γ ∈ Γ( ω (2) ) . Lemma 3.2 (Switching principle I) . For any finite graph G = ( V , E ) , pair ofdisjoint monomer sets M , M and { x, y } ⊂ V\ ( M (cid:116) M ) : Z (2) G ,K ( M (cid:116) { x, y } , M ; x ↔ y, C ) = Z (2) G ,K ( M , M (cid:116) { x, y } ; x ↔ y, C ) , (3.5) Z (2) G ,K ( M (cid:116) { x } , M (cid:116) { y } ; x ↔ y, C ) = Z (2) G ,K ( M (cid:116) { y } , M (cid:116) { x } ; x ↔ y, C ) (3.6) where C stands for any collection of other connection conditions among monomersin M (cid:116) M .Proof. Considering first the case C = ∅ (i.e. no other conditions), let Ω (2) ( M (cid:116){ x, y } , M ; x ↔ y ) be the set of double dimer covers for which there is a path γ ( x,y ) ∈ Γ with ∂γ ( x,y ) = { x, y } . The first assertion is based on the bijection Ω (2) ( M (cid:116) { x, y } , M ; x ↔ y ) → Ω (2) ( M , M (cid:116) { x, y } ; x ↔ y ) implemented by the symmetric difference (cid:52) of sets: ( ω , ω ) (cid:55)→ (cid:0) ω (cid:52) γ ( x,y ) , ω (cid:52) γ ( x,y ) (cid:1) . (3.7)This map reverses the “edge coloring” along the path γ ( x,y ) connecting x and y with the color indicating to which of the two dimer covers the edge belongs.The first identity thus follows immediately from the fact that the path weights areunchanged under a color-flip operation.The same switching argument implies also the second identity, and the gener-alization to more general condition C .The loop gas formulation of the double dimer model casts its correlation func-tions in terms of (discrete) path integrals, thereby bringing it closer to a broadrange of physics models. A more explicit version of this representation, whichcould be used for an alternative presentation of the analysis which follows, isstated in Appendix A. The switching principle allows a simple proof of the fact that boundary monomercorrelation functions have a Pfaffian nature on all planar graphs. The correspond-ing result for Ising model’s boundary spin-spin correlation functions goes backto [14]. Our proof parallels the more recent rederivation of that relation in [2].For the dimer model the following statement was derived in [25] in case ofthe infinite planar half-lattice for which the two-point function is given by (2.4).For other planar graphs, the theorem was recently established by different meansin [12].
Theorem 4.1 (Pfaffian boundary correlations) . For any finite planar graph G =( V , E ) the boundary values of the monomer correlation functions satisfy S n ( x , ..., x n ) = (cid:88) π ∈ Π n sgn( π ) n (cid:89) j =1 S ( x π (2 j − , x π (2 j ) ) ≡ Pf n ( S ( x i , x j )) (4.1) where M := { x , ..., x n } ranges over sequences of disjoint vertices positionedin a cyclic order along any boundary of G . Moreover, Π n is the collection ofpairings of { , ..., n } , and sgn( π ) is the pairing’s parity.Proof. Through a known characterization of Pfaffians (provable by an inductionargument) it suffices to show that for each n > and any cyclicly ordered se-quence of boundary sites S n ( x , ..., x n ) = Q n ( x , ..., x n ) (4.2)with Q n defined as: Q n ( x , ..., x n ) := n (cid:88) k =2 ( − k S ( x , x k ) S n − . ( (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n ) (4.3)At fixed k the term S ( x , x k ) S n − ( (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n ) is a sum of over con-figurations of the duplicated system, ω (2) ∈ Ω (2) ( { x , x k } , { (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n } ) ,which may be grouped according to the paths of Γ( ω (2) ) which connect to x and x k . These fall into two classes: the monomers x and x k may be connected toeach other by some γ ∈ Γ , or else each is connected to another monomer: Q n ( x , ..., x n ) ( Z G ,K ) = (4.4) n (cid:88) k =2 ( − k Z (2) G ,K ( { x , x k } , { (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n } ; x ↔ x k )+ n (cid:88) k =2 ( − k n (cid:88) l,m =2 k (cid:54) = l (cid:54) = m (cid:54) = k Z (2) G ,K (cid:18) { x , x k } , { (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n } ; x ↔ x m x k ↔ x l (cid:19) . Being based on combinatorial arguments, the above relation holds for arbitrarygraphs. It will now be combined with the following topological implication ofplanarity. For any planar graph, a pair of monomers { x i , x j } located along theboundary can be linked by one of the non-intersecting simple paths of Γ( ω (2) ) only if the two are either consecutively placed along the boundary or separated byan even number of other monomers. In other words, in the cases considered here: x i ↔ x j = ⇒ ( − i − j = − . (4.5)For the pair of sums on the right side of (4.4) this implies:i. In the first sum ( − k − = − , and hence n (cid:88) k =2 ( − k Z (2) G ,K ( { x , x k } , { (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n } ; x ↔ x k ) = n (cid:88) k =2 ( − k Z (2) G ,K ( ∅ , M ; x ↔ x k ) = n (cid:88) k =2 Z (2) G ,K ( ∅ , M ; x ↔ x k )= Z G ,K ( M ) Z G ,K = S n ( x , . . . , x n ) ( Z G ,K ) . (4.6)Here the first step is a consequence of the switching principle of Lemma 3.2.ii. In the second sum ( − k − l = − , and thus n (cid:88) k,l =2 m (cid:54) = k (cid:54) = l (cid:54) = m ( − k Z (2) G ,K (cid:18) { x , x k } , { (cid:26)(cid:26) x , x , ..., (cid:8)(cid:8) x k , ..., x n } ; x ↔ x m x k ↔ x l (cid:19) = 0 (4.7)due to the antisymmetry of the summands under the exchange of k with l asis apparent from the switching principle of Lemma 3.2.Upon insertion in (4.4) these relations prove (4.2), and through it the claimedPfaffian structure. In the context of planar Ising spin systems order-disorder correlation functionshave a Pfaffian structure throughout the bulk and reduce to simple correlations0functions in case of sites along the boundary. They have been recently discussed,from a pair of somewhat different perspectives, in [6] and [2]. To present a relatedconcept for the dimer model’s correlation functions we turn now to the dimeranalog of disorder operators.The definition of the disorder operators may be placed in the broader contextof gauge symmetries. For that let us first recall Kasteleyn’s observation [20] thatthe dimer model has the following Z gauge symmetry in the dependence of thepartition function Z G ,K on the kernel K .For subsets B ⊂ V let us denote ∂B := { [ x, y ] ∈ E | if exactly one of the two points is in B } (5.1)which forms the edge boundary of B .Next, for any edge set E ⊂ E let T E : C E → C E be the transformation of K which flips its signs over the edges in E , ( T E K ) b = (cid:40) − K b if b ∈ EK b otherwise. (5.2)The key observation now is that if E = ∂B for a set B ⊂ V then Z Λ ,T ∂B K = ( − | B | Z Λ ,K , (5.3)where | B | is the number of sites in B . For B containing a single site the relation(5.3) holds since in each dimer cover exactly one dimer is affected by the sign flip T ∂B . The general case follows by noting the commutative product relation T ∂B = (cid:89) x ∈ B T ∂ { x } (5.4)and taking the corresponding product of the single site case of (5.3).In view of the simplicity of the effect of T ∂B on the partition function (andalso on the expectations defined below), such mappings may be regarded as themodel’s gauge transformations.The disorder operators which are defined next may be viewed as partial gaugetransformations, given by T E where E is the collection of edges which are tra-versed by a line (cid:96) which has only transversal intersections with the edges of E ∂(cid:96) . The end-points of (cid:96) are associated with sites of the dual graph G ∗ , namely the faces of G in which theend points of (cid:96) lie. One may note that away from ∂(cid:96) the transformation locallyacts as if it could be associated with a gauge transformation – but it is not (unless ∂(cid:96) = ∅ ). Definition 5.1.
For a planar graph G = ( V , E ) with edge weights K : E (cid:55)→ C :i. The disorder operators τ (cid:96) are associated with site-avoiding, lines (cid:96) , . . . , (cid:96) n in the plane in which G is embedded. To each such line we associate thetransformation K (cid:55)→ T (cid:96) ∗ K where (cid:96) ∗ is the set of edges in E which are crossedby (cid:96) an odd number of times.ii. The expectation values of products of such disorder operators is defined as: (cid:104) n (cid:89) j =1 τ (cid:96) j (cid:105) G ,K := Z G ,T (cid:96) ∗ ◦···◦ T (cid:96) ∗ n K Z G ,K . (5.5)As an expression of the above mentioned gauge symmetry, the expectationvalue (cid:104) (cid:81) Nj =1 τ (cid:96) j (cid:105) K is a homotopy invariant under deformations of any (cid:96) j in theplane which preserve the line’s endpoints. More precisely, as a simple conse-quence of (5.3) we have: Proposition 5.2 (Homotopy invariance) . For any finite planar graph G = ( V , E ) ,edge weights K : E (cid:55)→ C and lines (cid:96) j , j ∈ { , . . . , n } , as in Definition 5.1,under deformations of each (cid:96) j in the plane which preserve the line’s endpoints theexpectation value functional ( (cid:96) , . . . , (cid:96) N ) (cid:55)→ (cid:104) (cid:81) nj =1 τ (cid:96) j (cid:105) G ,K is multiplied by ( − each time one deformed line is moved over a site of the planar graph. The above construction parallels the definition of disorder operators for theIsing model [18]. Disorder lines for the dimer-monomer model appear also in therecent discussion of the dimer model’s partition function in terms of Grassmannintegrals [3].
Our main concern in this paper will be canonical pairs of order-disorder variables,cf. Figure 2.2Figure 2: Order-disorder variables for a planar graph. Each of the ovals in thefigure encircles a pair consisting of a site x j ∈ G and a point, marked × , within anadjacent cell of the dual graph x ∗ j ∈ G ∗ . The disorder variables τ (cid:96) j are associatedwith lines (cid:96) j , each linking the corresponding × marked sites with a point in the grand central cell x ∗ . The disorder lines (cid:96) , (cid:96) , . . . are enumerated cyclicly in theorder of the lines’ emergence from the grand central x ∗ . The correlation functionassociated with such an array is defined in (5.5) Definition 6.1.
For a planar graph G = ( V , E ) with a set of edge weights K : E (cid:55)→ C , open-ended, site-avoiding, non-intersecting lines (cid:96) , . . . , (cid:96) n in the planein which G is embedded, together with disjoint sites x , . . . , x n ⊂ V are called acollection of canonical pairs of order-disorder variables in case:i. all lines have a common end-point x ∗ ∈ G ∗ , called the grand central , andii. the other endpoint of (cid:96) j is a face x ∗ j ∈ G ∗ adjacent to x j for all j ∈ { , . . . , n } .We call the canonical pairs of order-disorder variables cyclicly ordered if they arelabeled relative to their intersections with the edge boundary of x ∗ .The expectation values of products of order-disorder variables operators µ j := η x j τ (cid:96) j are defined as (cid:104) n (cid:89) j =1 µ j (cid:105) G ,K := Z G ,T (cid:96) ∗ ◦···◦ T (cid:96) ∗ n K ( { x , . . . , x n } ) Z G ,K . (6.1)Our main new result is:3 Theorem 6.2 (Pfaffian correlations) . For a finite planar graph G = ( V , E ) withedge weights K : E (cid:55)→ C , for any collection of canonical pairs of order-disordervariables p j = ( x j , (cid:96) j ) , j ∈ { , . . . , n } , ordered cyclicly relative to the grandcentral (cid:104) n (cid:89) j =1 µ j (cid:105) G ,K = (cid:88) π ∈ Π n sgn( π ) n (cid:89) j =1 (cid:104) µ π (2 j − µ π (2 j ) (cid:105) G ,K ≡ Pf n ( (cid:104) µ j µ k (cid:105) G ,K ) . (6.2)This result includes Theorem 4.1 as a special case. To see that, let us firstnote that for sites x j which lie along the boundary of the grand-central x ∗ , thecorresponding disorder sites may be chosen as x ∗ j = x ∗ . When the lines (cid:96) j donot cross any edge, as in this case, the operators τ (cid:96) j act as identity and may beomitted. Theorem 4.1 then emerges through the inverted picture of the plane inwhich the complement of the finite graph is viewed as a single cell (of potentiallylarge boundary).In case the monomers { x j − , x j } are pairwise adjacent, the disorder linesmay be chosen so that their actions are pairwise equivalent, and thus cancel eachother. In that case the pairwise product of two order-disorder variables reduces toa an ordinary product of monomers, i.e., a dimer µ j − µ j = η x j − η x j , so that (cid:10) n (cid:89) j =1 τ j (cid:11) G ,K = (cid:10) n (cid:89) j =1 η x j − η x j (cid:11) G ,K . (6.3)The proof of Theorem 6.2 is organized along the lines used to establish theboundary case, Theorem 4.1. However, the relevant topological considerationsare considerably more intricate. Defining, in analogy with Q n of (4.3), R n ( p , ...p n ) := n (cid:88) k =2 ( − k (cid:104) µ µ k (cid:105) G ,K (cid:104) (cid:89) j ∈{ (cid:1) , ,.., (cid:1) k,.., n } µ j (cid:105) G ,K , (6.4)(with p j := ( x j , (cid:96) j ) standing for an order-disorder variables) the Pfaffian structurewill be shown by proving that for each n and choice of order-disorder pairs: R n ( p , ...p n ) = (cid:104) (cid:89) j ∈{ ,.., n } µ j (cid:105) G ,K . (6.5)4At specified k the product of the order-disorder correlators is given by: (cid:104) µ µ k (cid:105) G ,K (cid:104) (cid:89) j ∈{ (cid:1) , ,.., (cid:1) k,.., n } µ j (cid:105) G ,K × Z G ,K = (6.6) (cid:88) ω (2) ∈ Ω (2) ( { x ,x k } , ( { (cid:26)(cid:26) x ,x ,.., (cid:26)(cid:26) x k ,..,x n } χ K ( ω ) χ K ( ω ) ( − ( ω | (cid:96) ,k ) ( − ( ω |L\ (cid:96) ,k ) where ( ω | (cid:96) ,k ) denotes the number of intersections of the edges of ω with twodisorder lines (cid:96) ,k := { (cid:96) , (cid:96) k } and likewise ( ω |L ) denotes the number of intersec-tions of the edges of ω with the collection of all disorder lines L := { (cid:96) , . . . , (cid:96) n } .The terms in the above sum can be split into two classes, according to whetherthe loop / path configuration Γ( ω (2) ) includes a path with ∂γ (1 ,k ) = { x , x k } ,or not. The corresponding partial sums will be studied through the followingquantities: W (2) G ,K ( { M , L } , { M , L } ; C ) := (cid:88) ω (2) ∈ Ω (2) ( M ,M ) (cid:2) ω (2) satisfies C (cid:3) χ K ( ω ) ( − ( ω | L ) χ K ( ω ) ( − ( ω | L ) (6.7)in which we specify a set of connections C of the involved monomer sets M , M .A key result here is the corresponding version of the switching lemma: Lemma 6.3 (Switching principle II) . For planar graphs, and the setup of Theo-rem 6.2, we have for any m (cid:54) = k (cid:54) = l (cid:54) = m : W (2) G ,K ( { p , p k } , { (cid:26)(cid:26) p , p , . . . , (cid:26)(cid:26) p k , . . . , p n } ; x ↔ x k )= ( − k W (2) G ,K ( ∅ , { p , . . . , p n } ; x ↔ x k ) (6.8) W (2) G ,K (cid:18) { p , p k } , { (cid:26)(cid:26) p , p , . . . , (cid:26)(cid:26) p k , . . . , p n } ; x ↔ x m x k ↔ x l (cid:19) = ( − k − l − W (2) G ,K (cid:18) { p , p l } , { (cid:26)(cid:26) p , p , ..., . . . , (cid:0)(cid:0) p l , . . . , p n } ; x ↔ x m x k ↔ x l (cid:19) (6.9) Proof.
The relation (6.8), which involves terms for which x ↔ x k , will be estab-lished through the switching transformation: ( ω , ω ) (cid:55)→ ( ω ∆ γ (1 ,k ) , ω ∆ γ (1 ,k ) ) (6.10)5Expanding the quantities W (2) (defined in (6.7)), which appear in (6.8), into sumsover ω (2) , the ratio of the corresponding terms is χ K ( ω ∆ γ (1 ,k ) ) χ K ( ω ∆ γ (1 ,k ) ) χ K ( ω ) χ K ( ω ) ( − ( ω ∆ γ (1 ,k ) |L ) ( − ( ω | (cid:96) ,k ) ( − ( ω |L\ (cid:96) ,k ) = ( − ( γ (1 ,k ) |L ) ( − ( ω (2) | (cid:96) ,k ) , (6.11)where the last step is by an elementary calculation in Z . The relation (6.8) thenfollows from the special case l = 1 through the lemma which is stated next. (Thisis where the model’s planarity plays a role.)The relation (6.9) concerns terms ω (2) for which x k ↔ x l for some l (cid:54) = 1 . Forthat we employ the switching transformation ( ω , ω ) (cid:55)→ ( ω ∆ γ ( k,l ) , ω ∆ γ ( k,l ) ) . (6.12)By a calculation similar to (6.11), the ratio of the corresponding contributions tothe sums which yield the two quantities W (2) in (6.9) is: ( − ( ω ∆ γ ( k,l ) | (cid:96) ,l ) ( − ( ω ∆ γ ( k,l ) |L\ (cid:96) ,l ) ( − ( ω | (cid:96) ,k ) ( − ( ω |L\ (cid:96) ,k ) = ( − ( γ ( k,l ) |L ) ( − ( ω (2) | (cid:96) ,l ) ( − ( ω (2) | (cid:96) ,k ) = ( − ( γ ( k,l ) |L ) ( − ( ω (2) | (cid:96) k,l ) . (6.13)The relation (6.9) then again follows from the next lemma.The topological statement which was quoted within the above proof is: Lemma 6.4 (Intersection parities) . In the planar graph setup of Proposition 6.3,for any ω (2) such that x k ↔ x l with respect to the corresponding loop / pathconfiguration Γ( ω (2) ) : ( − ( γ ( k,l ) |L ) ( − ( ω (2) | (cid:96) k,l ) = ( − k − l − . (6.14) Proof.
To establish this relation it is useful to join the open ended paths γ of Γ( ω (2) ) with the disorder lines corresponding to the paths’ edges into loops withonly transversal crossing. For this purpose, we employ the following construction.1. Join directly each x j with the endpoint x ∗ j of the corresponding disorder line (cid:96) j .62. Connect pairwise the other endpoints of the disorder lines within the grandcentral x ∗ , so that (cid:96) k is connected to (cid:96) l and the remaining lines are pairedconsecutively with respect to the cyclic ordering.Let σ ( k,l ) be the loop which includes γ ( k,l ) concatenated with (cid:96) k and (cid:96) l in theabove construction, and let Σ ( k,l ) stand for the collection of the other loops whichthe construction yields. Any two planar loops, simple or not, with transversalcrossings can intersect only even number of times (as can be deduced from theJordan curve theorem). Thus σ ( k,l ) has an even intersection with Σ ( k,l ) . The inter-sections within the grand central cell contribute to this the factor ( − k − l − , andthe rest is the parity of the intersections of γ ( k,l ) and (cid:96) k,l with the rest. Hence: − k − l − ( − ( γ ( k,l ) |L ) − ( γ ( k,l ) | (cid:96) k,l ) ] ( − ( ω (2) | (cid:96) k,l ) − ( γ ( k,l ) | (cid:96) k,l ) ]= ( − k − l − ( − ( γ ( k,l ) |L ) ( − ( ω (2) | (cid:96) k,l ) , (6.15)as claimed in (6.14).We are now ready to complete the proof of Theorem 6.2. Proof of Theorem 6.2.
Similarly as in the Proof of Theorem 4.1, it remains toshow that (cid:104) n (cid:89) j =1 µ j (cid:105) G ,K = R n ( p , ...p n ) . (6.16)The right side times ( Z G ,K ) may be rewritten as n (cid:88) k =2 ( − k W (2) G ,K ( { p , p k } , { (cid:26)(cid:26) p , p , . . . , (cid:26)(cid:26) p k , . . . , p n } ; x ↔ x k )+ n (cid:88) k =2 ( − k n (cid:88) l,m =2 k (cid:54) = l (cid:54) = m (cid:54) = k W (2) G ,K (cid:18) { p , p k } , { (cid:26)(cid:26) p , p , . . . , (cid:26)(cid:26) p k , . . . , p n } ; x ↔ x m x k ↔ x l (cid:19) = n (cid:88) k =2 W (2) G ,K ( ∅ , { p , . . . , p n } ; x ↔ x k ) . (6.17)Here the last line results from the switching Lemma 6.3. More precisely, the sec-ond sum on the left vanishes thanks to the antisymmetry in the k (cid:54) = l summationas is apparent from (6.9). Applying (6.8) to the first sum on the left yields the sumon the right, which coincides with T n ( p , . . . , p n ) .7 A A path integral representation
The loop gas formulation of the double dimer model, which is presented in Sec-tion 3, is of help in relating it to a broad range of physics models, for which relatedtechniques are of relevance. To highlight this picture, let us just state here the re-sulting path integral representation (in a discrete sense) of the model’s correlationfunction.Lemma 3.1 allows to classify the double-dimer cover configurations in termsof the loop-gas configuration Γ( ω (2) ) . Upon partial summation in (3.3) over theequivalence classes of configurations with common Γ( ω (2) ) one gets Z (2) G ,K ( M , M ) = (cid:88) Γ ∈ Ω ( L ) G ( M ,M ) n s (Γ) (cid:89) γ ∈ Γ χ K ( γ ) (A.1)where Ω ( L ) G ( M , M ) is the collection of loop / path configurations which are con-sistent with the conditions listed in Lemma 3.1, and χ K ( γ ) = (cid:81) b ∈ γ K b for each γ ∈ Γ . Next, summing over the loops of Γ , while keeping fixed the configura-tion’s the open-ended paths, one obtains a path representation of the monomercorrelation functions.For the monomer correlation function, which is defined in (2.3), this yields S ( x , x ) = ( Z G ,K ) − (cid:88) Γ ∈ Ω L G ( { x ,x } , ∅ ) n s (Γ) (cid:89) γ ∈ Γ χ K ( γ )= (cid:88) γ ∈ Ω A1 ∂γ = { x ,x } χ K ( γ ) (cid:18) Z G ,K ( V ( γ )) Z G ,K (cid:19) [ γ is odd ] , (A.2)where Ω A1 denotes the collection of simple paths on G .For a more general expression we use Γ P to refer to collections of non-intersectingsimple paths on the graph G , and denote by Ω A n the set of such path collections of n elements. The set of vertices which are covered by paths in Γ P will be denotedby V (Γ P ) , and the collection of the paths’ boundary points by ∂ Γ P = (cid:116) γ ∈ Γ P ∂γ .In these terms, (A.1) yields the following path representation. Proposition A.1 (Path integral for correlations) . For any finite graph G = ( V , E ) and disjoint sites { x , . . . , x n } ⊂ V the monomer correlation function admits therepresentation S n ( x , . . . , x n ) = (cid:88) Γ P = { γ ,...,γ n }⊂ Ω A n ∂ Γ P = { x ,...,x n } w K (Γ P ) (cid:89) γ ∈ Γ P [ γ is odd ] , (A.3)8 with the weight function w K (Γ P ) := (cid:18) Z G ,K ( V (Γ P )) Z G ,K (cid:19) (cid:89) γ ∈ Γ P χ K ( γ ) . (A.4) Acknowledgements
This work was supported in part by the NSF grant PHY-1305472. We thank HugoDuminil-Copin and Vincent Tassion for stimulating discussions of related topics,and Princeton University for hosting S. Warzel as a PU Global Scholar.
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