LLIMIT SHAPES FOR THE DIMER MODEL
NIKOLAI KUCHUMOV
Abstract.
We prove the existence of a limit shape for the dimer model on planarperiodic bipartite graphs with an arbitrary fundamental domain and arbitrary periodicweights. This proof is based on a variational principle that uses the locality of the modeland the compactness of the space of states.
Contents
1. Introduction and Summary 21.1. Introduction 21.2. Summary 41.3. Brief plan of the proofs 4Acknowledges 42. Dimers on graphs with boundaries and height functions 52.1. Dimers on graphs with boundaries 52.2. Dimers on bipartite graphs 52.3. Height function 72.4. Newton polygon 93. Boltzmann distribution on dimer covers 104. The cutting rule for the dimer model 114.1. Graphs with boundary conditions 114.2. The Cutting Rule 114.3. The cutting rule for surface graphs 125. Periodic graphs 145.1. Periodic graphs 145.2. Thermodynamic limit of the dimer model on the torus. 145.3. Surface tension 146. Thermodynamical limit of the dimer model on the plane 156.1. Asymptotic height functions 156.2. Approximations of domains 156.3. The Limit Shape Theorem 167. The proof of the variational principle 167.1. Convergence of height functions to the limit shape 167.2. The localization of the partition function 167.3. The Surface tension functional and the limit shape 178. The concentration lemma 178.1. Coupling lemma 199. Proofs of properties of height functions 209.1. The main construction for -Lipschitz functions 209.2. Piecewise linear approximations of asymptotic height functions 21 a r X i v : . [ m a t h - ph ] D ec NIKOLAI KUCHUMOV
Introduction and Summary
Introduction.
This paper is devoted to the limit shape phenomena in lattice modelsof equilibrium statistical mechanicsWe use an example of a periodic planar dimer model with an arbitrary fundamental domainand arbitrary periodic weights. Past results for arbitrarily planar domains were done fordomino tilings (the fundamental domain correspondent to lattice Z in the dimer model) andweights equal to one [CKP].The Dimer model is a stochastic model on finite graphs. We describe a set of configurations(dimer covers) in terms of discrete functions called height functions. When the size of thesystem is relatively small, the system behaves randomly. As the system grows, all heightfunctions gather around exactly one continuous function, called the limit shape .The history of studies of limit shape phenomena goes back to the famous work by A.Vershikand S.Kerov on asymptotics for the Plancherel measure on Young diagrams [VK1] using avariational principle and the following works,[VK2], [V]. Figure 1.
The limit shape for the Young diagrams according to the Plancherel measure. There are several lecture notes on modern studies of the limit shape phenomena, see [AO] [K] andreferences there.
IMIT SHAPES FOR THE DIMER MODEL 3
In more recent studies of limit shape phenomena the key example is random dominotilings of Aztec diamond and the arctic circle theorem, [JPS], [CEP] and Figure 2.. Laterthe technique was generalized to the arbitrary regions in [CKP].Aztec diamond of size 4 Large scale behavior.
Figure 2.
The arctic circle theorem on the left with frozen regions near theboundary. There are four types of dominos according to the standard bipartitestructure on Z that are marked by four colorsThen limit shape was found for 3d Young diagrams (also known as a plane partition) usingWolf crystal contraction, [CK], see Figure 3.An example of a plane partition. The Wolf crystal around which plane partitionsconcentrate at large scales. Figure 3.
The Wolf crystal for the plane partitions.Later on there was a generalization of works on asymptotics of Young diagrams for de-formed Plancherel measure,[NO].
NIKOLAI KUCHUMOV
Also, there was a work on a dimer model that generalized domino tilings (correspondent tothe Z case of the dimer model) and built a connection to algebraic geometry, [KOS], [KO1],[KO2].Then there were works with purely algebraic methods that give alternative proofs of knowntheorems (i.e. the arctic circle theorem) and many new results, see [BG], [BF] and referencesthere.In the last five years, several studies on limit shape phenomena of six vertexes were done,using algebraic methods for stochastic six-vertex model and a correspondent new type ofintegrable PDE was found [RS], [BCG].1.2. Summary.
The plan of the paper is as follows: first comes introduction, in the secondsection we discuss basic definitions of a dimer model, in the third section we define theprobability distribution on a set of the dimer covers, then we define the cutting rule thatplays the crucial role in the proofs. In the next section we discuss the Z -periodic graphswith which we will work through the rest of the paper. In the fifth section we define thethermodynamic limit of the dimer model on a plane and formulate the main result of thepaper, The Limit Shape Theorem.Exactly the same proof works for the six vertex model as well. It will be presented in ournext paper.1.3. Brief plan of the proofs.
The proof of the limit shape theorem consists of three parts, a variational principle , which we prove in the seventh section and in the eighth section. Thesecond part is The Density lemma that we prove in the tenth section Proofs of properties ofheight functions. The third part of the proof is the Surface Tension theorem that is the keytheorem in the proof of the main theorem.A variational principle is a general analytic statement. We prove it under assumption ofthe Concentration lemma and the Surface Tension theorem. The Concentration lemma is aprobabilistic measure-theoretical statement which states that all height functions concentratearound an average height function.In the second part we formulate and prove several properties of height functions, which leadus to the Density lemma. The idea of this part is that height functions are discrete analogsof Lipschitz functions, so the statements for the height functions and asymptotic heightfunctions are almost the same including the proofs. Then we prove the Density lemma thatstates that for each asymptotic height function there exists a sequence of normalized heightfunctions that converges to it.In the third part, we prove several auxiliary propositions that lead us to the Surface Tensiontheorem. The idea is to approximate a square by a torus, then approximate a triangle bysquares and approximate an arbitrary domain by triangles.
Acknowledges
We are grateful to Institut Henri Poincaré for hospitality during the trimester "Combina-torics and interactions", where the part of the work was done. Also we would like to thankAnatoly Vershik for the excellent introduction to the limit shape phenomena and support,thank Nicolai Reshetikhin for the suggest to look at this problem and inspiring talk in PDMIRAS in 2017 and thank Vadim Gorin for the references. For useful comments and discus-sions we would like to thank Dmitry Chelkak, Fedor Petrov, Pavel Zatitskii, and Vladimir
IMIT SHAPES FOR THE DIMER MODEL 5
Fock. For the help in finding typos we would like to thank Masha Smirnova and Igor Skut-senya. Especially we would like to thank Pavlo Gavrilenko for reading the manuscript andIlia Nekrasov without whose support and remarks the work would be impossible.2.
Dimers on graphs with boundaries and height functions
In this section we partly follow [CR] and [GK].2.1.
Dimers on graphs with boundaries. A graph with a boundary is a finite graph Γ together with a set ∂ Γ of valence one vertices. We will refer to such verteces as boundaryvertices and the other vertices as internal .A dimer cover D on a graph with a boundary (Γ , ∂ Γ) is a choice of edges of Γ , calleddimers, so that each vertex, that is not a boundary vertex, is adjacent to exactly one dimer.Note that some of the boundary vertices may be adjacent to a dimer of D , and some maynot. Dimer covers are also known as dimer configurations or perfect matchings.We will divide these partitions of boundary vertices into two groups - matched and non-matched and refer to them as boundary conditions of dimer covers on Γ .One can parametrize boundary condition by choosing a set of non-matched boundaryvertices, let us use notation δD for this set. δD := { v | v ∈ ∂ Γ , v is not matched by D } (1)Let us call a set of the dimer covers on (Γ , ∂ Γ) by D (Γ) and denote the set of the dimercovers with fixed boundary conditions by D (Γ; δD ) := { D ∈ D (Γ) | δD = δD } (2)2.2. Dimers on bipartite graphs.
Bipartite structure.
We will always assume that our graph is a surface graph , thatis a graph embedded into a compact oriented surface S whose faces, i.e. the connectedcomponents of S − Γ , are contractible. By embedding of graph with a boundary into surface S we mean such embedding without self-intersections Γ (cid:44) −→ i S that i (Γ) ∩ ∂S = i ( ∂ Γ) andthe complement of Γ \ ∂ Γ in S \ ∂S consists of open 2-cells.A bipartite structure on a graph Γ is a partition of its set of vertices into two groups, sayblacks and whites, such that no edge of Γ joins two vertices of the same group.A bipartite structure induces an orientation on the edges of Γ , called the bipartite orien-tation: simply orient all the edges from the white vertices to the black ones.Now our graph is a cell complex and we will use usual boundary operator ∂ and standardnotation for chain complex with the bipartite orientation. Also we will use following notationfor expressions with the boundary operator.Let us denote by [ v ] the 0-chain correspondentto a vertex v , similar for edges and faces. Suppose that [ e ] is an edge between vertices [ v ] and [ v ] , then we will use following notations: ∂ ([ e ] ) = [ v ] − [ v ] = (cid:88) i =1 , sgn ( v i )[ v i ] ∈ C (Γ , Z ) , (3)where sgn ( v ) = 1 is for the black vertex v , and sgn ( v ) = − is for the white vertex.Equivalently, a bipartite structure can be regarded as a 0-chain NIKOLAI KUCHUMOV
Figure 4.
An orientation of an edge β = (cid:88) v sgn ( v )[ v ] ∈ C (Γ , Z ) (4)where the sum is over all vertices v of Γ . Figure 5.
An example of a dimer cover on a bipartite graph.2.2.2.
Dimer covers.
Using bipartite orientation, a dimer cover D ∈ D (Γ , ∂ Γ) can now beregarded as a 1-chain with Z -coefficients D = (cid:88) e ∈ D sgn ( e )[ e ] ∈ C (Γ; Z ) . (5)The condition that D is a dimer cover means that ∂D = (cid:88) v sgn ( v )[ v ] (6)where the sum is taken over all internal vertices and some boundary vertices of Γ .2.2.3. Boundary conditions and decomposition cycles.
One can regard boundary conditionsas an element in C ( ∂ Γ; Z ) , let D be a dimer cover, δD = (cid:88) v ∈ δD sgn ( v )[ v ] . (7) One can rewrite it as ∂D = (cid:80) v − black vertex [ v ] − (cid:80) v − white vertex [ v ] . IMIT SHAPES FOR THE DIMER MODEL 7
Then one can notice that there is a relation between a dimer cover, its boundary conditionsand bipartite structure on the graph. This relation is a set-theoretic statement with respectto bipartite orientation. β = (cid:88) v is not adjacent to D sgn ( v )[ v ] + (cid:88) v is adjacent to D sgn ( v )[ v ] = ∂D + δD. (8)Let D, D (cid:48) be dimer covers with boundary conditions δD and δD (cid:48) . One can look at thedifference D − D (cid:48) in 1-chains for two dimer covers D and D (cid:48) .Then after substitution of (7) we get that ∂ ( D − D (cid:48) ) = δD − δD (cid:48) . In case of equal boundaryconditions D − D (cid:48) is a 1-cycle. Otherwise it is true only in relative 1-chains, C (Γ , ∂ Γ; Z ) .Let us call D − D (cid:48) decomposition cycles of dimer covers D and D (cid:48) . Figure 6.
An example of two dimer covers D and D (cid:48) . Dimers from D are insolid lines and dimers from D (cid:48) in traced lines.2.3. Height function.
In this section we want to make a bijection between dimer covers of Γ and some functions on faces of Γ . Somehow we want to map a dimer cover to its «height».However, on non simply-connected surfaces there is no global height function and it is notactually a «height», but rather «The Penrose stairs», see Figure 7. Figure 7. «The Penrose stairs».We will consider two cases — graphs embedded into the torus and graphs embedded intothe two-dimensional disk.
NIKOLAI KUCHUMOV
Case of the disk.
Let Γ be a graph with a boundary embedded into disk D . Werecall that Γ induces a cell decomposition of D . We denote by boundary cells such cells thatcontain boundary vertices.Let us take two dimer covers, D and D (cid:48) . Because D − D (cid:48) is 1-cycle (rel ∂ Γ) there is anelement σ D,D (cid:48) ∈ C (Γ , ∂ Γ; Z ) such that ∂σ D,D (cid:48) = D − D (cid:48) . Then let us define 2-cochain h D,D (cid:48) by the following formula: σ D,D (cid:48) = (cid:88) f h D,D (cid:48) ( f )[ f ] ∈ C (Γ , ∂ Γ; Z ) (9)where the sum goes over all faces of Γ . We can regard cocycle h D,D (cid:48) as a function on faces of Γ . We will call such functions as height functions .One can notice that a height function is simply a function of level for decomposition cycles,so it is uniquely defined by D, D (cid:48) up to an additive constant. Hence, one can normalize allheight functions by setting h D,D (cid:48) ( f ) = 0 for some fixed face f . Note that a height functionsatisfy Lipschitz condition in some sense. For any two faces y and x values of a heightfunction at this faces differ at most by the length on the dual graph between the faces. Figure 8.
Decomposition cycles for Dimer covers from Figure 6 and corre-spondent height function.One can think about boundary conditions for dimer covers in terms of height functions.Let us look at two dimer covers D , D (cid:48) and their height function h D,D (cid:48) . A boundary heightfunction of D and D (cid:48) is a restriction of the height function to boundary faces. Fix dimercover D (cid:48) and look at different D . It is clear that one can reconstruct boundary conditionsof D using boundary height function and δD (cid:48) . Simply because the height function changesacross edges where D and D (cid:48) are different from each other. So it is sufficient to change aboundary condition for D (cid:48) along every edge where the height function changes.Also note that for any three dimer configurations D , D (cid:48) and D (cid:48)(cid:48) on Γ , the following cocycleequality holds: h D,D (cid:48) + h D (cid:48) ,D (cid:48)(cid:48) = h D,D (cid:48)(cid:48) . (10) IMIT SHAPES FOR THE DIMER MODEL 9
Later on we will work with height functions h D,D (cid:48) for a fixed reference dimer configuration D (cid:48) . For regular graphs, there is an alternative definition of height functions called absoluteheight functions which we review in the appendix.2.3.2. Case of a torus.
In case of a torus height function it can be defined as a function onlylocally.because of monodromy along not simply-connected cycles . Note that in this casethere are no boundary vertices. Look at Figure 6 for an example. Figure 9.
An example of a height function for two dimer covers on a torus.Note that is has monodromy along vertical cycles and along horizontalones.2.4. Newton polygon.
Because [ D ] − [ D (cid:48) ] is 1-cycle, it defines a homology class. Due to H ( T , Z ) (cid:39) Z the homology class can be identified with a pair of integers ( s D , t D ) after thechoice of a basis ( γ , γ ) in H ( T , Z ) .Each s D and t D are intersection numbers of [ D ] − [ D ] (cid:48) with γ and γ , respectively. Inother words it is a monodromy of a corresponding height function along the cycles.We will call this pair of integers ( s D , t D ) a slope of dimer cover D . Note that it dependson D (cid:48) which is fixed.In case of a graph Γ embedded into a torus, a set of slopes of the all dimer covers is denotedby S (Γ) , S (Γ) := { ( s D , t D ) ∈ Z , D ∈ D (Γ) } (11)It is a finite set of points in Z and it is uniquely defined up to a change of D (cid:48) that shifts theset of slopes.The Newton polygon for the Γ is a convex hull of the set of slopes, N Γ := Conv ( S (Γ)) (12)For example, see Figure 10 for the case of square grid.Note that N G is defined up to a linear shift, so if / ∈ N G we can make a proper linear shiftto fix it. For the case of not simply-connected region the height function is not a function, but rather a «sectionof Z bundle». It defines a function locally on every simply-connected component. So it is uniquely definedup to an action of π ( S ) , which adds for each loop monodromy along it. Figure 10.
An example of the Newton polygon for square grid on the rightand corresponding D (cid:48) on the left3. Boltzmann distribution on dimer covers
Let (Γ , ∂ Γ) be a graph with a boundary. A weight system on (Γ , ∂ Γ) is a map from theset of dimer covers D (Γ) to positive numbers R > . A weight system w defines a Boltzmanndistribution (also known as a Gibbs measure) on the set of dimer covers. For D ∈ D (Γ) letus define its probability by (13), P ( D ) := w ( D ) Z ( w ; Γ) (13)where D ∈ D (Γ) and Z ( w ; Γ) is a normalization constant called the partition function: Z ( w ; Γ) = (cid:88) D ∈ D (Γ) w ( D ) . (14)We shall focus on a particular type of weight systems called edge weight systems . Let usassign to each edge e of Γ a positive real number w ( e ) called the weight of the edge e . Theassociated edge weight system on D (Γ) is given by w ( D ) = (cid:89) e ∈ D w ( e ) (15)where the product goes over all edges contained in D .In statistical mechanics, these weights are called Boltzmann weights. Their physical mean-ing can be expressed by the following formula: w ( e ) = exp (cid:18) − E ( e ) kT (cid:19) , (16)where E ( e ) is the energy of the dimer occupying the edge e , T is the absolute temperatureand k is the Boltzmann constant. Remark 1.
Boltzmann distribution on dimer covers induces a probability distribution on theset of corresponding height functions. Probability of a height function h D is just probabilityof the dimer cover D .We will denote by ¯ h corresponding expectation value of height function and by ¯ h ( v ) its valueat face v . IMIT SHAPES FOR THE DIMER MODEL 11 The cutting rule for the dimer model
Graphs with boundary conditions.
Let (Γ , ∂ Γ) be a graph with a boundary and letus fix a boundary height function χ . Then we will call such a pair a graph with a boundarycondition and denote by (Γ , χ ) .Let D (Γ , χ ) be a set of dimer covers on Γ with a boundary height function χ . Let us denotea set of height functions with the given boundary condition χ by H (Γ , χ ) . Usually we willbe interested in partition functions associated with (Γ , χ ) . Z (Γ , χ ) := (cid:88) D ∈ D (Γ ,χ ) w ( D ) (17)It is straightforward that the partition function of (Γ , ∂ Γ) is the sum of Z ((Γ , χ )) over allboundary height functions: Z (Γ , ∂ Γ) = (cid:88) χ Z (Γ , χ ) (18)4.2. The Cutting Rule.
Let (Γ , ∂ Γ) be a graph with a boundary, and let us fix an edge a ∈ E (Γ) . Let us denote by (Γ a , ∂ Γ a ) a graph with a boundary obtained from (Γ , ∂ Γ) asfollows: cut the edge a into two edges a and a , and set ∂ Γ a = ∂ Γ ∪ { v , v } , where v and v are the new valence one vertices adjacent to a and to a , respectively. See on Figure 11.Note that there is a natural map φ a : D (Γ , ∂ Γ) (cid:55)→ D (Γ a , ∂ Γ a ) that simply cuts in two thedimer of D that belongs to the edge a . Figure 11.
An example of the cutting procedureThen we want to find a «pushforward» φ ∗ of weight systems with respect to the cuttingprocedure described above. Thus we want to construct a weight system on D (Γ a , ∂ Γ a ) sothat it has the following property: w ( D ) = φ ∗ w ( φ ( D )) (19)It means that all dimer covers that come from Γ will have the same weight.Then the partition function of (Γ , ∂ Γ) can be identified with a part of the partition functionof (Γ a , ∂ Γ a ) that comes from (Γ , ∂ Γ) . Z (Γ , ∂ Γ) = (cid:88) D ∈ D (Γ ,∂ Γ) w ( D ) = (cid:88) D ∈ D (Γ ,∂ Γ) φ ∗ w ( φ ( D )) = (cid:88) D (cid:48) ∈ φ ( D (Γ ,∂ Γ)) φ ∗ w ( D (cid:48) ) (20)There is a general way to construct φ ∗ w :For all edges that are not coming from cutting the edge a weights remain as they are on Γ . For the case of edges a (cid:48) and a (cid:48)(cid:48) , that come from cutting edge a , it is sufficient to take suchweights that w ( a ) = w ( a ) w ( a ) . That way the condition will be true.General way to do it is to take w ( a (cid:48) ) = (cid:112) w ( a ) t and w ( a (cid:48)(cid:48) ) = (cid:112) w ( a ) t − for t > . Later onwe will use t := (cid:112) w ( a ) to make w ( a (cid:48) ) = w ( a ) and w ( a (cid:48)(cid:48) ) = 1 .Note that due to locality of the cutting procedure cuts of different edges commute in asense that we obtain the same graph with a weight system. This means that the cuttingprocedure can be extended for a family of edges A . Just iterate the procedure for all edges a ∈ A . The corresponding map φ will be just a composition of maps φ a for edges a ∈ A , thesame for φ ∗ and the Cutting rule will still holds.4.3. The cutting rule for surface graphs.
The cutting of graphs with boundaries.
Let (Γ , χ ) be a graph with a boundary conditionembedded into surface S and ρ be a simple curve in S that is “in general position” with respectto Γ , in the following sense: • it is disjoint from the set of vertices of Γ ; • it intersects the edges of Γ transversally; • its intersection with any given face of Γ is connected.Let S ρ be the surface with the boundary obtained by an open cutting S along ρ . Also let Γ ρ := Γ A ( ρ ) be a graph with a boundary obtained by cutting (Γ , ∂ Γ) along the set A ( ρ ) ofedges of Γ that intersect ρ .Obviously, Γ ρ is a surface graph with a boundary. We will say that it is obtained by thecutting Γ along ρ .Let us denote by φ ρ the map of the dimer covers and by w ρ the weightsystem on Γ ρ obtained from the weight system on Γ . Figure 12.
An example of a cutting along a curve Z (Γ , χ ) = (cid:88) D ∈ D (Γ ,χ ) w ( D ) = (cid:88) D (cid:48) ∈ φ γ ( D (Γ ,χ )) w ρ ( D (cid:48) ) (21)4.3.2. Cutting graphs with boundary conditions.
The next step is to cut graphs with bound-aries, to do it we need to set a boundary height function along Γ ρ . Basically we want todescribe the set φ ( D (Γ , χ )) to rewrite the cutting rule in terms of height functions.Suppose that we have a graph with a boundary condition (Γ , χ ) . There are two types offaces on Γ ρ : each face is either an old face that comes from a face in Γ or a new face obtainedfrom cutting a face of Γ intersected by the curve ρ . IMIT SHAPES FOR THE DIMER MODEL 13
For old faces we can leave old values of χ . For new faces one may take boundary conditionsthat are obtained by cutting a dimer cover on Γ which we will parametrize by a boundaryheight function χ ρ .In terms of boundary height function it means the following: There are faces that intersectwith ρ , we denote the set of such faces by F ( ρ ) . There is a natural map ψ from F ( ρ ) to pairsof boundary faces on Γ ρ that are obtained by cutting faces from F ( ρ ) . Then we have oneboundary value for a face f ∈ F ( ρ ) and a pair of values for ψ ( f ) = ( f , f ) . The conditionthat χ ρ is obtained by cutting along ρ means that χ ( f ) = χ ρ ( f ) = χ ρ ( f ) . See Figure 13
Figure 13.
An example of a cutting of boundary condition.
Proposition 1 (The Cutting rule) . Suppose we cut a graph with a boundary condition (Γ , χ ) into two graphs Γ (cid:48) and Γ (cid:48)(cid:48) .Then we can express the partition function of (Γ , χ ) in terms of Γ (cid:48) and Γ (cid:48)(cid:48) . Z (Γ , χ ) = (cid:88) χ ρ Z (Γ (cid:48) , χ (cid:48) ρ ) Z (Γ (cid:48)(cid:48) , χ (cid:48)(cid:48) ρ ) (22) Here χ ρ means a pair of the boundary height functions ( χ (cid:48) ρ , χ (cid:48)(cid:48) ρ ) that are obtained from χ bycutting along ρ . Remark 2.
The cutting rule has the following combinatorial explanation: We need to sumup over all dimer covers. One way to do it is to calculate partition functions Z ( h ( ρ )) withgiven height function h ( ρ ) along ρ . And then to sum up Z ( h ( ρ )) over all h ( ρ ) . The result isthe same because we just permute terms in a finite sum.Then, we can interpret each Z ( h ( ρ )) as a product of two partition functions because ρ cuts Γ into two graphs where dimer covers are independent. Thus until we fix the boundarycondition along ρ to be h ( ρ ) , we are calculating the original sum Z ( h ( ρ )) . In case ρ cuts the graph into two graphs the «equality» means that the boundary height functions agreeup to an additive constant. Periodic graphs
Periodic graphs.
In this section we follow 2nd and 3rd paragraphs of [KOS] .Let G be a Z -periodic planar graph. By this we mean that G is a bipartite graph embeddedinto the plane R (we denote the embedding by φ ) so that translations in Z act by color-preserving isomorphisms of φ ( G ) – isomorphisms which preserve bipartite structure: mapblack vertices to black vertices and white to white.Let G ( n ) be the quotient of G by the action of n Z . It is a finite bipartite graph on a torus. G (1) is called a fundamental domain. An example is Z itself with fundamental domain onFigure 10 .Later on we will need finite subgraphs in G and call them planar periodic graphs. G willbe fixed in all following statements and theorems. Note that periodic planar graphs are alsographs with boundaries. We will denote Newton polygon corresponding to G (1) by N G .We will fix a weight system on G (1) and then continue it to a weight system on G byperiodicity. Thermodynamic limit of the dimer model on the torus.
Let us look at G ( n ) ,one can calculate its partition function Z ( G ( n )) in limit as n → ∞ .Because actual partition function grows rapidly it is convenient to normalize it, F := lim n →∞ n − log( Z ( G ( n ))) . (23)This limit exists and the answer is the following [KOS]: F = (cid:90) (cid:90) | z | = | w | =1 πi ) log P ( z, w ) dz dwzw , (24)Here P ( z, w ) is a Laurent polynomial in z and w , with Newton polygon N G . It dependsonly on the fundamental domain and the weight system on it.For the case of square lattice with all weights equal 1 the polynomial is the following: P ( z, w ) = (1 + z ) z + (1 + w ) w (25)5.3. Surface tension.
For fixed ( s, t ) ∈ R we denote by D s,t ( G ( n )) the set of dimer coverson G ( n ) , that have a slope ( (cid:98) ns (cid:99) , (cid:98) nt (cid:99) ) . .Consider the normalized partition function of dimer covers with the fixed slope s, tZ s,t ( T n ) = n − log (cid:88) D ∈ D s,t ( G ( n )) w ( D ) . (26)This partition function is called the surface tension. It has a limit as n → ∞ and the answeris lim n →∞ n − log( Z s,t ( G ( n ))) = σ ( s, t ) (27)Here σ is minus Legendre transform of Ronkin function of P ( z, w ) , R ( B x , B y ) := (cid:90) (cid:90) | z | = e Bx , | w | = e By log P ( z, w ) dzdwzw (28) Weight systems on G ( n ) can be obtained by the same construction. D s,t ( G ( n )) is not empty for ( s, t ) ∈ N G IMIT SHAPES FOR THE DIMER MODEL 15
From the corollary 3.7 from [KOS], σ is a concave function .6. Thermodynamical limit of the dimer model on the plane
Let Ω be a compact connected simply-connected domain in R . In this section we will lookat sequences of increasing graphs embedded into Ω and formulate the limit shape theoremfor it.In following paragraphs we assume that all functions are bounded real-valued functions on Ω , we will denote a set of this functions on Ω by B (Ω) . We recall that there is a standardnorm on B (Ω) , (cid:107) f (cid:107) := max x ∈ Ω | f ( x ) | . Suppose that we have a graph with a boundary (Γ , ∂ Γ) embedded into Ω . Then heightfunctions on (Γ , ∂ Γ) can be treated as bounded functions on Ω assigning its value at point x to be h ( f ) if x is contained in one face f and average of values h ( f ) over all faces f containing x .6.1. Asymptotic height functions.
Due to the Lipschitz condition on height functionsone might expect that in a limit they will converge to continuous Lipschitz functions.Let
Lip (Ω) be the space of all Lipschitz functions on Ω . Lip (Ω) := { f ∈ B (Ω) |∃ C > ∀ x, y ∈ Ω | f ( x ) − f ( y ) | ≤ C | x − y |} . (29)For each function f such a minimal constant C from the definition is called the Lipschitzconstant for the function f and f is called C -Lipschitz in this case. Lipschitz functions arecontinuous and differentiable almost everywhere due to Rademacher’s theorem.Let us define a set of asymptotic height functions by H (Ω) := { f ∈ Lip (Ω) | ∇ f ∈ N G almost everywhere } . Later on we will prove The Density Lemma, that justifies this definition.Let us take an asymptotic height function χ that is fixed on ∂ Ω . We will call such a pair (Ω , χ ) a domain with a boundary condition and we will look at asymptotic height functionsthat coincide with χ on ∂ Ω , let us denote the set of such asymptotic height functions by H (Ω , χ ) .Note that N G is defined up to a linear shift, so if / ∈ N G we can make a proper linear shiftto fix it.6.2. Approximations of domains.
Let d n : R → R be a dilatation d n : ( x, y ) (cid:55)→ ( n − x, n − y ) . Let us look at rescaled embedding φ n := d n ◦ φ , and denote rescaled G by G n := φ n ( G ) . Also we need to rescale height functions on Ω n by a factor n − , let us denotethem by η n and its expectation values by ¯ η n . We will call them normalized height functions .Let (Ω , χ ) be a domain with a boundary condition. Then we will call a sequence of graphswith boundary conditions { Γ n , χ n } an approximation of (Ω , χ ) (where { χ n } are normalizedboundary height functions) if(1) Γ n ⊂ G n ∩ Ω (2) each Γ n admits at least 1 dimer cover with normalized boundary height function χ n . (3) (cid:107) χ n − χ (cid:107) → as n → ∞ .(4) Γ n tends to Ω with respect to Hausdorff distance , d H (Ω , Γ n ) → , as n → ∞ . In our notations of σ we follow [CKP], so they differ from the notations in [KOS] by the sign of σ . Here d H means Hausdorff distance, d H ( X, Y ) = inf { (cid:15) ≥ X ⊆ Y (cid:15) and Y ⊆ X (cid:15) } , where X (cid:15) is (cid:15) -neighborhood of X . Actually we will assume that the sum of areas of faces of Γ n converges to the euclidean area of Ω . The Limit Shape Theorem.Theorem 1 (The limit shape theorem) . Let (Ω , χ ) be a domain with a boundary conditionand (Ω n , χ n ) be an approximation of (Ω , χ ) , then lim n →∞ n − log Z (Ω n , χ n ) = (cid:90) (cid:90) Ω σ ( ∇ g ) dxdy where g is the maximizer of the functional F ( h ) := (cid:82)(cid:82) Ω σ ( ∇ h ) dxdy on the set H (Ω , χ ) .Moreover, let η n be a random height function on (Ω n , χ n ) . Then we have the convergencein probability for η n , that is for each c > P ( (cid:107) η n − g (cid:107) > c ) → as n → ∞ . So all the height functions converge to the «limit shape» g pointwise in probability. The proof of the variational principle
In this section we prove the The Limit Shape Theorem under assumptions of the Concen-tration Lemma and the Surface Tension Theorem. First we prove that all height functionsconverge to the limit shape, then we show that the partition function is localized around thelimit shape and finally notice that the limit shape is the maximizer of the surface tensionfunctional.7.1.
Convergence of height functions to the limit shape.
Let (Ω n , χ n ) be an approxi-mation of a domain with a boundary condition (Ω , χ ) .Consider the sequence of average height functions, { ¯ η n } . By the density lemma there is asequence of asymptotic height functions { g n } , such that (cid:107) g n − ¯ η n (cid:107) ≤ Cn . Due to compactnessof H (Ω , χ ) { g n } has a convergent subsequence, let us denote its limit by g . Without loss ofgenerality we suppose that convergent subsequence is { g n } itself. Now we denote by B C ( ¯ η n ) balls of radius C around ¯ η n .From the Concentration lemma and its corollary we get that P ( η n / ∈ B C ( ¯ η n )) ≤ n exp( − KnC / (30)for some real positive constant K .Consider balls of the radius Cn / around { ¯ η n } , let us call them B / ( ¯ η n ) . Note that { g n } liein these balls due to Cn / ≥ Cn and similar bound takes place. The difference between B C ( ¯ η n ) and B / ( ¯ η n ) is that the new sequence of balls becomes smaller as n → ∞ and all sequencesof height functions lying in the balls converge to the same limit as g n which is g . P (cid:0) η n / ∈ B / ( ¯ η n ) (cid:1) ≤ n exp( − Kn / C / → as n → ∞ (31)for K > by the Density lemma.So all height functions converge pointwise in probability to g .7.2. The localization of the partition function.
Let us recall an expression for proba-bility that a random normalized height function lies in a set S , P ( η n ∈ S ) = Z (Ω n |S ) Z (Ω n ,χ n ) where Z (Ω n |S ) is a partition function where we sum only dimer covers with normalized heightfunctions lying in the set S . Let us take S = B δ ( g ) for sufficiently small fixed δ > that wesent to in the end of the proof. IMIT SHAPES FOR THE DIMER MODEL 17 P ( η n ∈ B δ ( g )) + P ( η n / ∈ B δ ( g )) (32) lim n →∞ Z (Ω n | δ, g ) Z (Ω n ) = 1 − lim n →∞ P ( η n / ∈ B δ ( g )) (33)Then ( ) says that probability in the right hand side vanishes as n → ∞ . Taking logarithmof (33) and multiplying by n − we get that lim n →∞ n − log( Z (Ω n )) = lim n →∞ n − log( Z (Ω n | δ, g )) . (34)One can express the right hand side by the Surface Tension Theorem, lim n →∞ n − log Z (Ω n ) = lim n →∞ n − log Z (Ω n | δ, g ) = F ( g ) + o δ (1) (35)where o δ (1) → as δ → , so after taking limit δ → we get the second part of the proof.7.3. The Surface tension functional and the limit shape.
By proposition 2.4 from[CKP] there exists the unique maximizer of the surface tension functional on H (Ω , χ ) .Then g is the maximizer of the surface tension functional, because F ( g ) is proportionalto lim n →∞ n − log Z ( g, δ ) , but all the height functions concentrate around g by the (31) .8. The concentration lemma
In this section we follow section «6.2. Robustness.» from [CEP].
Claim 1.
Let (Γ , ∂ Γ) be a graph with a boundary. Let us fix a boundary condition and takea face v on Γ .Then h ( v ) is a random variable and the following estimate for an expectation value is trueLet us take a > , then P (cid:0) | h ( v ) − ¯ h ( v ) | > a ·√ m (cid:1) < − a / (36) where m — is the distance on Γ ∗ from v to the nearest boundary face, m = dist ( v, ∂ Γ) .Proof. Let us recall that H (Γ , χ ) is a set of height functions on (Γ , ∂ Γ) . Consider a probabilityspace ( H (Γ , χ ) , Ω , P ) , where Ω := 2 H ( − ,χ ) is σ -algebra of subsets of H (Γ , χ ) , and P is the Gibbsmeasure on H (Γ , χ ) .Let ρ be the shortest path on Γ from ∂ Γ to the face v , ρ := ( x , x . . . x m ) , where x ∈ ∂ Γ and x m = v .We consider a family of equivalence relations on H (Γ , χ ) : h ∼ k h if and only if, h ( x ) = h ( x ) for all x = x , · · · , x k , so height functions agree at first k points of path ρ . It isstraightforward that from h ∼ k h (cid:48) follows h ∼ k − h (cid:48) , thus we may consider a decreasingfiltration of σ -algebras, F k in Ω : F k := { Equivalence classes under ∼ k } (37) F k − ⊂ F k (38)Because boundary conditions are fixed we have F = H (Γ , χ ) . Let us take the followingsequence of conditional expectation values: M k := E ( h ( v ) | F k ) . (39) M k is an expectation value of a height function at face v with random values at first k − points of the path ρ (values at these points define an equivalence class in F ) . Notethat M = ¯ h ( v ) because there are no random values and M m = h ( v ) because we leavevalues totally random according to the Gibbs measure. Because F k is a filtration, M k is amartingale: E ( M k +1 | F k ) = M k , (40)which is due to the tower rule: if G and G are two two σ -algebras such that G ⊂ G , wehave E [ E [ X | G ] | G ] = E [ X | G ] .Also, let us show that | M k − M k +1 | ≤ .We need to take an equivalence class in F k and then prove the inequality | M k − M k +1 | ≤ . A value of a height function from an equivalence class from F k in x k is fixed by theequivalence class and a value in x k +1 is a convex combination of the two possible values: h ( x k ) − and h ( x k ) with coefficients proportional to the partition function of the dimercovers with this value at x k +1 .Note that expectation values for the equal boundary conditions are equal.On the other hand a value of a height function from F k +1 at x k +1 is fixed by its equivalenceclass and is either h ( x k ) − or h ( x k ) for the same h ( x k ) ∈ Z . So M k and M k +1 are expectationvalues of height functions with boundary conditions that differ less than .By the coupling lemma it is true for expectations.Note that M k is a martingale such that | M k − M k +1 | ≤ . Applying Azuma inequality[Azuma] for M k , we get that P ( | M m − M | > a ·√ m ) < − a / (41)that is the same as P ( | h ( v ) − ¯ h ( v ) | > a ·√ m ) < − a / . (42) (cid:3) Corollary 1.
Moreover if we take an approximation of a domain with a boundary condition (Ω , χ ) we get the following bound for normalized height function. Let η n be a random heightfunction on Ω n , then (1) P ( | η n ( v ) − ¯ η n ( v ) | > a ) < − a n ) (43)(2) P ( η n / ∈ B δ ( ¯ η n )) < Kn exp( − δ n )) (44) for some K > and sufficiently small δ . Which obey the Gibbs measure. We recall that the value at the first face of the path ρ is fixed by the boundary condition and randomvalues start from the second face, thus M is an expectation value of h ( v ) . In our parametrization of ρ = { x , · · · , x k } where x ∈ ∂ Γ We do not change σ -algebra. Two possible values could be h ( x k ) − and h ( x k ) depending on an orientation. IMIT SHAPES FOR THE DIMER MODEL 19
Proof.
Dividing by n inequality in the Concentration lemma we get that P ( | η n ( v ) − ¯ η n ( v ) | > n − a ·√ m ) < − a / (45)After some rescaling we get the following expression from 43, P ( | η n ( v ) − ¯ η n ( v ) | > c ) < − c n ) . (46)Then to obtain probability that η n / ∈ B δ ( g ) we find P ( η n ∈ B δ ( g )) . To do it we need tomultiply probabilities that | η n ( v ) − ¯ η n ( v ) | ≤ δ for all faces v of graph Γ : P ( η n ∈ B δ ( ¯ η n )) = (cid:89) v − face of Γ P ( | η n ( v ) − ¯ η n ( v ) | ≤ δ ) (47)Then each factor is equal to − P ( | η n ( v ) − ¯ η n ( v ) | > δ ) , that we can estimate by using theequality − P = exp(log(1 − P )) and Taylor expanding of log . Let us denote P ( | η n ( v ) − ¯ η n ( v ) | >δ ) by p . Then the bound of the whole product is exp(( − p + o ( p )) × n K ) where we usethe fact that in the domain there are around Kn vertices for some K > . Finally we bound − p by − − δ n/ using 46 and get the following expression forthe probability that a random normalized height function lies in δ -ball around the averagenormalized height function. P ( η n ∈ B δ ( ¯ η n )) = exp( − Kn exp( − δ n/ . (48)It is clear that it converges to one. After using the inequality − x ≤ exp( − x ) and the smallchange of variables we get . (cid:3) Coupling lemma.Claim 2.
Let (Γ , ∂ Γ) be a graph with a boundary and let f and g be two boundary heightfunctions on Γ . We denote by ¯ h g an average height function on (Γ , g ) , ¯ h f on (Γ , f ) . WeSuppose that f ≤ g , then ¯ h f ≤ ¯ h g pointwise.Proof. Let H (Γ , g ) and H (Γ , f ) be sets of height functions on Γ with boundary height func-tions f and g . Denote induced Gibbs measures by µ f , µ g .Let us prove that f ≤ g ⇒ ¯ h f ≤ ¯ h g . To do this we need to build a coupling of measures µ f and µ g . It is a probability measure π on H (Γ , g ) × H (Γ , f ) , such that its projection on H (Γ , g ) gives µ g and the same is true respectively for f . The most important constraint isthat P π ( h , h ) = 1 for only pairs of height functions such that h ≤ h . In case we havesuch a measure, we get ¯ h f ≤ ¯ h g pointwise, because ¯ h f and ¯ h g can be computed using π dueto the fact that each measure can be obtained as a projection of π .Let us prove it using induction by number of internal vertices of Γ . Base of inductionwhere the set of internal vertices is empty is trivial. Moreover the case of f = g is trivial and it is sufficient to prove for f < g at some boundary face v . Note that h f ( v ) ≤ h g ( v ) + 1 . We can bound n − / √ m by (cid:112) Diam (Ω) + o ( n ) because the length of a path is less then the maximallength of a path that is bounded by the diameter of the domain Ω (in Euclidean metric) and n − / is adsorbeddue to rescaling of the domain (length of a path between fixed points in Ω is proportional to n after rescaling).Then we define new constant c := a/ × n − / √ m × Diam (Ω) log(1 − p ) = − p + o ( p ) , and the bound for the factor is exp(log(1 − p )) = exp( − p + o ( p )) . Here we have the following change of variables P ( η n ∈ B δ ( ¯ η n )) = 1 − P ( η n / ∈ B δ ( ¯ η n )) ,so we can applythe inequality exp( − Kpn ) ≥ − Kpn = 1 − P ( η n / ∈ B δ ( ¯ η n )) in case of equal boundary conditions expectations are just the same. Let us pick an internal face w adjacent to v . There are two possible values for h g ( w ) - h and h − for some h ( respectively h (cid:48) and h (cid:48) − for h f ( w ) for some h (cid:48) ). µ f = αµ + f + (1 − α ) µ − f (49) µ g = βµ + g + (1 − β ) µ − g (50)where α is proportional to the partition function of the dimer covers with the extra boundarycondition f + and β is proportional to the partition function with the extra boundary condi-tion f − at w . And µ ± f (respectively g ) is the Gibbs measure on height functions on (Γ , ∂ Γ) with an extra boundary condition at w . And thus we can use our induction hypothesis toconclude that the coupling exists for any pair measures with an extra boundary condition at w .Then let us consider the case of β = 1 . In this case we have two couplings π + and π − between µ + f , µ − f and µ + g . We can take the coupling to be a superposition: π = απ + +(1 − α ) π − . µ f = αµ + f + (1 − α ) µ − f (51) µ g = βµ + g (52)The case of arbitrary β is just a consequence of the previous one. µ f (53) µ g = βµ + g + (1 − β ) µ − g (54)We know from the previous case that there exist couplings of µ f with both µ + g and µ − g . Thenwe can just take a superposition of µ f with the µ + g and µ − g with the weights β and − β . (cid:3) Proofs of properties of height functions
Here we formulate and prove several propositions about height functions, all these state-ments are quite the same and the simplest case is -Lipschitz functions where the structureis easier to understand and that we prove in the next section .9.1. The main construction for -Lipschitz functions. Let (Ω , χ ) be a domain witha boundary condition. One can ask a question, under which constraints on χ it admits acontinuation to a -Lipschitz function on Ω . Proposition 2.
Suppose that we have a domain with a boundary condition (Ω , χ ) . Then χ admits a continuation to a -Lipschitz function if the following inequality holds, | χ ( x ) − χ ( y ) | ≤ | x − y | , ∀ x, y ∈ Ω . (55) Proof.
Due to the fact that the pointwise minimum of -Lipschitz functions is a -Lipschitzfunction, we can construct an extension by the following formula, h ( x ) := min y ∈ ∂ Ω χ ( x ) + | x − y | (56)We know that h is a -Lipschitz function and we need to show that h coincides with χ on ∂ Ω . To do this we can prove that χ ( x ) ≤ h ( x ) ≤ χ ( x ) , ∀ x ∈ ∂ Ω .The first inequality archives the equality for the function from the family correspondentsto the point y = x . N G = D . IMIT SHAPES FOR THE DIMER MODEL 21
The second inequality χ ( x ) ≤ h ( x ) holds for all functions among which we take the mini-mum, χ ( x ) ≤ χ ( y ) + | x − y | (57)It is just the Lipschitz condition 55. (cid:3) In following sections we repeat almost the same propositions thrice.9.2.
Piecewise linear approximations of asymptotic height functions.
In this sub-section we recall piecewise linear approximations of Lipschitz functions that we use in theproofs.Let us take (cid:96) > and take a triangular lattice with equilateral triangles of side (cid:96) . We mapan asymptotic height function h ∈ H (Ω) to a piecewise linear approximation, that is linearon every triangle, moreover it is the unique linear function that agrees with h at vertices ofthe triangle. Let us denote this approximation of h by ˆ h . Then the following statements aretrue by the Lemma 2.2,from [CKP], Claim 3.
Let h ∈ H (Ω) be asymptotic height function and let (cid:15) > . Then for sufficientlysmall (cid:96) > , on at least − (cid:15) fraction of the triangles in the (cid:96) -mesh of triangles that intersect Ω we have the following property: (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) ≤ (cid:96)(cid:15) . The density lemma and the criteria of existence.
Here we prove a sequence ofauxiliary propositions that lead us to the Density Lemma and the Criteria of existence.9.3.1.
Criteria for asymptotic height functions.
Before the propositions we need to introducethe support function of N G . It is a function θ : R → R defined by the following formula, θ ( x ) := min (cid:16) λ ≥ | xλ ∈ N G (cid:17) . (58)It is useful to keep in mind the case of 1-Lipschitz functions, where N G is just a unit disk and θ ( x ) = | x | . The support function has a value on ∂N G and its slope lays on ∂N G , so θ isan asymptotic height function. Its value at x = (0 , is and it is the maximal asymptoticheight function among such asymptotic height functions that have this property in the sensethat for any such asymptotic height function φ the following inequality holds, φ ( x ) ≤ θ ( x, ) ∀ x ∈ R . (59)Note that we can rewrite it for any asymptotic height function ψ substituting φ ( x ) = ψ ( x ) − ψ ( y ) ψ ( x ) − ψ ( y ) ≤ θ ( x, y ) , ∀ x, y ∈ R (60)Let us call it the Support Lipschitz condition .Suppose that we have a domain with a boundary condition, (Ω , χ ) . One can ask a question,under which constraints on χ it has a continuation to an asymptotic height function on Ω . Proposition 3.
Let (Ω , χ ) be a domain with a boundary condition. Then, χ admits a con-tinuation to an asymptotic height function if it satisfies the following inequality, ∀ x, y ∈ ∂ Ω | χ ( x ) − χ ( y ) | ≤ θ ( x − y ) (61) which becomes usual -Lipschitz condition for N G = D ) Proof.
Let us take a family of asymptotic height functions, { χ ( y ) + θ ( x − y ) } , x ∈ Ω and y ∈ ∂ Ω . Then, let us take the pointwise minimum over this family. h max ( x ) := min y ∈ ∂ Ω ( χ ( y ) + θ ( x − y )) (62) h max is an asymptotic height function because the pointwise minimum of asymptotic heightfunctions is again an asymptotic height function. So we need to show that h max agrees with χ on ∂ Ω .To do this let us prove that χ ≤ h max ≤ χ on the boundary of the domain. Let x be anarbitrary point x ∈ ∂ Ω , the first inequality χ ( x ) ≤ χ ( y ) + θ ( x − y ) . (63)That is just (61). The second inequality becomes an equality for the asymptotic heightfunction from the family correspondents to the point x = y . (cid:3) Due to (61) h max is the maximal extension of χ to an asymptotic height function Ω . Theminimal extension, h min has the similar form, h min ( x ) := max y ∈ ∂ Ω ( χ ( y ) − θ ( x − y )) . (64)9.4. The criteria for height functions.
Let (Γ , χ ) be a graph with a boundary condition.One can a question under which assumptions on χ it extends to a height function on Γ .Before formulating the criteria we need to introduce the support height function that is aheight function defined on the whole G .Let x ∈ G and let us look at all height functions that have the value at face x and letus take the pointwise maximum among such height functions. Note that a height function ofperiodic dimer covers of G are defined up to an additive constant that we can pick so that avalue at x is . ˆ θ ( x, y ) := max { h ( x ) | h is a height function such that h ( y ) = 0 } . (65) Proposition 4.
Let (Γ , χ ) be a graph with a boundary. Then χ admits an extension to aheight function on Γ if the following inequality holds, χ ( x ) − χ ( y ) ≤ ˆ θ ( x, y ) (66) Proof.
Let us take a family of height functions, { χ ( y ) + ˆ θ ( x, y ) } and let us take the pointwiseminimum over this family. h max ( x ) := min y ∈ ∂ Γ ( χ ( y ) + ˆ θ ( x, y )) (67) h max is a height function because the pointwise minimum of height functions is again a heightfunction. So we need to show that h max agrees with χ on ∂ Γ .To do it let us prove that χ ≤ h max ≤ χ on the boundary faces. Let x be an arbitraryboundary face, x ∈ ∂ Ω . It is sufficient to show that there exists a height function from thefamily such that it satisfies the first inequality for the continuation that is the following, χ ( x ) ≤ χ ( y ) + ˆ θ ( x, y ) . (68) IMIT SHAPES FOR THE DIMER MODEL 23
It is the statement that the height function χ ( x ) − χ ( y ) that has a value at y is less thanthe maximal height function with this property, χ ( x ) − χ ( y ) ≤ +ˆ θ ( x, y ) . (69)The second inequality becomes an equality for the point x = y . (cid:3) Note that due to (66) h max is the maximal extension of χ to a height function Γ . Theminimal extension of χ can be constructed by almost the same way as the maximal. Let usdefine the minimal extension h min , h min ( x ) := max y ∈ ∂ Γ ( χ ( y ) − ˆ θ ( x, y )) . (70)9.5. Convergence of the maximal extensions.
Notice that the strategy of proof of thecriteria is almost the same. The key point is the use of the support functions. To understanda convergence of maximal extensions we need to understand a convergence of the supportfunctions. Before the proposition we need to normalize the support height function, θ n := ˆ θn . (71) Proposition 5.
Let us consider a support height function θ n ( , y ) and the asymptotic heightfunction θ ( , y ) , where a face y contains (0 , . Then there is the following convergence, θ n ( , y ) → θ ( , y ) as n → ∞ , (72) Proof.
By definition, θ and θ n have the same values at . And to prove a convergence weneed to understand that the slope of θ n converges to the slope of θ .Let us take an arbitrary direction in R , the slope of θ along the direction defines a pointon ∂N G , denote the point ( s, t ) .From Proposition 3.2 in [KOS] we know that there exists a sequence of dimer covers of G with the normalized slopes converge to ( s, t ) as n → ∞ , denote the sequence D n .By the definition of θ n its slope is not less than the slope of D n . So the slope of θ n convergesto the slope θ . (cid:3) Now we are ready to prove a convergence of the maximal extensions.
Proposition 6.
Suppose that (Ω , χ ) is a domain with a boundary condition, and (Γ n , χ n ) itsapproximation. Let η max n be the maximal extension of χ n . Then η max n → h max as n → ∞ (73) Proof.
Let us look at expressions of the maximal extensions, h max ( x ) := min y ∈ ∂ Ω ( χ ( y ) + θ ( x, y )) η max n ( x ) := min y ∈ ∂ Γ ( χ n ( y ) + θ n ( x, y )) . (74)To prove the convergence we need to show that the pointwise minimum of χ n ( y ) + θ n ( x, y ) converges to the pointwise minimum of χ ( p )+ θ ( x − p ) . Because (Γ n , χ n ) is an approximation, χ n ( y ) converges to χ ( y ) and θ n ( x, y ) converges to θ ( x − y ) . So if χ ( p ) + θ ( x − p ) archives itsminimum at some point p , χ n ( y ) + θ n ( x, y ) must archive its minimum at a face that contains So θ n ( , y ) is defined at (0 , with the value . p . Otherwise it will converge to χ ( z ) + θ ( x, z ) for some point z , that is bigger then a valueof χ + θ at point p . (cid:3) Density lemma.
Now we are ready to prove the density lemma. Before the formulationand its proof note that for a domain with a boundary condition (Ω , χ ) there is a tautologicalway to express an asymptotic height function f in terms of its own values on Ω . ˆ f ( x ) := min y ∈ Ω ( f ( y ) + θ ( x, y )) (75)Again we need to show that that ˆ f ≤ f ≤ ˆ f , first inequality f ≤ f ( y ) + θ ( x, y ) is just theconsequence of the Support-Lipschitz condition. And the equality for the second inequalityis archived at for the function correspondents to y = x . Theorem 2 (The Density lemma) . Let (Ω n , χ n ) be an approximation of (Ω , χ ) . Then forevery h ∈ H (Ω , χ ) there exists a sequence of normalized height functions η n , such that (cid:107) h − η n (cid:107) ≤ Cn , where C is a positive constant.Vice versa, for every normalized height function η n on (Ω n , χ n ) there exists an asymptoticheight function h ∈ H (Ω , χ ) such that (cid:107) h − η n (cid:107) ≤ n − C .Proof. The strategy is to use the tautological expression for function h and its analogue forheight functions.Let us take a family of points F := { p f } on Ω , where f is a face on Ω n and for every face f there exists a point from the family that contains exactly one point from the family.Let us define the height function that approximates f . ˆ η n ( x ) := min y ∈{ p f } f ∈ Ω ( f ( y ) + θ n ( x, y )) , (76)where f ( y ) is a value of f at a point from the family { p f } corresponding to the face y . Weclaim that ˆ η n approximates f . The proof is almost the same as for the maximal extension.Let us show that ˆ η n ( x ) ≤ f ( x ) ≤ ˆ η n ( x ) .Notice that we have to show it up to an error of order o ( n − ) .Note that f ( y ) + θ n ( x, y ) converges to f ( y ) + θ ( x, y ) as n → ∞ . Thus the second inequalitybecomes the Support Lipschitz condition.The equality for the first inequality is achieved at the face containing point x . If x ∈ F the equality is exact, otherwise it is true up to an error of order n − which is f ( x ) − f ( x (cid:48) ) where x (cid:48) is a point from the family correspondent to the face x .It possible that ˆ η n will have a wrong boundary condition, so we need to balance it betweenthe maximal and the minimal height functions on Ω n . η n := max( h nmax , (min( h min , ˆ η n ))) (77)The proof of the second part of the statement is almost the same, let us define an asymptoticheight function that approximates η n . f ( x ) := min y ∈{ p f } f ∈ Ω η n ( x ) + θ ( x, y )) . (78) (cid:3) On the other points of the face, a value of χ + θ is o ( n − ) -close to χ ( p ) + θ ( x − p ) which is o ( n − ) -closeto χ n ( y ) + θ n ( x, y ) . We can add a term Cn − for sufficiently large C to make the first inequality true, while the limit as n → ∞ will be the same. IMIT SHAPES FOR THE DIMER MODEL 25
The Surface tension theorem.
Finally, we can formulate the main auxiliary theorem of the proof.
Theorem 3.
Let Ω be a compact, connected, simply-connected domain in R . Then let (Ω .χ ) be the domain with a boundary condition g ∈ H (Ω , χ ) and A n = (Ω n , χ n ) be anapproximation of (Ω , χ ) . We recall that Z ( g, δ | A n ) is a partition function of configurationswith height functions δ -close to g . Then lim δ → lim n →∞ n − log Z ( g, δ | A n ) = (cid:90) Ω σ ( ∇ g ) dxdy (79)Before the proof we need two lemmas, the Square lemma and the Triangle lemma . Beforethe Square lemma we need to prove an auxiliary proposition.10.1. The Slide lemma.
Suppose that we have a unit square S and a smaller square oflength − (cid:96) inside S with the same center( see Figure 14). Let ψ be a linear function on thesmaller square and let χ be an asymptotic height function that δ -close to a linear functionwith the same slope as ψ . Proposition 7 (The slide lemma) . In the above notations there is an extension of ψ to anasymptotic height function on ( S , χ ) for sufficiently small δ and a proper choice of (cid:96) . Figure 14.
A unit square containing a square of size − (cid:96) Proof.
We need to extend ψ to an asymptotic height function ˆ ψ that coincides with χ on ∂ S . We will construct a Lipschitz extension and then choose parameters to make it anasymptotic height function. First we make a shift of all functions by the linear function withthe same slope as ψ . Then we have zero on the smaller square and the boundary conditionon S bounded in modulus by δ , without loss of generality let us denote the shifted boundarycondition by the same letter χ . Let us connect two values on every ray by the unique linear function that has a value onthe boundary of the smaller square and a value of χ on ∂ S . Let us denote the continuationby h . Figure 15.
A continuationWe have to prove that h is an asymptotic height function. To do this we need to checkthat ∇ h ( x ) ∈ N G .On each ray ∇ h ( x ) is a constant equal to χ ( x ) (cid:96) and in direction orthogonal to the ray, h has the same slope as χ . Let us choose (cid:96) = √ δ , then for sufficiently small δ the slope afterinverse shift lies on N G . (cid:3) Remark 3.
Note that the area of continuation is of order √ δ . The square lemma.Proposition 8 (The square lemma) . Let ( S , χ ) be a domain with a boundary condition,where S is a unit square and χ is δ -close to a linear function with the slope ( s, t ) .Suppose that ( S n , χ n ) is an approximation of ( S , χ ) and let g ∈ H ( S , χ ) be δ -close to alinear with the slope ( s, t ) . We recall that Z ( g, δ |S n ) is the partition function of S n where wetake into account only configurations with height functions δ -close to g . Then, lim δ → lim n →∞ n − log Z ( g, δ |S n ) = σ ( s, t ) (80)In this proposition all graphs will be subgraphs of G n for a fixed n > which we hold untilthe end of the proposition and then send it to infinity. Before the proof we need to introducetwo sequences of graphs embedded into the torus. T n ± l ( n ) := φ n ( G/ ( n ± l ( n )) Z ) (81)Where l ( n ) is a sequence such that lim n →∞ l ( n ) n = (cid:96) . Note that not all boundary height functions χ are possible to appear. For example in case of ( s, t ) ∈ ∂N G and s, t ≥ height functions on the north boundary are less than zero and on the south are more than zerodue to the fact that the slope before the linear shift lies in N G . IMIT SHAPES FOR THE DIMER MODEL 27
Proof.
We want to approximate the partition function of S n by the partition functions of T n ± l ( n ) that include dimer cover of the slope ( s, t ) (let us denote the partition function ofconfiguration with this slope by Z ( s,t ) ). The weights of the dimer covers are the same, so wehave to show that the number of such dimer covers is approximately the same. Let us recallthat D s,t ( T n − l ( n ) ) is the set of dimer configurations on T n − l ( n ) that have the slope ( (cid:98) ns (cid:99) , (cid:98) nt (cid:99) ) .We want to show that there are two inclusions, D s,t ( T n − l ( n ) ) ⊆ D ( S n | δ ) ⊆ D s,t ( T n + l ( n ) ) . (82)Then we can conclude the following relation between the partition functions , (cid:88) D ∈ D s,t ( T n − l ( n ) ) w ( D ) ≤ (cid:88) D ∈ D ( S n | δ ) w ( D ) ≤ (cid:88) D ∈ D s,t ( T n + l ( n ) ) w ( D ) . (83)And after the normalization, n − log( Z s,t ( T n − l ( n ) )) ≤ n − log( Z ( S n | δ )) ≤ n − log( Z s,t ( T n + l ( n ) ) . (84)We know that lim n →∞ n − log( Z ( s,t ) ( T n ) = σ ( s, t ) from 26.So we can conclude that lim n →∞ ( n ± l ( n )) × ( n ± l ( n )) − n − log( Z s,t ( T n ± l ( n ) )) = σ ( s, t ) × lim n →∞ ( n ± l ( n )) n = σ ( s, t )(1+ o ( δ / ) . (85)Let us prove the first inclusion in (82) , to do it we have to show that each height functionon ( T s,tn − l ( n ) ) extends to a height function on ( S n | δ, g ) . Note that T s,tn − l ( n ) is an approximation ofthe smaller square inside S (let us denote the smaller square by S ) with the linear boundarycondition χ that has the slope ( s, t ) . First let us choose the extension of the boundarycondition on ∂ S to an asymptotic height function on S . From the previous proposition weknow that there is a choice of (cid:96) such that the extension of the boundary condition to anasymptotic height function on S exists.Then using the density lemma we can build a sequence of height functions on ( S n , χ n ) suchthat on T n − l ( n ) the sequence coincides with the approximatively linear height function.To prove the second inclusion we have to show that each height function on ( S n | δ, g ) extendsto a height function on ( T s,tn + l ( n ) ) . The proof is almost the same as for the first inclusion exceptthe way of using the previous proposition. We have to invert the slide lemma to the case wherethe boundary condition on the boundary of the bigger square is linear with the slope ( s, t ) and the boundary condition on the boundary of the smaller square is bounded in modulusby δ . (cid:3) Remark 4.
The Square lemma also works for tori of different sizes. Let us look at the squareof size k × k with embedded G n ,let us denote the image of the embedding by S n ( k ) . Doing thesame as for the normalization of tori we get an area factor, lim n →∞ ( kn ) × ( kn ) − n − log( Z ( s,t ) ( S n ( k )) = σ ( s, t ) × k . (86) Note that the weight of dimer covers and the weight of their extension are not the same. However,the dimer covers differ only on the area of continuation that is of order δ . Thus the weights differ by C Area = C (cid:48) exp( o ( δ )) The extension is the same, we make the shift by the linear function with the slope ( s, t ) and on each raybetween boundaries we connect two boundary conditions by the unique linear function that coincides withthe boundary conditions. The triangle lemma.Claim 4.
Let Ω be an equilateral triangle in R and χ ∈ H (Ω) . Let g ∈ H (Ω , χ ) be anasymptotic height function δ -close to a linear with the slope ( s, t ) and let A n = (Ω n , χ n ) bean approximation of (Ω , χ ) . lim δ → lim n →∞ n − log Z ( g, δ | A n ) = σ ( s, t ) (87) Proof.
Let us take a square lattice with the mesh (cid:96) and use the cutting rule for the curve ρ obtained from the intersection of the square lattice and Ω . Z (Ω n , χ n ) = (cid:88) χ ρ Z ( χ ρ ) = (cid:88) χ ρ (cid:89) j Z ( S j , χ jρ ) , (88)where S j is the square from the Square lattice with the boundary height function χ jρ .We have two types of squares: included squares that do not intersect ∂ (Ω) and excludedsquares that intersect ∂ (Ω) . We want to build an upper and a lower bounds for the partitionfunction of (Ω n , χ n | g, δ ) , let us denote the upper bound by Z L (Ω n ) and the lower bound by Z U (Ω n ) . We estimate included squares using the Square lemma and make a rough bound forthe excluded squares. Figure 16.
An intersection of a triangle with a square lattice. Includedsquares in gray and excluded ones in red.10.3.1.
The lower bound.
We want to make a lower bound of sum from the cutting rule, letus take only one summand correspondent to a boundary height function χ n . Here we estimate the excluded squares by the minimal weight of an edge (let us denote itby w min ) to the power of number of edges in these squares. log Z (Ω n , χ n ) ≥ log Z L (Ω n ) := log w min S ex + (cid:88) k log Z kn ( s, t ) , (89) Where we take (cid:96) ≤ δ such that (cid:96) = o ( δ ) Existence of such height function follows from the density lemma applied to g . A boundary heightfunction is a restriction of the height function that approximates g . IMIT SHAPES FOR THE DIMER MODEL 29 where S ex is the total number of edges in the excluded squares that we estimate by o ( (cid:96) ) .And Z kn ( s, t ) is the partition function of the square S k . After the normalization we have lim n →∞ n − log Z kn ( s, t ) = σ ( s, t ) ×A ( S k )+ o ( δ ) from the square lemma and remark 1. Finally,the lower bound is, n − log Z L (Ω n ) = (cid:88) k n − log Z kn ( s, t ) + o ( δ ) . (90)10.3.2. The upper bound.
In the upper bound we estimate the sum in the Cutting Rule bytaking the maximal summand times the number of summands, which is the number of theall boundary height functions.We bound the excluded squares by the maximal weight of the edge to the power of totalnumber of edges. For the included squares we estimate the same way as for the lower bound.Let us bound the number of boundary height function by o ( n ) Z (Ω n , χ n ) ≤ Z U (Ω n ) := w o ( (cid:96) )max × o ( n ) × (cid:88) j n − log Z jn ( s, t ) (91)Finally after taking the limit as n → ∞ the first summands differ from each other by o ( δ ) . Thus, both estimates differ from σ ( s, t ) × Area of the triangle by o ( (cid:96) ) . So after taking limitas (cid:96) → and δ → we have the proposition. (cid:3) The proof of the Surface Tension theorem.
In this section we will use a triangularlattice with a mesh (cid:96) and peacewise linear approximations of Lipschitz functions from Claim3. The idea of the proof is almost the same as for the triangular lemma.
Theorem 4.
Let Ω be compact, connected, simply-connected domain in R and χ ∈ H (Ω) .Let g be an asymptotic height function g ∈ H (Ω , χ ) and let A n = (Ω n , χ n ) be an approxima-tion of (Ω , χ ) . Then lim δ → lim n →∞ n − log Z ( g, δ | A n ) = (cid:90) Ω σ ( ∇ g ) dxdy (92) Proof.
From theDensity lemma we have a sequence of normalized height function { η n } thatconverges to g .Take (cid:96) small enough such that (cid:107) g (cid:96) − g (cid:107) ≤ δ/ on − δ/ fraction of the triangles .Let us use the Cutting Rule for curve ρ obtained from the intersection of the triangularlattice of the mesh (cid:96) with the domain. Note that the curve cuts Ω n into triangles withboundary conditions along ρ , let us denote the triangles by { T j } and their boundary heightfunctions by { χ jρ } . (See figure 17) Z (Ω n , χ n ) = (cid:88) χ ρ Z ( χ ρ ) = (cid:88) χ ρ (cid:89) j Z ( T j , χ jρ ) , (93)There are two types of triangles { T j } . The first type is triangles that do not intersect theboundary of the domain and where g (cid:96) is δ/ -close to g . The second type consists of trianglesthat intersect the boundary of the domain or where g (cid:96) does not approximate g .Then we want to make an upper and a lower bounds for the normalized partition function, n − log Z (Ω n , χ n | δ ) . In both cases we estimate two types of triangles separately. For the firsttype we can use claim 4 and for the second we make a rough estimate. Then after takinglimit as n → ∞ the normalized estimates will differ by o ( δ ) . It corresponds to a choice (cid:15) = δ/ in claim 3. Figure 17.
Intersection of Ω n with a triangular lattice. Triangles of the firsttype in gray and triangles of the second type in yellow and red.10.4.1. The lower bound.
In a lower bound we have to include some height functions that δ -close to g . To do this we can take only one summand from (93) corresponding to oneboundary height function χ ρ . Let us take the boundary height function obtained from therestriction of η n , without loss of generality let us denote it χ ρ .Let us estimate the triangles of the first type by the product that includes only trianglesof this type. We can bound the triangles of the second type by the weight of one heightfunction with the minimal weight, that is the minimal weight of an edge to the power of totalnumber of edges in triangles of the second type (let us denote it by S ). Z (Ω n , χ n ) = (cid:89) j Z ( T j , χ jρ ) ≥ Z L := (cid:89) k Z ( T k , χ kρ ) w S min (94)Let us estimate the triangles of the first type using triangle case to count δ/ -close heightfunctions to make sure that we include only height functions δ -close to g . For triangles ofthe first type we have the following, lim n →∞ n − log (cid:89) j Z ( T j , χ jρ | δ/
3) = (cid:88) k σ ( s k , t k ) × A ( T k ) + o ( δ ) , (95)where ( s k , t k ) is a slope of g (cid:96) on the boundary of the triangle T k and A ( T k ) is the area ofthe triangle T k .Finally, the lower bound after taking the limit as n → ∞ is the following, (cid:88) j σ ( s j , t j ) A ( T j ) + log w min S + o ( δ ) , (96)where we bound the total number of edges in triangles of the second type by the area of suchtriangles that we denote by S .10.4.2. The upper bound.
We can use almost the same strategy to make an upper bound.First, we have to include all height function δ -close to g . Let us estimate the triangles ofthe first type by the same way, but count δ -close height functions. For the triangles of thesecond type we make a rough estimate taking the maximal weight of an edge to the powerof the number of edges in this triangles multiplied by a number of summands in the cuttingrule that is o ( n ) . Note that
S → as δ → because the fraction of the triangles of the second type is o ( δ ) . IMIT SHAPES FOR THE DIMER MODEL 31 Z (Ω n , χ n ) ≤ Z U := (cid:89) j Z ( T j , χ jρ ) = (cid:48) (cid:89) j Z ( T j , χ jρ ) w S ( T j )max o ( n ) (97)And after taking limit as n → ∞ the normalized upper bound is the following, (cid:48) (cid:88) j σ ( s j , t j ) A ( T j ) + log w max S + o ( δ ) . (98)Both bounds are F ( g (cid:96) ) + o ( δ ) that differs from F ( g ) by o ( δ ) from the lemma 2.2 from[CKP]. Thus, after taking the limit as δ → we have the theorem. (cid:3) Appendix: Absolute height functions
Ribbon graph structure.
Every surface graph is also a ribbon graph: a ribbon graphis a graph with an additional structure given by, for each vertex, a cyclic order of the edgesat this vertex. For the case of bipartite graphs we can fix an orientation simply orient edgesat every black vertex clock-wise and at white vertices counter clock-wise.An oriented path on a ribbon graph is an oriented path on a dual graph that preservescyclic order at every vertex i.e. it goes in clock-wise direction at every black vertex andcounter-clock-wise at white one. See an example on Figure 18.
Figure 18.
Examples of oriented paths.10.6.
Absolute height functions.
Due to the dependence of h D,D (cid:48) and N Γ on D (cid:48) it is moreconvenient to define an absolute homology class for D . One way to do it is to fix a 1-chain Φ such that [ D ] − Φ is a cycle.It can be done for every bipartite graph without self-intersections, see theorem 3.3 on page28 in [GK]. We will focus on regular graphs, i.e. graphs with the same valence N for allvertices.In this case one can pick Φ = (cid:80) e N [ e ] . Obtained height functions are called absolute heightfunctions . (cid:82) Ω σ ( ∇ g (cid:96) ) dxdy = (cid:80) j σ ( s j , t j ) However, one needs to change coefficients of homology from Z to Q or to multiply coefficients by N tomake them integers. One can notice that absolute height functions obey the local rule. It states that aroundeach vertex they increase with the respect to cyclic order around this vertex: around blackvertices they increase in clock-wise direction and around white vertices in counter-clock-wisedirection. See an example on Figure 19 .
Figure 19.
The local rule for an absolute height functionMoreover, it is not hard to check that every function on faces that satisfy the local rule isan absolute height function of some dimer cover.
Proposition 9.
There is a bijection between absolute height functions and functions on facesthat satisfy the local rule.
Also note that absolute height functions are Lipschitz functions in a sense described below,which is a consequence of the local rule.Let π ( f , f ) be the length of the shortest oriented path on dual graph of Γ that connectsfaces f and f . Proposition 10.
Every absolute height function satisfies modified Lipschitz condition: h D ( f ) − h D ( f ) ≤ N π ( f , f ) (99) Remark 5.
It is important that there are such functions that satisfy Lipschitz condition,but are not absolute height functions. For example, a constant function on faces satisfiesLipschitz condition, but does not satisfy the local rule.
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