Domino Tilings of the Torus
Abstract
We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be represented by height functions; for a torus given by a lattice , these functions exhibit arithmetic quasiperiodicity. The additive constants determine the flux of the tiling, which can be interpreted as a vector in the dual lattice . We give a characterization of the actual flux values, and of how corresponding tilings behave. We also consider domino tilings of the infinite square lattice; tilings of tori can be seen as a particular case of those. We describe the construction and usage of Kasteleyn matrices in the counting problem, and how they can be applied to count tilings with prescribed flux values. Finally, we study the limit distribution of the number of tilings with a given flux value as a uniform scaling dilates the lattice .
itemize \BeforeBeginequation \BeforeEndequation \BeforeBegintheo \BeforeEndtheo \BeforeBeginprop \BeforeEndprop \authorRImpellizieri, Fillipo de Souza Lima \resumeThe author obtained the degree of Bacharel em Matemática from PUCRio in July 2013 \advisorNicolau Corção Saldanha \advisorRSaldanha, Nicolau Corção \titlebrCoberturas do Toro por Dominós 11 September \monthN09 2015 \cityRio de Janeiro \CDD510 \departmentMatemática \programMatemática \programbrMatemática \schoolCentro Técnico Científico \universityPontifícia Universidade Católica do Rio de Janeiro \uniPUCRio \jury \jurymemberCarlos TomeiDepartamento de Matemática – PUCRio \jurymemberJuliana Abrantes FreireDepartamento de Matemática – PUCRio \jurymemberMarcio da Silva Passos TellesInstituto de Matemática e Estatística – UERJ \jurymemberRobert David MorrisInstituto de Matemática Pura e Aplicada – IMPA \schoolheadJosé Eugenio Leal \acknowledgment To CNPq, FAPERJ and PUCRio, for making this research possible. To my advisor, Prof. Nicolau Corção Saldanha, for his boundless patience, eagerness to help and valuable insight. To the Department of Mathematics, especially Creuza, Fred and Renata, for helping me get here and putting up with my fickleness. To my jury, for their appreciation, understanding and precious feedback. In particular, I would like to thank Carlos Tomei for his spirit and enthusiasm, and Marcio Telles for his dedication. To my professor Ricardo Earp, for his guidance and caring. To my friends, for their positivity and reassurance. In particular, to my graduation friends Felipe, Leonardo and Gregory for holding onto this ‘academic friendship’. To my family, for their patience, unwavering support and cherished love. \keywordsbr \keydominó \keycobertura \keytoro \keyreticulado \keyfluxo \keyflip \keyfunção altura \keymatriz de Kasteleyn \abstractbr Consideramos o problema de contar e classificar coberturas por dominós de toros quadriculados. O problema de contagem para retângulos foi estudado por Kasteleyn e usamos muitas de suas ideias. Coberturas por dominós de regiões planares podem ser representadas por funções altura; para um toro dado por um reticulado , estas funções exibem quasiperiodicidade aritmética. As constantes aditivas determinam o fluxo da cobertura, que pode ser interpretado como um vetor no reticulado dual . Damos uma caracterização dos valores de fluxo efetivamente realizados e de como coberturas correspondentes se comportam. Também consideramos coberturas por dominós do reticulado quadrado infinito; coberturas de toros podem ser vistas como um caso particular destas. Descrevemos a construção e uso de matrizes de Kasteleyn no problema de contagem, e como elas podem ser aplicadas para contar coberturas com valores de fluxo prescritos. Finalmente, estudamos a distribuição limite do número de coberturas com um dado valor de fluxo quando o reticulado sofre uma dilatação uniforme. \tablesmodenoshow \makenoidxglossaries\newglossaryentrydual graph name=dual graph, sort=dual graph, description=The graph obtained from a quadriculated region by substituting each of its squares by a vertex and joining neighboring vertices by an edge; see page Domino Tilings of the Torus \newglossaryentrymatching name=(perfect) matching, sort=matching, description=A perfect matching is a set of edges on a graph in which each vertex features exactly once. Corresponds to a tiling. See page Domino Tilings of the Torus \newglossaryentrybipartite name=bipartite, sort=bipartite, description=A graph whose vertices can be separated into two disjoint sets such that every edge joins a vertex from to . See page Domino Tilings of the Torus \newglossaryentryblack and white condition name=black and white (condition), sort=black and white, description=The requirement that a region’s dual graph be bipartite. Alternatively, the requirement that every domino on a tiling consist of one square of each color. Implies an equal number of black squares and white squares. See pages Domino Tilings of the Torus, 5 and 5 \newglossaryentryflip name=flip, sort=flip, description=A move on a tiling that exchanges two dominoes tiling a square by two dominoes in the only other possible configuration \newglossaryentryheight function name=height function, sort=height function, description=An integer function that encodes a tiling. For a tiling of a quadriculated planar region , it is defined on the vertices of ; see Section 1. For a tiling of a quadriculated torus , it is quasiperiodic and defined on (sometimes called its toroidal height function); see Section 4 \newglossaryentryKasteleyn matrix name=Kasteleyn matrix, sort=Kasteleyn matrix, description=A modified adjacency matrix whose determinant counts tilings of a region. For Kasteleyn matrices of planar regions, see Section 2. For an overview of Kasteleyn matrices on the square torus, see page Domino Tilings of the Torus; for a detailed exposition on more general tori, refer to Chapter Domino Tilings of the Torus \newglossaryentryPfaffian name=Pfaffian, sort=Pfaffian, description=The determinant of a skewsymmetric matrix can be written as the square of a polynomial in ’s entries. This polynomial is ’s Pfaffian \newglossaryentryi name=, sort=i, description=The imaginary unit \newglossaryentryDn name=, sort=dn, description=A square fundamental domain for the square torus . See page Domino Tilings of the Torus \newglossaryentryTn name=, sort=Tn, description=The square torus . See pages Domino Tilings of the Torus \newglossaryentrycrossflip name=crossflip, sort=crossflip, description=A flip involving a crossover domino. See page Domino Tilings of the Torus \newglossaryentrycrossover domino name=crossover domino, sort=crossover domino, description=On a fundamental domain of a torus , a domino that crosses a side that’s been identified with another side. See page Domino Tilings of the Torus \newglossaryentryflux name=flux, sort=flux, description=A flux of a tiling counts crossover dominoes with a sign; see page Domino Tilings of the Torus for an overview, and Chapter Domino Tilings of the Torus for a detailed exposition. For a torus , the flux may also be thought as an element of the affine lattice ; see Section 6 \newglossaryentryLaurent polynomial name=Laurent polynomial, sort=Laurent polynomial, description=A Laurent series with finitely many nonzero coefficients \newglossaryentryPhi name=, sort=zPhi, description=The mod 4 prescription function on the infinite square lattice . See page 4 \newglossaryentryQuasiperiodicity name=(arithmetic) quasiperiodicity, sort=quasiperiodicity, description=A function satisfying for some and all \newglossaryentryL name=, sort=Lattice, description=A lattice. See page Domino Tilings of the Torus \newglossaryentryLstar name=, sort=Latticedual, description=The dual lattice of , given by . See page Domino Tilings of the Torus \newglossaryentryLsharp name=, sort=Latticesharp, description=The translate of in that contains all flux values of tilings of . See Section 6 \newglossaryentrydisjointUnion name=, sort=zunion, description=The union of two disjoint sets and \newglossaryentryvalid lattice name=valid lattice, sort=valid lattice, description=A lattice whose vectors have integral coordinates that are the same parity. See Section 5 \newglossaryentryE name=, sort=even, description=The lattices and of vectors whose coordinates are respectively both even and both odd \newglossaryentryTL name=, sort=TL, description=The torus , where is a lattice. See page 5 \newglossaryentryt name=, sort=t, description=A tiling \newglossaryentryphi name=, sort=zphi, description=A flux value \newglossaryentryedge name=edge, sort=edge, description=An edge on a graph, or an edge on the boundary of a square of a quadriculated region \newglossaryentryedgepath name=edgepath, sort=edgepath, description=A sequence of neighboring vertices (either on a graph or on a quadriculated region) \newglossaryentryFL name=, sort=FL, description=The set of all flux values of tilings of . See Section 7 \newglossaryentryv1 name=, sort=zvnorm1, description=For , the norm \newglossaryentryvinfinity name=, sort=zvnorminfinity, description=For , the infinity norm } \newglossaryentryhmax name=, sort=hmax, description=The height function that is maximal over height functions on that are 0 at the origin. See page 7.2 \newglossaryentryhmin name=, sort=hmin, description=The height function that is minimal over height functions on that are 0 at the origin. See page 7.5 \newglossaryentrydoublyinfinite name=doublyinfinite, sort=doublyinfinite, description=A staircase edgepath or domino staircase that is infinite in both directions \newglossaryentryH0R name=, sort=H0R, description=The set of height functions on that are 0 at the origin. See page 7 \newglossaryentrygammavw name=, sort=zgammavw, description=The set of edgepaths in joining to and that respectively respect and reverse colorinduced edge orientation. See pages 7 and 7 \newglossaryentrylgamma name=, sort=lgamma, description=The length of an edgepath \newglossaryentryedgeprofile name=edgeprofile (type), sort=edgeprofile, description=The colorinduced orientation of edges round a vertex, one of two types. See page 7 \newglossaryentryordered sum name=ordered sum (representation), sort=ordered sum, description=A unique representation of edgepaths in or sets. See 7 \newglossaryentryhvphimax name=, sort=hmaxvphi, description=The height function that is maximal over height functions on that take the value at . See 7 \newglossaryentryhLphimax name=, sort=hmaxLphi, description=The height function that is maximal over toroidal height functions of with flux . See page 7 \newglossaryentryflipconnected name=flipconnected, sort=flipcon, description=A quadriculated region or set of tilings such that any two distinct tilings can be joined by a sequence of flips. For flipconnectedness on simplyconnected planar regions, see Corollary 1.6 in Section 1. For a discussion of flipconnectedness on tori, refer to Chapter Domino Tilings of the Torus \newglossaryentryflipisolated name=flipisolated, sort=flipiso, description=A tiling which admits no flips, or a set of tilings in which no two distinct tilings can be joined by a sequence of flips \newglossaryentrystaircase name=staircase, sort=staircase, description=For domino staircases, see page 8. For staircase edgepaths, see page 8. For types of staircases, see pages 8 and 10 \newglossaryentrywindmill name=windmill (tiling), sort=windmill, description=A tiling of the infinite square lattice that admits no flips and consists entirely of infinite domino staircases that are never doublyinfinite. See page 8 \newglossaryentrybrick wall name=brick wall, sort=brick wall, description=A tiling that uses only one type of domino (vertical or horizontal) and consists entirely of doublyinfinite domino staircases. See page 8 \newglossaryentryrugged rectangle name=rugged rectangle, sort=rugged rectangle, description=A type of quadriculated region. See page 8 \newglossaryentryrugged quadrant name=(cardinal) rugged quadrant, sort=rugged quadrant, description=A type of quadriculated region. See page 8 \newglossaryentryvLeL name=, sort=zvLeL, description=An equivalence class of a vertex , edge or doublyinfinite domino staircase . See pages 9, Domino Tilings of the Torus and 10 \newglossaryentryLKasteleynsigning name=Kasteleyn signing, sort=LKasteleynsigning, description=An assignment of plus and minus signs to equivalence classes of edges. See page Domino Tilings of the Torus \newglossaryentryLKasteleynsigninguniversal name=Universal Kasteleyn signing, sort=universalKasteleynsigning, description=A Kasteleyn signing that applies to any valid lattice. See page Domino Tilings of the Torus \newglossaryentrygz2 name=, sort=gz2, description=The dual graph of the infinite square lattice. See page Domino Tilings of the Torus \newglossaryentryLflip name=flip (round ), sort=Lflip, description=A flip round each vertex in . See page 9 \newglossaryentrystairflip name=stairflip, sort=stairflip, description=A move on a tiling that exchanges a doublyinfinite domino staircase by a doublyinfinite domino staircase in the only other possible configuration. See page 10 \newglossaryentryLstairflip name=stairflip (on ), sort=Lstairflip, description=A stairflip on each doublyinfinite domino staircase in . See page 10 \newglossaryentrystairL name=, sort=stairL, description=The set of equivalence classes of doublyinfinite domino staircases in . See page 10 \newglossaryentrystairLk name=, sort=stairLk, description=The set of equivalence classes of type doublyinfinite domino staircases in . See page 10 \newglossaryentrystairLkverthor name=, sort=stairLkverthor, description=The sets of equivalence classes of type doublyinfinite domino staircases in whose dominoes are all vertical, or all horizontal. See page 10 \newglossaryentryxiL name=, sort=zxiL, description=The stairflip operator. See page 10 \newglossaryentryxiLkexclusive name=exclusive, sort=zxiLkexclusive, description=A type of subset of . See page 10 \newglossaryentryQk name=, sort=Qk, description=A component of . See pages 10 and 10 \newglossaryentrycycleflip name=cycle, cyle flip, sort=cycle flip, description=Simultaneously representing two tilings of on the same fundamental domain induces a decomposition of the domain into disjoint domino cycles. A cycle flip uses these cycles to go from one tiling to the other. For details, see page 11 \newglossaryentrycycle interior name=interior of a cycle, sort=interior of a cycle, description=The union of the interior of domino paths of a closed cycle; see page 11 \newglossaryentrycycle exterior name=exterior of a cycle, sort=exterior of a cycle, description=The intersection of the exterior of domino paths of a closed cycle; see page 11 \newglossaryentryctot1 name=, sort=ct0t1, description=The set of cycles formed by the superposition of tilings and . See page 11 \newglossaryentryshort name=short, sort=short, description=A vector of a lattice such that for all \newglossaryentryparameter name=(cycle) parameter, sort=parameter, description=A short vector in , unique up to sign, associated to an open cycle. See page 12 \newglossaryentryquasicycle name=()quasicycle, sort=quasicycle, description=A type of ()quasiperiodic function. See page 12 \newglossaryentrypseudoflux name=, sort=zphi de gamma, description=The pseudoflux of the quasicycle ; see page 12 \newglossaryentrycompatible name=compatible (with ), sort=compatible, description=A tiling of a torus is compatible with a quasicycle if it contains every other domino of ; see page 12 \newglossaryentryargument name=argument (function), sort=argument, description=A function that can be defined for certain edgepaths. See page 12 \newglossaryentryflux connecting name=fluxconnecting (basis), sort=fluxconnecting, description=A basis of such that the moves connect . See page 13 \newglossaryentryKast name=, sort=Kaste, description=The Kasteleyn weight of the edge . For Kasteleyn weights, refer to the construction of Kasteleyn matrices (Section 2 and Chapter Domino Tilings of the Torus) \newglossaryentryDL name=, sort=DL, description=A specific choice of rectangular fundamental domain for a lattice . See page Domino Tilings of the Torus \newglossaryentryDab name=, sort=Dab, description=A set of points in a translate of . See page 14 \newglossaryentryop0p1 name=, sort=op0p1, description=The oriented weight of the edge . See page 14 \newglossaryentryqquotient name=, sort=zquotient, description=The quotient of by \newglossaryentryz2modLstar name=,sort=zquotientstar, description=The group \newglossaryentryFLq name=, sort=FLq, description=The space of complex functions on that are periodic with parameter . See page 15 \newglossaryentrypsi name=, sort=zpsi, description=An isomorphism between a space of periodic functions and . See page 15 \newglossaryentryl0 name=, sort=Latticez, description=The lattice spanned by ; the lattice that contains every valid lattice. See page 15 \newglossaryentryl0bw name=, sort=Latticezbw, description=The affine lattices and , respectively. Equivalently, respectively the set of black vertices and the set of white vertices of . See page 15 \newglossaryentryBLq name=, sort=BLq, description=The space of complex functions on that are periodic with parameter . See page 15 \newglossaryentryWLq name=, sort=WLq, description=The space of complex functions on that are periodic with parameter . See page 15 \newglossaryentryDbDw name=, sort=DbDw, description=Ordered bases of and , associated to the construction of a Kasteleyn matrix. See page 15 \newglossaryentryKKd name=, sort=KKD, description=A Kasteleyn matrix. The subindex in refers to the matrix constructed from Kasteleyn weights in a fundamental domain ; see page 15 \newglossaryentryKE name=, sort=KKE, description=A diagonal Kasteleyn matrix, obtained from representing in exponential bases. See page 15 \newglossaryentryfrestra name=,sort=zfrestra, description=The restriction of to \newglossaryentryEbEw name=, sort=EbEw, description=Ordered bases of and , called exponential bases, associated to a diagonalization of . See page 15 \newglossaryentryXAB name=, sort=XAB, description=The change of basis from to . See page 15 \newglossaryentryrho name=, sort=zrho, description=A complex phase related to the determinants of and . See page 15 \newglossaryentryrho12 name=, sort=zrho12, description=Complex phases associated to . See page 16 \newglossaryentryeven lattice name=even lattice, sort=even lattice, description=A valid lattice that does not contain points with odd coordinates. See page 16 \newglossaryentryodd lattice name=odd lattice, sort=odd lattice, description=A valid lattice that contains points with odd coordinates. See page 16
Capítulo \thechapter Introduction
Tilings of planar regions by dominoes (and also lozenges) can be thought of as perfect matchings of a corresponding graph. In this sense, the enumeration of matchings was studied as early as 1915 by MacMahon [macmahon1984combinatory], whose focus was on plane partitions. Also around the time, chemists and physicists were interested in aromatic hydrocarbons and the behavior of liquids. Hereafter, I will refer to perfect matchings simply by ‘matchings’.
Research on dimers in statistical mechanics had a major breakthrough in 1961, when Kasteleyn [kasteleyn1961statistics] (and, independently, Temperley and Fisher [temperley1961dimer]) discovered a technique to count the matchings of a subgraph of the infinite square lattice. He proved that this number is equal to the Pfaffian of a certain ,matrix associated with . Not much later, Percus [percus1969one] showed that when is bipartite, one can modify so as to obtain the number from its determinant (rather than from its Pfaffian). James Propp [propp1999enumeration] provides an interesting overview of the topic on his ‘Problems and Progress in Enumeration of Matchings’.
In the early 90s, more advances were made and gave new impetus to research. Conway [conway1990tiling] devised a grouptheoretic argument that, in many interesting cases, may be used to show that a given region cannot be tessellated by a given set of tiles. In a related work, Thurston [thurston1990] introduced the concept of height functions: integervalued functions that encode a tiling of a region. With them, he presented a simple algorithm that verifies the dominotileability of simplyconnected planar regions.
In 1992, Aztec diamonds were examined by Elkies, Kuperberg, Larsen and Propp [elkies1992alternating], who gave four proofs of a very simple formula for the number of domino tilings of these regions. Later, probability gained importance with the study of random tilings, and Jockush, Propp and Shor [jockusch1998random, cohn1996local] proved the Arctic Circle Theorem. This framework was further generalized in the early 2000s by Kenyon, Okounkov and Sheffield [kenyonokounkov2006dimers, kenyonokounkov2006planar], whose work relates random tilings to Harnack curves and describes the variational problem in terms of the complex Burgers equation.
While now much is known about tilings for planar regions, higher dimensions have proven less tractable. Randall and Yngve [randall2000random] examined analogues of Aztec diamonds in three dimensions for which many of the twodimensional results can be adapted. Hammersley [hammersley1966limit] makes asymptotic estimates on the number of brick tilings of a dimensional box as all dimensions go to infinity. In his thesis, Milet [milet2015domino] studied certain threedimensional regions for which he defines an invariant that can be interpreted under knot theory.
This dissertation was motivated by the observation of a certain asymptotic behavior in the statistics of domino tilings of square tori. We elaborate: consider a quadriculated torus, represented by a square with sides of even length and whose opposite sides are identified. A domino is a rectangle. Below, we have a tiling of the torus which also happens to be a tiling of the square.
Because in the torus opposite sides are identified, we may also consider tilings with dominoes that ‘cross over’ to the opposing side.
The flux of a tiling is an algebraic construct that counts these crossover dominoes, with a sign; one may think of it as a pair of integers. In the next figure, we assign the positive sign when a white square is to the right of the blue curve or when a black square is above the red curve (and the negative sign otherwise). Hence, their fluxes are , and , where the first integer counts horizontal dominoes crossing the blue curve and the second integer counts vertical dominoes crossing the red curve.
We may thus count tilings of tori by flux. In the model, we have:
For a total of 272 tilings. Observe the proportion of total tilings by flux:
Now we repeat the process for different square tori:
In each case, tilings with flux comprise almost half of all tilings of the square torus. For other values of flux in the table it may not be as apparent, but as increases the proportions stabilize.
Theorem.
As goes to infinity, the proportions converge to a discrete gaussian distribution. More specifically, for each , as goes to infinity the proportion relative to flux tends to
The formula for the rather curious constant can be derived from thetafunction identities; see Yi [yi2004theta]. For comparison, we provide the previous table, together with the limit value given by the formula above:
We will not prove this theorem in this dissertation.
Nevertheless, it motivated us to study domino tilings of the torus and the underlying combinatorial and algebraic structures involved. We expect the content of this text lays the groundwork for writing a proof of this theorem in the future.
That said, results of this kind are not new to physicists, and in fact neither to mathematicians. Boutillier and de Tilière [boutillier2009loop] derived explicit formulas for the limit proportions in the honeycomb model of the torus (in this model, a matching may be thought of as a lozenge tiling of the torus). They interpret matchings as loops (see Cycles and cycle flips, Section 11) and study the asymptotic behavior of corresponding winding numbers. Although their methods differ from ours, parallels can be drawn.
In Chapter Domino Tilings of the Torus, we examine domino tilings of quadriculated planar, simplyconnected regions. We discuss how the study of domino tilings is related to the problem of determining perfect matchings of a graph, and present the idea of blackandwhite colorings (so our equivalent graphs are bipartite). In Section 1 we explore two concepts, as well as their relations. A flip is a move on a tiling that exchanges two dominoes tiling a square by two dominoes in the only other possible configuration. A height function is an integer valued function on the vertices of the squares of a tiling that encodes . Later, these concepts will be generalized to the torus case, and many results of this section (like a characterization of height functions, or the flipconnectedness of these regions) admit adaptation.
Section 2 details the construction of Kasteleyn matrices and explains how their determinants can be used to count domino tilings of a region. Finally, Section 3 contains a worked, classical example: the problem of enumerating domino tilings of the rectangle.
In Chapter Domino Tilings of the Torus begins our study of the torus; we initially consider the square torus with side length . The notion of flux is introduced here, and an overview of how Kasteleyn matrices can be adapted is provided. We also supply a figure with all tilings of the square torus.
Section 4 extends height functions to this scenario by interpreting as a quotient , where is the lattice generated by . Moreover, we show the flux manifests in the arithmetic quasiperiodicity of height functions: they satisfy for some and all .
In Section 5, we consider more general tori by allowing other lattices in the quotient. These are called valid lattices: their vectors have integral coordinates that are the same parity. This condition is necessary for the resulting graph to be bipartite.
Chapter Domino Tilings of the Torus further investigates the flux. In Section 6, we describe how the flux can be thought of as an element of the dual lattice . More precisely, we show there is a translate of in that contains all flux values; we call this affine lattice .
Section 7 provides our first theorem. For a valid lattice , let be the set of all flux values of tilings of ; the inner product identification allows us to regard as a subset of . Consider also the (filled) square with vertices .
Theorem 1 (Characterization of flux values).
.
The proof is given by two separate propositions, each showing one inclusion. Much of the technical work here relates to the description of maximal height functions (given a base value at a base point).
In Chapter Domino Tilings of the Torus, we discuss how flipconnectedness extends to the torus. Flips preserve flux values, so of course the situation must be unlike that of Section 1. It turns out that for flux values in the interior of , its tilings are flipconnected, but for flux values in the boundary, none of its tilings admit any flips: they are flipisolated!
In order to show that, Section 8 is devoted to understanding tilings that do not admit flips, and contains our second theorem. It is a fairly independent section, requiring only that the reader be familiar with (maximal) height functions and flips; see Sections 1 and 7.
Theorem 2 (Characterization of tilings of the infinite square lattice).
Let be a tiling of . Then exactly one of the following applies:

admits a flip;

consists entirely of parallel, doublyinfinite domino staircases;

is a windmill tiling.
The proof (and theory leading up to it) delves into properties of domino staircases and staircase edgepaths.
Tilings of the torus can be seen as periodic tilings of . Section 9 combines this observation with Theorem 2 to obtain relations between the shape of a tiling and its flux. The final result is the above description of flipconnectedness on the torus. For a survey of flipconnectedness on more general surfaces, see Saldanha, Tomei, Casarin and Romualdo [saldanha1995].
In Chapter Domino Tilings of the Torus, we go into detail about the construction of a Kasteleyn matrix for the torus. Some of its entries are monomials in , or Laurent monomials in , so its determinant is a Laurent polynomial in . We show that each monomial in counts tilings with a flux given by the exponents of . Later, Chapter Domino Tilings of the Torus will consider these variables as complex numbers on the unit circle. Moreover, Section 10 examines the structure of tilings with flux in the boundary of , primarily through a move called stairflip, that exchanges a doublyinfinite domino staircase by the only other one.
Chapter Domino Tilings of the Torus elaborates on how the signs of monomials in are assigned. The main tool here are cycles and cycle flips. Cycles are obtained by representing two tilings simultaneously, and cycle flips use them to go from one tiling to the other.
In the end of Section 13, we exhibit an oddoneout pattern for signs over , and use it to show that the total number of tilings can be given as a linear combination of where each coefficient is either or .
Chapter Domino Tilings of the Torus revisits techniques used in Section 3 and refines them for the calculation of Kasteleyn determinants of the torus. In Section 14, we examine the case of , for which we can compute all eigenvalues. Section 15 interprets as linear maps on spaces of quasiperiodic functions, allowing us to exhibit bases for which they are diagonal. Studying the change of basis, we are able to relate the determinant of the original matrix to that of its diagonal version.
Finally, Section 16 makes explicit calculations on these determinants and investigates the effects of scaling uniformly. This leads to our third and last theorem, which relates the Laurent polynomials (from Kasteleyn determinants) for and by a simple product formula.
Let be defined by , where is the diagonal Kasteleyn matrix for and .
Theorem 3.
For any positive integer and reals
Intuitively, Theorem 3 says can be obtained from determinants of by considering all th roots of and of . Product formulas of this kind have been encountered by Saldanha and Tomei [saldanha2003tilings] in their study of quadriculated annuli.
Capítulo \thechapter Definitions and Notation
This will be a short chapter detailing definitions and conventions used in the dissertation.
The imaginary unit will be denoted by the boldface .
A lattice is a subgroup of that is isomorphic to and spans (as a real vector space). An equivalent description is that a lattice is the (additive) group of all integer linear combinations of a basis of ; in this case, we say is generated by . Notice different bases may generate the same lattice.
The dual lattice of is , the set of homomorphisms from to . Observe that, under addition, is a group. Moreover, we may identify an element with a unique via (for all ). This allows us to see as an additive subgroup of , so that is itself a lattice. Under this representation, it is easy to see that . We will generally not make a distinction between and .
Given a basis of , its dual basis is , where (). Geometrically, this means is perpendicular to and its length is determined by the equality . It is a straightforward exercise to check that generates , and that . We can also make explicit calculations; let and . Then
Notice that because is a basis of , is always nonzero, so the dual basis is welldefined.
A fundamental domain for a lattice is a set such that, for all , the affine lattice intersects exactly once. Another way to think of this is as follows: acts on by translation, so the orbit of any under (that is, the set of images of under ) is the affine lattice . Hence, contains exactly one point from each orbit: it is a visual realization of the representatives of each orbit. It is easily seen that is partitioned by the sets .
For a lattice generated by , the fundamental domain is usually the parallelogram , but we will generally prefer other kinds of fundamental domain (discussed in Section 5).
Consider the infinite square lattice . A quadriculated region is a union of (closed, filled) unit squares with vertices in . We say two squares are adjacent if they share an edge. A domino is a union of two adjacent unit squares, that is, a rectangle with vertices in . A (domino) tiling of a quadriculated region is a collection of dominoes on with pairwise disjoint interiors and such that every unit square of belongs to a domino in .
A torus is a quotient ; we may represent it by a fundamental domain of whose boundary has appropriate identifications. If the fundamental domain is chosen to be a quadriculated region, we say the torus is a quadriculated torus. A tiling of a quadriculated torus is much like that of its fundamental domain, except dominoes account for boundary identifications. Alternatively, a tiling of a quadriculated torus is an periodic tiling of the infinite square lattice.
Capítulo \thechapter Domino tilings on the plane
Let be a finite, simplyconnected, quadriculated planar region. A domino is a 21 rectangle made of two unit squares. Is it possible to tile entirely using only domino pieces? In how many ways can this be done?
For instance, if has an odd number of squares, then there is no domino tiling of . If is the 23 rectangle below…
…then there are exactly three distinct domino tilings of :
The first observation is this problem can be converted to a dual problem on graph theory. This conversion associates to the region a graph (’s dual graph) obtained by substituting each square of by a vertex and joining neighboring vertices by an edge (horizontally and vertically, but not diagonally). On a domino tiling level, each domino corresponds to an edge on : the edge joining the two vertices whose associated squares that are tiled by that domino.
For instance, the region and the graph in Figure 3 are dual. Likewise, the domino tiling of and the subgraph in Figure 4 are dual.
In this context, a question on domino tilings of can be translated naturally into a question on the matchings of . A matching of a graph is a set of edges on with no common vertex. If two vertices on are joined by an edge of , we say matches those vertices. A perfect matching of a graph is a matching of that matches all vertices on .
Now we may translate the opening questions: ‘Is it possible to tile by dominoes?’ becomes ‘Is there a perfect matching of the dual graph ?’; and ‘In how many ways can this be done?’ becomes ‘How many perfect matchings does the dual graph have?’. Henceforth, unless explicitly stated, we shall use matchings when referring to perfect matchings. Nonperfect matchings do not interest us in this study.
The second observation is that these constructions lend themselves naturally to the concept of bipartite graphs. A graph is bipartite if its vertices can be separated into two disjoint sets and so that every edge on joins a vertex in to a vertex in . In this case, the sets and are called a bipartition of . With this in mind, we may return to our initial problem and consider a prescribed ‘bipartition’ on : we assign the label ‘black’ to an initial square, then assign the label ‘white’ to its neighbors, and so on in alternating fashion. Naturally, the vertices of the dual graph inherit the labels.
At this point, notice every domino in a tiling of must be made of a single black square and a single white square. Hence, a necessary condition for to admit a domino tiling is that the number of black squares and the number of white squares be equal. Observe, however, that it is not sufficient.
We point out that we will generally think of as embedded on the region , with each vertex lying on the center of its corresponding square and each edge a straight line.
1 Flips and height functions
We now introduce the concept of flips. To that end, notice a 22 square can be tiled by two dominoes in exactly two ways: by using both dominoes vertically, or by using both dominoes horizontally.
Consider two adjacent parallel dominoes forming a 22 square. A flip of these two dominoes consists in substituting the domino tiling of the square they form by the only other domino tiling of that same square. Naturally, the concept of flip is transferred to the graph treatment of the problem.
Of course, given a domino tiling of a planar region , the execution of a flip takes us to a new domino tiling of . Following this train of thought, a natural question might be whether two given domino tilings of can be joined by a sequence of flips. To answer this question, we will investigate the height function of a domino tiling of .
We highlight the distinction between an edge on a graph and an edge on a quadriculated region : the latter refers to an edge on the boundary of a square on . Similarly, an edge on a domino tiling of is an edge on (of a square, not of a domino) that does not cross a domino (it has not been ‘erased’ to produce said domino).
We choose once and for all the clockwise orientation for black squares; the other orientation is assigned to white squares. This choice induces an orientation on each edge on . Notice it is consistent: along an edge where two squares meet, each square will have a different orientation and thus the orientations induced on the edge will agree.
Now, choose a base vertex on and assign an integer value to it; we will always choose a base vertex in the boundary of the region and we will always assign the value to it. This is the value takes on . We now propagate that value across all vertices of as follows. For each vertex joined to by an edge on , that edge may point from to or from to , depending on its orientation as defined above. In the first case, is assigned the integer value ; otherwise, it is assigned the integer value .
By connectivity, this process defines the height function on each vertex of , but it may not be clear whether or not the definition is consistent. It’s easy to verify consistency on a single domino, as the image below shows.
Consistency for a general simplyconnected planar region can be proved as follows: starting from a vertex on which is welldefined (for instance, the base vertex ), suppose we wish to check consistency on another vertex, say . Consider then two different edgepaths and on joining those vertices; these paths can be seen as the boundary of a region tiled by dominos. The area of that region is thus welldefined. Now, incrementally deform onto , with each step producing a new region with less area than the previous one through the removal of a domino. Here, consistency on a single domino ensures each step is consistent with the previous one. Finally, the simplyconnectedness of guarantees this process can fully deform onto .
We provide a simple example of this process below.
With these conventions, given a domino tiling and a base vertex of a blackandwhite quadriculated region , the height function of is welldefined. An example of height function can be seen in the following image; the marked vertex is the base vertex.
This provides a constructive definition of height functions, but we highlight now some of their properties.
Proposition 1.1.
Let be a blackandwhite quadriculated region. Fix a base vertex (independent of choice of tiling). Then (1) the values a height function takes on and (2) the mod 4 values a height function takes on all of are all independent of choice of tiling.
Demonstração.
Remember that, regardless of the choice of tiling , an edge on is an edge on . Since we have already proved consistency, (1) is automatic.
For (2), let and be vertices on joined by an edge . Observe that the orientation of depends only on the region and not on choice of tiling; assume then that is oriented from to . The constructive definition implies a change in height function along occurs in one of the following ways:

If is on the tiling , then .

If is not on the tiling , then .
Notice that in both cases has the same mod 4 value. The same occurs when is oriented from to . By connectivity, we are done. ∎
Proposition 1.1 allows us to fully characterize height functions of tilings of a region .
Proposition 1.2 (Characterization of height functions).
Let be a blackandwhite quadriculated region. Fix a base vertex . Then an integer function on the vertices of is a height function (of a tiling of ) if and only if satisfies the following properties:

has the prescribed values on .

has the prescribed mod 4 values on all of .

changes by at most 3 along an edge on .
Demonstração.
Proposition 1.1 and its proof guarantee that any height function satisfies the listed properties. We will now show that if an integer function on the vertices of satisfies those properties, it is the height function of a tiling on . To that end, we will construct a tiling that realizes one such function .
On , whenever two vertices joined by an edge have values that differ by 3, erase that edge (thus producing a domino). We claim the result is a domino tiling on . Indeed, properties (2) and (3) ensure each square on will have exactly one of its sides erased. Furthermore, by (1) that side will never occur on . It’s easy to see this yields a domino tiling of ; furthermore, by construction this tiling’s height function is . ∎
From now on, for any blackandwhite quadriculated region , assume the base vertex is fixed independently of choice of tiling.
Another interesting and perhaps less obvious property of height functions is that the minimum of two height functions is itself a height function.
Proposition 1.3.
Let be a blackandwhite quadriculated region and , be two domino tilings of with corresponding height functions , . Then is a height function on .
Demonstração.
Indeed, by Proposition 1.2, it suffices to show that changes by at most along an edge on . This is trivially verified on vertices and joined by an edge whenever or on both and . Suppose this is not the case; furthermore, suppose without loss of generality , and that the edge joining them points from to .
The edge’s orientation implies if the edge is on and otherwise (). Since , the only possibility that realizes is the edge being on and not on , so that and . Now, because and mod 4 values are prescribed, the difference must be for some positive integer , so that .
Finally, can now be rewritten as , or simply . This is a contradiction, implying only the cases when or on both and can occur. ∎
Corollary 1.4 (Minimal height function).
Let be a blackandwhite quadriculated region. If can be tiled by dominoes, then there is a minimal height function.
Along a 22 square tiled by dominoes, it’s easy to verify that height function values are distributed so that the center vertex is a local maximum or minimum. Furthermore, applying a flip changes a local maximum vertex to a local minimum vertex, and viceversa, leaving other values unchanged. Figure 11 illustrates this phenomenon.
Together with Corollary 1.4, an application of this technique provides the following result.
Proposition 1.5.
Let be a blackandwhite quadriculated region with minimal height function . Let be a height function associated to the domino tiling of . Then there is a flip on that produces a height function with on one vertex of .
Demonstração.
Consider the difference . By Proposition 1.2, it is 0 along the boundary and takes nonnegative values on . Let be the set of vertices of on which is maximum, and choose a vertex that maximizes . Notice by hypothesis is nonempty, and does not intersect . We assert that is a local maximum of .
Suppose were not a local maximum of , that is, suppose there were a vertex joined to by an edge so that . There are two cases:

is on and points from to , so that .

is not on and points from to , so that .
Remember edge orientation does not depend on choice of tiling (and thus does not depend on the height function considered).
In case (1), if is on the associated minimal tiling , and otherwise. Neither can occur: the first contradicts maximizing (since ), and the latter contradicts maximizing (since ).
Case (2) is similar: if is on , and otherwise. The first contradicts maximizing , and the latter contradicts maximizing .
Whatever the situation, we derive a contradiction, implying must indeed be a local maximum. Since is not on , we can perform a flip round . This makes it a local minimum while preserving the values takes on all other vertices of and completes the proof. ∎
Because the situation is finite, Proposition 1.5 essentially tells us any tiling of a region can be taken by a sequence of flips to the tiling that minimizes height functions over tilings of . A simple but important corollary follows.
Corollary 1.6 (Flipconnectedness).
Let be a blackandwhite simplyconnected quadriculated region tileable by dominoes. Then any two distinct tilings of can be joined by a sequence of flips.
2 Kasteleyn matrices
A Kasteleyn matrix ‘encodes’ a quadriculated blackandwhite region in matrix form, and its construction is similar to that of adjacency matrices.
Given one such region , we can obtain an adjacency matrix of from its dual graph as follows: enumerate each black vertex (starting from 1), and do the same to white vertices. Then if the th black vertex and th white vertex are joined by an edge, and 0 otherwise.
Consider now an adjacency matrix and the combinatorial expansion of its determinant:
(1) 
In the expansion above, each nonzero term of the form can be seen as corresponding to a matching of . In fact, the term is nonzero if and only if each factor in the product is 1, in which case the th black vertex is joined by an edge to the white vertex. Since is a permutation on , the collection of these edges is by construction a set of edges on in which each vertex of features exactly once. The observation follows.
Of course, the correspondence goes both ways. This means that, except for sgn(), det() counts the number of matchings of (and thus also the domino tilings of ). How do we get past the sign?
The obvious way would be to consider ther permanent of
but permanents lack a number of interesting properties when compared to determinants, and are also much more costly to compute.
The answer is precisely the Kasteleyn matrix : an altered adjacency matrix in which some entries are replaced by . Its construction is similar to the ordinary adjacency matrix, except some edges on are assigned the value rather than . This distribution of minus signs can be done in many ways, but the following observation explains the general principle behind it: a flip on a matching of always changes the sign of the corresponding permutation in (1). This is because, on a permutation level, applying a flip amounts to multiplying the original permutation by a cycle of length 2.
With this in mind, the distribution of minus signs over edges on is made so that the sign change in a permutation caused by a flip is always counterbalanced by a sign change on the corresponding product of entries of . Such a distribution ensures that applying a flip does not change the ‘total’ sign of the term
in (1). And since we’ve shown that any two distinct domino tilings of (and thus matchings of ) can be joined by a sequence flips, this means the sum in (1) is carried over identically signed numbers. In other words, for a Kasteleyn matrix of a region , is the number of domino tilings of .
An easy, convenient way of distributing minus signs over edges on is assigning them to all horizontal edges in alternating lines (say, all odd lines, or all even lines). This way, a square in the dual graph will always contain exactly one negative edge (either the topmost or the bottommost horizontal line), so that a flip always will always produce a sign change on the product of entries of .
We highlight that in his original paper [kasteleyn1961statistics], flipconnectedness (or more generally, flips) was not a part of Kasteleyn’s exposition. His methods were combinatorial but he employed Pfaffians.
Below, we show an example of construction of a Kasteleyn matrix. In the corresponding dual graph, negative edges are red and dashed.
3 A classical result: domino tilings of the rectangle
We will end this chapter by using our methods to provide a classical result: the counting of domino tilings of an blackandwhite rectangular region, . Of course, if both and are odd, that number is 0; we assume then that is even.
Let be ’s dual graph with minus signs assigned to all horizontal edges in even lines, from which we obtain the corresponding Kasteleyn matrix . Rather than compute the determinant of , we will consider the matrix ; it’s clear that .
can be seen as a double adjacency matrix of , acting as a linear map on the space of formal linear combinations of vertices. It takes a vertex to the sum of vertices that are joined to via an edgepath of length two on the graph. Notice edge sign and vertex multiplicity (when a vertex can be reached from via two distinct edgepaths) are taken into account.
Because edgepaths considered have length two, takes white vertices to white vertices and black vertices to black vertices. Another way of thinking this is as follows: when interpreting the Kasteleyn matrix as a linear map (like above), our general construction of the Kasteleyn matrix implies it goes from the space of white vertices to the space of black vertices . Of course, this also means . It then becomes clear by the definition of that it is a colorpreserving map. This essentially means acts independently on and .
Consider the grid below, so that each vertex of is identified by a double index .
In the obvious notation, vertices of at least two units away from the boundary satisfy . The coefficient in comes from moving forward then backwards in each cardinal direction; notice that a negative edge traversed this way will account for two minus signs, so the end result is always positive. Vertices of the form do not feature because each of them can be reached via exactly two distinct edgepaths with necessarily opposite signs.
The formula can be extended to all vertices of as follows. Put if it is immediately outside the boundary of ; then, for each line of zerovertices, reflect through that line and set a vertex obtained this way to be minus the vertex from which it was reflected. We then repeat this process, so that in the end will be defined for all .
Figure 15 is a visual representation of this extension; in it, each gray square is a zero vertex. More generally, one such extension can be succinctly represented by the relations:
(2) 
With this, the formula for now holds not only on all of , but also on all of . Notice the space on which acts is still dimensional, since coordinates on the original vertices of propagate to all vertices of via the relations above. We will now compute .
We will always refer to the imaginary unit by the boldface . Let and . For all with and , let
Notice