Domino Tilings of the Torus

# Domino Tilings of the Torus

Fillipo de Souza Lima Impellizieri
###### 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 .

\keydomino \keytiling \keytorus \keylattice \keyflux \keyflip \keyheight function \keyKasteleyn matrix
\AfterBegin

## 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 group-theoretic 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: integer-valued functions that encode a tiling of a region. With them, he presented a simple algorithm that verifies the domino-tileability of simply-connected 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 two-dimensional 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 three-dimensional 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 cross-over 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:

 FluxTilings∣∣ ∣ ∣∣(0,0)132∣∣ ∣ ∣∣(0,±1),(±1,0)32∣∣ ∣ ∣∣(±1,±1)2∣∣ ∣ ∣∣(0,±2),(±2,0)1

For a total of 272 tilings. Observe the proportion of total tilings by flux:

 FluxProportion∣∣ ∣ ∣∣(0,0)0.48529∣∣ ∣ ∣∣(0,±1),(±1,0)0.11765∣∣ ∣ ∣∣(±1,±1)0.00735∣∣ ∣ ∣∣(0,±2),(±2,0)0.00368

Now we repeat the process for different square tori:

 Flux4×46×610×1016×16∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,0)0.485290.489890.494360.49564∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,±1),(±1,0)0.117650.110820.105750.10411∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(±1,±1)0.007350.014160.018200.02053∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,±2),(±2,0)0.003680.002530.001410.00109

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

 2⋅Γ(34)2√(6+4√2)⋅π⋅exp(−12(i2+j2))

The formula for the rather curious constant can be derived from theta-function identities; see Yi [yi2004theta]. For comparison, we provide the previous table, together with the limit value given by the formula above:

 Flux4×46×610×1016×16Limit∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,0)0.485290.489890.494360.495640.49629∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,±1),(±1,0)0.117650.110820.105750.104110.10317∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(±1,±1)0.007350.014160.018200.020530.02145∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣(0,±2),(±2,0)0.003680.002530.001410.001090.00093

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, simply-connected 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 black-and-white 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 flip-connectedness 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 flip-connectedness 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 flip-connected, but for flux values in the boundary, none of its tilings admit any flips: they are flip-isolated!

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:

2. consists entirely of parallel, doubly-infinite domino staircases;

3. is a windmill tiling.

The proof (and theory leading up to it) delves into properties of domino staircases and staircase edge-paths.

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 flip-connectedness on the torus. For a survey of flip-connectedness 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 doubly-infinite 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 odd-one-out 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 well-defined.

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, simply-connected, 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. Non-perfect 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 simply-connected planar region can be proved as follows: starting from a vertex on which is well-defined (for instance, the base vertex ), suppose we wish to check consistency on another vertex, say . Consider then two different edge-paths 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 well-defined. 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 simply-connectedness 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 black-and-white quadriculated region , the height function of is well-defined. 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 black-and-white 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 black-and-white 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:

1. has the prescribed values on .

2. has the prescribed mod 4 values on all of .

3. 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 black-and-white 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 black-and-white 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 black-and-white 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 vice-versa, 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 black-and-white 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 non-empty, 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:

1. is on and points from to , so that .

2. 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 (Flip-connectedness).

Let be a black-and-white simply-connected 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 black-and-white 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:

 det(A)=∑σ∈Snsgn(σ)n∏i=1Ai,σ(i) (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

 perm(A)=∑σ∈Snn∏i=1Ai,σ(i),

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

 sgn(σ)n∏i=1Ki,σ(i)

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], flip-connectedness (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 black-and-white 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 edge-path of length two on the graph. Notice edge sign and vertex multiplicity (when a vertex can be reached from via two distinct edge-paths) are taken into account.

Because edge-paths 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 color-preserving 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 edge-paths 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 zero-vertices, 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:

 ⎧⎪⎨⎪⎩v0,j=vm+1,j=vi,0=vi,n+1=0v−i,j=vi,−j=−vi,jvi+2m+2,j=vi,j+2n+2=vi,j (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

 v(k,l)=∑i,j4 sin(i⋅k⋅πm+1)sin(j⋅l⋅πn+1)⋅vi,j=∑i,j(ζik1−ζ−ik1)(ζjl2−ζ−jl2)⋅vi,j

Notice