Dimers on Rail Yard Graphs

Dimers on Rail Yard Graphs

Cédric Boutillier Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris cedric.boutillier@upmc.fr Jérémie Bouttier Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette and Département de Mathématiques et Applications, École normale supérieure, 45 rue d’Ulm, F-75231 Paris Cedex 05 jeremie.bouttier@cea.fr Guillaume Chapuy LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13 guillaume.chapuy@liafa.univ-paris-diderot.fr Sylvie Corteel LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13 corteel@liafa.univ-paris-diderot.fr  and  Sanjay Ramassamy Mathematics Departement, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912 sanjay_ramassamy@brown.edu
July 9, 2019
Abstract.

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson-fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions.

Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick’s theorem.

Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimer model. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.

We acknowledge financial support from the Ville de Paris via Projet Émergences “Combinatoire à Paris” (JB, GC, SC), from the Monahan Foundation via a Monahan Science Fellowship (SR) and from the Agence Nationale de la Recherche via the grants ANR-08-JCJC-0011 “IComb” (SC), ANR-10-BLAN-0123 “MAC2” (CB), ANR-12-JS02-0001 “Cartaplus” (JB, GC), ANR-14-CE25-0014 “GRAAL” (JB). JB acknowledges the hospitality of LIAFA. SR acknowledges the hospitality and financial support of the Erwin Schödinger Institute in Vienna, where part of this work was done.

1. Introduction

The two-dimensional dimer model is arguably the most studied exactly solvable model in statistical mechanics (note that it encompasses, in a sense, the equally well-known two-dimensional Ising model), see for instance [Kas67, Chapter 5] for a review of the seminal works of Kasteleyn, Temperley and Fisher, and the introduction of [All14] for a nice survey of the more recent literature. Dimer configurations are also known as perfect matchings in combinatorics and theoretical computer science. Actually, perhaps the oldest exact solution of a 2D dimer model is MacMahon’s enumeration of plane partitions [Mac04], as these were later identified with lozenge tilings, or alternatively dimer configurations on the hexagonal lattice.

Kasteleyn’s method allows to reduce the problem of computing the partition function (and the correlation functions) of the dimer model on any finite weighted planar graph (assuming that the dimers interact only through their hard-core repulsion) to the evaluation of a determinant (or Pfaffian) whose size is linear in the number of vertices of the graph. Under the usual assumption that the graph is periodic in two directions, one can then evaluate this determinant and take the thermodynamic limit to obtain the free energy, study phase transitions, etc. In this paper, we consider a dimer model on a new family of graphs, called rail yard graphs, which are periodic in one direction but not in the other.

One of our motivations is that the rail yard graph dimer model encompasses both the plane partitions mentioned above and another celebrated model, namely domino tilings of the Aztec diamond [EKLP92a, EKLP92b] (corresponding to, roughly speaking, dimer configurations on the portion of the square lattice fitting into a large square tilted by ). What relates these two models is that they can be seen as Schur processes [OR03], that is to say random sequences of integer partitions whose transition probabilities are given by Schur functions. If the relation between plane partitions and Schur processes was explicited by Okounkov and Reshetikhin, the case of the Aztec diamond appears implicitly in [Joh02] and has, to the best of our knowledge, remained in such implicit form until [BCC14], of which this paper is a continuation (see below). The interest of making the connection between dimer models and Schur processes explicit is that it allows to use an operator formalism coming from the boson-fermion correspondence (see the references given at the beginning of Section 3) which is both powerful and intuitive, as the operators are nothing but transfer matrices or observables satisfying some particularly simple commutation relations. Furthermore it allows us to say that the RYG dimer model forms another situation, besides the 2D Ising model [Dub11], where “bosonization” works at an exact discrete level. The rail yard graph dimer model corresponds essentially to the most general Schur process with nonnegative transition probabilities.

Before describing our work in more detail, let us further discuss some history and background behind it. Bender and Knuth [BK72] made the link between plane partitions and the Robinson-Schensted-Knuth correspondence, see also [Sta99, Chapter 7]. Okounkov [Oko01, Oko02] used the boson-fermion correspondence to define the so-called Schur measure over integer partitions, and study its correlation functions. The Schur process [OR03, OR07] is a time-dependent version of this measure, that can also be viewed as a system of particles with certain dynamics. It contains as a special case a generalization of plane partitions, namely plane partitions with an evolving “back wall”. Numerous papers followed on this subject [Bor07, BF14, Bor11, BMRT12, BBB14, BF15] and on its extension to the Hall-Littlewood and Macdonald cases [Vul07, FW09, Vul09, Oka10, CSV11, BC14].

In [BCC14], three authors of the present paper introduced a general class of domino tilings called steep tilings, encompassing both tilings of the Aztec diamond and the so-called pyramid partitions [Ken05, You09]. It was shown in [BCC14] that steep tilings also correspond to Schur processes and, using the vertex operator formalism, their partition functions (of the “hook formula” type) were computed for a variety of boundary conditions. Since both (generalized) plane partitions and steep tilings are special instances of the Schur process, it is then natural to ask if there is a more general model of tilings or dimer coverings that would reformulate the Schur process in full generality, at least when the number of underlying parameters is finite. Such a model was sketched in [BCC14, Section 7], that can be viewed as a preliminary attempt at what we reach in the present paper.

The rail yard graphs (RYG) that we introduce here are infinite bipartite plane graphs, obtained by the “concatenation” of column-shaped elementary graphs, and come with a family of admissible dimer coverings. The RYG dimer model is then a probability measure over such coverings. The elementary graphs can be of four types that correspond to the four possible types of “atomic” transitions in the Schur process. For the special families of RYG that correspond to the special families of Schur processes considered in [OR03, BCC14], we recover generalized plane partitions and steep tilings, respectively. As we hope will be apparent in this paper, RYG provide a nice and natural formulation of the Schur process in terms of dimers, in a well-adapted system of coordinates, much simpler than the one from [BCC14, Section 7]. Having shown the correspondence between rail yard graphs and Schur process, we can then apply the same classical tools as in [OR03] to get explicitly the partition functions in a nice (hook-type) product form. Even more, we can interpret these partition functions in terms of a combinatorial parameter related to the flip operation on coverings.

Beyond the partition function, we compute all the dimer correlation functions, which requires the introduction of suitably defined observables (or constrained transfer matrices) that enable us to localize, in the algebraic setting, a given set of dimers. To prevent any confusion, let us note that the particle correlations computed in [OR03] for the Schur process, when translated in terms of RYG, give only a special case of this result. Indeed, as we will see, there are three kinds of dimers in a RYG, and particles correspond to one of the three kinds (so in our setting the correlations results of [OR03] only describe correlations between dimers of the first kind). Once the observables are constructed, we use classical fermionic tools such as Wick’s formula to evaluate the correlation functions in an explicit determinantal form. We also make the connection with the general Kasteleyn theory: it is a general fact that correlations between dimers on plane bipartite graphs have a determinantal form, underlaid by an inverse of the so-called Kasteleyn matrix of the model. For RYG, we show that the determinantal form we obtain by our approach indeed gives an inverse Kasteleyn matrix, as was remarked in [OR07] for the case of skew plane partitions, see also [BF14, Section 5]. Our approach generalizes both this case and that of the Aztec diamond (for the so-called weighting), treated previously in [CY14] by a very tricky and somehow mysterious calculation. As further applications concerning the Aztec diamond, we rederive the so-called edge-probability generating function and biased creation rate, and the arctic circle theorem using the general saddle-point techniques of [OR03].

We now present the structure of the paper. Section 2 is devoted to the basic definitions (rail yard graphs in Subsection 2.1, their dimer coverings in Subsection 2.2, flips in Subsection 2.3) and to the statement of our main results, namely the expression for the partition function (Subsection 2.4) and for the dimer correlation functions (Subsection 2.5). Section 3 introduces bosonic operators (Subsection 3.1) that act as transfer matrices in the RYG dimer model (Subsection 3.2), allowing to compute efficiently the partition function (Subsection 3.3). Section 4 considers fermionic operators (Subsection 4.1) that play the role of observables in the RYG dimer model (Subsection 4.2). Rewriting the correlation functions in the “Heisenberg picture” (Subsection 4.3), we derive their expression in the form of a determinant (Subsection 4.4), before making the connection with Kasteleyn’s theory (Subsection 4.5). Section 5 discusses the previously known cases: plane partitions and lozenge tilings (Subsection 5.1) and steep domino tilings (Subsection 5.2). In Section 6 we address the specific case of the Aztec diamond, for which we provide new derivations of the edge-probability generating function and biased creation rate (Subsection 6.1), of the inverse Kasteleyn matrix (Subsection 6.2) and of the arctic circle theorem (Subsection 6.3). Concluding remarks are gathered in Section 7. Some auxiliary material is given in the appendix: a combinatorial proof of the bosonic-fermionic commutation relations (Appendix A) and a rederivation of Wick’s formula (Appendix B).

After the completion of this work, Alexei Borodin and Senya Shlosman informed us (by private communication) that graphs similar to those described in Section 2.1 were suggested to them by Richard Kenyon, providing a dimer interpretation of the -paths considered in [BS10].

2. Basic definitions and main results

2.1. Rail yard graphs

We start by defining the underlying graph of our dimer model. We fix two integers such that , and denote by the set of integers between and . We then consider two binary sequences indexed by the elements of :

  • the LR sequence ,

  • the sign sequence .

The rail yard graph associated with the integers and , the LR sequence and the sign sequence , and denoted by , is the bipartite plane graph defined as follows. Its vertex set is , and we say that a vertex is even (resp. odd) if its abscissa is an even (resp. odd) integer. Each even vertex , , is then incident to three edges: two horizontal edges connecting it to the odd vertices and , and one diagonal edge connecting it to

  • the odd vertex if and ,

  • the odd vertex if and ,

  • the odd vertex if and ,

  • the odd vertex if and .

Hopefully, this explains our motivations for using the symbols and . Drawing the edges straight, the graph is indeed bipartite and plane by construction. For an edge, we write to mean that is the even endpoint of , and its odd endpoint. For a vertex, we will denote by its abscissa, and by its ordinate.

Figure 1. The rail yard graph associated with the LR sequence and the sign sequence (with , ). It is infinite and periodic in the vertical direction, but finite in the horizontal direction.

Figure 1 displays the rail yard graph associated with the LR sequence and the sign sequence (with , ). Observe that a rail yard graph is infinite and -periodic in the vertical direction. When , the LR and sign sequences both consist of a single element, and the corresponding rail yard graph, which is said elementary, is of one of four possible types, see Figure 2. Given two rail yard graphs and such that , we define their concatenation by taking the union of their vertex and edge sets. It is nothing but the rail yard graph where and denote the concatenations of the LR and sign sequences. Clearly, a general rail yard graph is obtained by concatenating elementary ones.

Figure 2. The four elementary rail yard graphs , , , and (up to horizontal translation)

The left boundary (resp. right boundary) of a rail yard graph consists of all odd vertices with abscissa (resp. ). Vertices which do not belong to the boundaries are said inner. When drawn in the plane, the graph delimits some faces, and the bounded ones are called inner faces. Note that inner faces may be incident to , or edges. Finally, observe that our definition works equally well if we take and/or , thus considering infinite LR and sign sequences. In that case, the rail yard graph “fills” either the whole plane or a half-plane, boundaries being sent to infinity.

2.2. Admissible and pure dimer coverings

We now turn to the characterization of the configurations of our dimer model. Given a rail yard graph with finite, an admissible dimer covering is a partial matching of this graph such that:

  • each inner vertex is covered (i.e. matched),

  • there exists an integer such that: any left boundary vertex is covered for and uncovered for , any right boundary vertex is covered for and uncovered for ,

  • only a finite number of diagonal edges are covered.

A pure dimer covering is an admissible dimer covering for which the second property above holds for : in other words the uncovered vertices are precisely the left boundary vertices with negative ordinate and the right boundary vertices with positive ordinate (see Figure 4). The fundamental dimer covering is the pure dimer covering where no diagonal edge is covered (it is not difficult to check its existence and uniqueness e.g. by induction on ). Observe that any admissible dimer covering coincides with the fundamental dimer covering outside a finite region. An elementary dimer covering is an admissible dimer covering of an elementary rail yard graph (see Figure 3).

Figure 3. Some elementary dimer coverings. The underlying elementary rail yard graph has type in the first two cases, and , , and , in the three others, from left to right. Outside of the displayed region, the configuration is identical towards the top (resp. bottom) on each horizontal level to what it is on the topmost (resp. bottommost) displayed level.
Figure 4. A pure dimer covering of the rail yard graph of Figure 1.

Similarly to rail yard graphs, admissible dimer coverings behave nicely with respect to concatenation. More precisely, consider two rail yard graphs and which are concatenable (i.e. ) and let be their concatenation. Let and be admissible dimer coverings of respectively and : we say that and are compatible if, for each , the vertex is covered in if and only if it is not covered in . In that case, by taking the union of and , we obtain an admissible dimer covering of , which we denote by . Conversely, any admissible dimer covering can be decomposed as the concatenation of elementary dimer coverings which are sequentially compatible.

It is also interesting to consider the limiting cases and/or , which requires a slight adaptation of our definitions. An admissible (resp. a pure, resp. the fundamental) dimer covering is then a matching such that each inner vertex is covered, and such that there exists finite integers such that :

  • inside the strip , we see an admissible (resp. a pure, resp. the fundamental) dimer covering in the previous sense,

  • outside this strip, all covered edges are horizontal.

(Note that this definition works in all situations: it coincides with the previous one when are both finite.)

Our motivation for considering pure dimer coverings of rail yard graphs is that we recover several well-known dimer models as specializations. For instance, taking , a LR sequence of the form and a sign sequence of the form , the corresponding pure dimer configurations are in bijection with domino tilings of the Aztec diamond of size . We also recover plane partitions and so-called pyramid partitions, which requires taking and : plane partitions are obtained by taking a constant LR sequence and a sign sequence of the form , while pyramid partitions are obtained by taking an alternating LR sequence () and the same sign sequence. We will discuss these specializations in greater detail in Section 5.

2.3. Flips

We now define a local transformation on admissible coverings called the flip. Let be a rail yard graph, be an admissible covering of , and let be an inner face of . If exactly half of the edges bordering belong to , then removing these edges from and replacing them by the other edges bordering gives another admissible covering of . The operation that replaces by is called the flip of the face , see Figure 5.

Figure 5. Examples of flips. All these flips are positive when the transformation is made from left to right.

We say that the flip of an inner face is positive if after performing the flip, the edges of that belong to the covering are oriented from odd to even vertices in counterclockwise direction around . The flip is negative otherwise. For example, each flip displayed on Figure 5 is positive when performed from left to right. It follows from [Pro02, Theorem 2] that the positive flip relation endows the set of all pure coverings of a given rail yard graph with a distributive lattice structure. In particular, each rail yard graph has a unique minimal pure covering from which all other ones can be reached using positive flips only. The minimal covering is the only pure covering on which no negative flip is possible. Using this criterion one easily checks that the minimal covering coincides with the fundamental covering defined above. The flip distance between two coverings is the minimal number of flips needed to go from one to the other. When one of the two coverings is the fundamental one, the flip distance is realized by a sequence that uses positive flips only.

2.4. Enumeration

Our main enumerative result is an expression for the partition function of the RYG dimer model, which we now define. Consider a rail yard graph , and a sequence of formal variables , where possibly or . The weight of an admissible dimer covering of is then defined as

(1)

where is the number of diagonal dimers in column (i.e. the number of covered diagonal edges incident to an even vertex with abscissa ). This weight is well-defined since is finite by the definition of an admissible dimer covering. The partition function of the multivariate RYG dimer model, denoted , is then the sum of the weights of all pure dimer coverings of .

Theorem 1.

The partition function of the multivariate RYG dimer model reads

(2)

where

(3)
Remark 2.

The partition function is always a well-defined power series in the ’s: indeed, all but finitely many factors contribute a factor to the coefficient of a given monomial in (2).

An interesting specialization is the -RYG dimer model: given a formal variable , we attach to each configuration a weight with its flip distance to the fundamental one. As explained in Section 3.3 below, this can be achieved by taking, for all , if , and if , with an indeterminate. A caveat is that, when or is infinite, this specialization may be ill-defined since an infinite number of monomials in the ’s might specialize to the same monomial . A sufficient condition for the specialization to be well-defined is the following finiteness condition on the sign sequence:

  • if , then there exists finite such that for all ,

  • if , then there exists finite such that for all .

(This condition is essentially necessary, because any initial run of or final run of in the sign sequence does not contribute to the partition function, and can be removed without loss of generality: any pure dimer covering coincides with the fundamental dimer covering in the corresponding regions.)

Theorem 3.

Assuming that the finiteness condition holds, the partition function of the -RYG dimer model is

(4)
(5)

The proofs of Theorems 1 and 3 are given in Section 3.3.

Remark 4.

The product form (4) is strongly reminiscent of a hook-length formula, upon interpreting the sign sequence as describing the shape of a (possibly infinite) Young diagram, see Figure 6. The finiteness condition ensures that there are finitely many “hooks” of a given length, and hence that (4) is a well-defined formal power series in .

Figure 6. The interpretation of Theorem 3 as a hook-length formula. Displaying the sign sequence as a lattice path whose horizontal (resp. vertical) steps correspond to (resp. to ), one obtains a Young diagram whose boxes are indexed by pairs such that , , . The quantity is the “hook-length” of the box. In (4), each box gives rise to a multiplicative factor whose value or is determined by comparing the two terms and of the LR sequence .

2.5. Correlations

So far we have introduced the partition function of the RYG dimer model, which depends on a sequence of formal variables in general and on a single variable in the flip specialization. For a probabilistic or statistical physics interpretation, one shall rather consider the ’s or as nonnegative real numbers such that the sum of the weights over all pure dimer coverings is convergent. As apparent from Theorem 1, this is the case if and only if

(6)

and, when or is infinite,

(7)

In the -RYG dimer model, (6) is satisfied whenever , and (7) amounts to the finiteness condition defined before.

Assuming that the RYG dimer model is well-defined, that is to say (6) and (7) are satisfied, we may interpret as the probability of the pure dimer covering . For a finite set of edges, we denote by the probability that all the edges of are covered by a dimer. Our main probabilistic result is an explicit determinantal expression for , which requires to introduce some notations. For two integers, we set

(8)

and

(9)

Note that all the products in (8) are convergent by (7), hence is a meromorphic function on the whole complex plane, whose all zeros and poles are on the real axis.

For two vertices of such that is even and is odd, we set

(10)

where the contours must satisfy the following conditions: (i) should encircle and all the negative poles of , but not the positive ones; (ii) should encircle and all the positive zeros of , but not the negative ones; (iii) and should not intersect, and should surround if and only if . We shall check in Section 4.4 that the assumptions (6) and (7) imply that such contours always exist but, at this stage, let us mention their intuitive interpretation: is obtained by extracting the coefficient of in , when we treat each factor as a power series in , each factor as a power series in , each factor as a power series in , each factor as a power series in , and finally we expand if , or if .

We are now ready to state our theorem. Recall that, for an edge , and are assumed to be respectively the even and the odd endpoint of .

Theorem 5 (Dimer correlations).

Let be a finite set of edges of , with . Then, we have

(11)

with the number of horizontal edges in whose right endpoint is at an even abscissa, with the number of diagonal edges in in column , and defined as in (10).

The proof of Theorem 5 is given in Section 4, where we also prove that the infinite matrix , with rows indexed by even vertices and columns by odd vertices , is an inverse of the Kasteleyn matrix of the rail yard graph for a suitable Kasteleyn orientation (see Theorem 17). Applications will be discussed in Section 5.

3. Bosonic operators

The purpose of this section is to establish Theorems 1 and 3. This is done naturally by the transfer-matrix method, which here consists in decomposing the pure dimer coverings we want to enumerate into a sequence of compatible elementary dimer coverings. It turns out that the transfer matrices are isomorphic to certain operators arising in the so-called boson-fermion correspondence. For more details on this latter subject, we refer the reader to one of the many references available in the mathematical physics literature, for instance [Kac90, Chapter 14], [MJD00], [Oko01, Appendix A], [OR07], [Tin11] and [AZ13].

We start by giving the necessary reminders in Section 3.1, then make the connection with rail yard graphs in Section 3.2, and finally complete the proofs of Theorems 1 and 3 in Section 3.3.

3.1. Reminders

An integer partition, or partition for short, is a nonincreasing sequence of integers which vanishes eventually. The size of a partition is . We say that two partitions and are interlaced, and we write or , if we have

(12)

In the well-known pictorial representation in terms of Young diagrams, this means that the skew shape is a horizontal strip, see e.g. [Sta99, Chap. 7] for more precise definitions. To a partition we may associate its conjugate , whose Young diagram is the image of that of by a reflection along the main diagonal. In more explicit terms, we have . Note that the relation amounts to

(13)

The bosonic Fock space, denoted , is the infinite dimensional Hilbert space spanned by orthonormal basis vectors where runs over the set of integer partitions. Here we will use the bra-ket notation so that denotes the dual basis vector. For a formal or complex variable, we introduce the operators whose action on basis vectors reads

(14)

These operators are sometimes called (half-)vertex operators. Let us mention that, in the literature, is often denoted , see e.g. [Oko01], while is sometimes denoted [You10, BCC14]. Observe that we have

(15)

where denotes the empty partition. Note also that (resp. ) is the dual of (resp. ), and that (resp. ) is conjugated to (resp. ) via the involution of sending to .

Remark 6.

For , the product is clearly well-defined, because its coefficient between two states and involves only a finite sum. The same is true for when (observe that the “intermediate” partitions cannot get too large). Infinite sums arise when considering with , but its coefficients are power series in and , which are convergent for as apparent from the following proposition.

Proposition 7 (Commutation relations).

For , we have

(16)

while commutes with , and commutes with .

Proof.

See for instance [You10, Lemma 3.3] for an algebraic proof, and [BBB14, Section 3] for a bijective proof of (16). The celebrated Bender-Knuth involution [BK72, pp. 46-47] implies that commutes with for or . That commutes with is also well-known, but for completeness let us here sketch a short proof: for two partitions , one sees easily that the two sets and are nonempty if and only if is a skew shape containing no square. In that case, both sets have the same cardinality , where is the number of connected components of , and one easily constructs a bijection between them proving the wanted commutation relation. Another byproduct of this bijection is that

(17)

Given two symbols and (called respectively black and white marbles), a Maya diagram [MJD00] is an element of such that is eventually equal to for and to for . It then not difficult to check that the quantity

(18)

is a finite integer, and we call it the charge of . Let be the positions of in enumerated in decreasing order, and let : it is easily seen that is a partition and that the correspondence is one-to-one, the pair being called a charged partition. Observe that we may extend the involution to charged partitions by setting , and this corresponds on Maya diagrams to performing a reflection across and exchanging and : in other words, is sent to such that for all . By a slight abuse, we still denote by the Maya diagram of charge corresponding to the empty partition.

The fermionic Fock space, denoted , is the infinite dimensional Hilbert space spanned by orthonormal basis vectors where runs over the set of all Maya diagrams. For , let denote the subspace spanned by Maya diagrams of charge , so that . By the bijection between Maya diagrams and charged partitions, each may be canonically identified with . This defines the action of the bosonic operators and on , leaving each subspace invariant (by a slight abuse we keep the same notations for the operators acting on this larger space, and note that the commutations relations of Proposition 7 remain valid). We now end this section devoted to reminders, leaving the discussion of fermionic operators to Section 4.1.

3.2. Interpretation as transfer matrices for RYGs

The purpose of this section is to explain how the operators may be identified with dimer transfer matrices. The key observation is that Maya diagrams describe the boundary states in our model. More precisely, let us consider an admissible dimer covering of . We define the left boundary state of by setting, for all ,

(19)

It is a Maya diagram by the definition of an admissible dimer covering. Similarly, the right boundary state of is the Maya diagram defined by

(20)

for all . See Figure 7(a). A pure dimer covering has both boundary states equal to . Note that, if is a rail yard graph which is concatenable after , and if is an admissible dimer covering of , then and are compatible if and only if . We may now state the main result of this section.

Proposition 8 (Transfer matrix decomposition).

Given a rail yard graph with finite, and two Maya diagrams and , the sum of the weights (1) of all admissible dimer coverings of with left boundary state and right boundary state is given by

(21)

In particular, the partition function reads

(22)
Proof.

Note that (22) follows from (21) by taking , which amounts to considering pure dimer coverings.

We first verify (21) for , i.e. when is an elementary rail yard graph. Let us here treat the case , (displayed third on Figure 2(c)) and leave the other cases to the reader. Let (resp. ) be the positions of in (resp. ) enumerated in increasing order. Then, we claim that both sides of (21) are equal to if the two conditions

(23)
(24)

hold, and that both sides vanish otherwise. Indeed, on the one hand, it is not difficult to check (see Figure 7) that there is at most one elementary dimer configuration with prescribed boundary states and , and that there is exactly one such configuration (containing diagonal dimers) if and only if (23) and (24) hold. On the other hand, let and be the integer partitions associated with the Maya diagrams and : the quantity is equal to if the two conditions

(25)
(26) and have the same charge

hold, and it vanishes otherwise. But we have and for all hence, in view of (13), we find that the conditions (23)-(24) amount to (25)-(26), and then that as wanted.

Figure 7. (a) The rules (19)-(20) that define the two Maya diagrams and . (b) An admissible dimer covering of the elementary rail yard graph . White marbles are numbered as in the proof of Proposition 8. (c) If the left boundary vertex at ordinate is covered by a horizontal (resp. diagonal) dimer, then the right boundary vertex at ordinate (resp. ) is necessarily uncovered. Therefore there is a white marble at this ordinate on the right boundary, and induction implies that it is the -th white marble on the right boundary. This proves that and that is equal to the number of diagonal dimers. Moreover, since dimers incident to the left boundary can be recovered from the knowledge of the sequences and , it is clear that there is at most one dimer configuration with given boundary states and , and that there is one if and only if (23) and (24) hold. (d) Condition (23) can be interpreted by saying that white marbles are “jumping downwards” by  or . The cases of the elementary graphs , , and have similar interpretation, respectively with white marbles jumping upwards, black marbles jumping upwards, and black marbles jumping downwards, in all cases by or .

It remains to verify (21) for , which may be easily done by induction: it suffices to observe that, if is the rail yard graph obtained by removing the last “strip” of , then any admissible dimer covering of with boundary states is uniquely decomposed into a pair formed by an admissible dimer covering of with boundary states , for some Maya diagram , and an elementary dimer covering with boundary states , such that . ∎

3.3. Proof of enumeration results and computation of the partition function

We are now ready to prove Theorems 1 and 3. The first one is a direct consequence of the formalism developed above, whereas the second deserves an inspection of the different types of flips in rail yard graphs.

Proof of Theorem 1.

We simply have to evaluate the right-hand side of (22), which can be done as in in [OR07, Section 4.1], [You10, Section 4] or [BCC14, Section 5.1]: first observe that, by (15), for any , any , one has:

(27)

the being formal variables.

Now, by applying successively the commutation relations of Proposition 7, one can transform (22) into a scalar product of this form, up to a multiplicative prefactor, by moving to the left all the operators such that . In order to do that, we have, for each such that and , to transform the product of operators into the product . For each such transformation, we obtain a multiplicative contribution given by (16), and the result follows. ∎

Proof of Theorem 3.

As explained before the statement of Theorem 3, we have to prove that, specializing for the weights (1) to if and if amounts to attaching to each configuration a weight , where is the flip distance to the fundamental configuration.

First, this is true for the fundamental configuration that receives a weight in both cases. Second, since by Propp’s theory (recalled in Section 2.3) each shortest path from the fundamental state to any configuration is realized using positive flips only, it is enough to check that, in this specialization, each positive flip increases the weight of a configuration by a factor of .

Figure 8. (a) The 16 possible face types of rail yard graphs are obtained by matching one of the half-face types on the left with one on the right. (b-c) The dimer configuration around half-faces before (b) and after (c) a positive flip.

Consider an inner face in a rail yard graph. Then is made by the union of two half-faces as shown on Figure 8(a). Each of these two half-faces is incident to a diagonal edge, one in column , and one in column , in the sense of Section 2.4, for some . Then, a case inspection (see Figure 8(b-c)) shows that the following is true: when performing a positive flip on , the number of diagonal dimers on column increases (resp. decreases) by if (resp. ), and the number of diagonal dimers on column decreases (resp. increases) by if (resp. ).

Therefore this flip multiplies the weight (1) of the configuration by a factor of:

(28)

where we identified with respectively. But, in the specialization we are considering, we have for all , so (28) is equal to and the proof is complete. ∎

4. Fermionic operators

The purpose of this section is to establish Theorem 5. We start in Section 4.1 by recalling the definitions and basic properties of fermionic operators. In Section 4.2, we show that these operators can be used to construct constrained transfer matrices, that is transfer matrices enumerating dimer configurations containing a given subset of edges. We rewrite the product of constrained transfer matrices in another convenient form in Section 4.3, and complete the proof of Theorem 5 in Section 4.4. Finally, we elucidate the connection with Kasteleyn’s theory in Section 4.5.

4.1. Reminders

Recall that the fermionic Fock space , introduced at the end of Section 3.1, is the infinite dimensional Hilbert space spanned by orthonormal basis vectors where runs over the set of all Maya diagrams. For , we define the fermionic operators and (also called creation/annihilation operators) through their action on a basis vector by

(29)

where is the Maya diagram obtained from by inverting the color of the marble on site . Observe that the operators and are adjoint to one another. In particular, (resp. ) is the orthogonal projector on the space spanned by Maya diagrams with (resp. ). Fermionic operators obey the following well-known canonical anticommutation relations:

Proposition 9.

For any and in , we have

(30)

Here denotes the anticommutator of and : .

Proof.

Easy. ∎

Define the fermionic generating functions

(31)

Proposition 9 translates into

(32)

where is the formal Dirac delta function. It is straightforward to check that

(33)

Here the leftmost equal signs correspond to formal identities, but the rightmost equal signs require to treat and as complex variables. Let us also mention a lesser-known fact about the action of the involution on the fermionic operators (recall that is the involution that maps a charged partition to , hence can be seen as acting on ).

Proposition 10.

For , we have

(34)

where is the charge operator (acting on as the multiplication by ).

Proof.

Follows from the fact that, for any integer and any Maya diagram of charge , we have

(35)

Last but not least, we have the following commutation relations between bosonic and fermionic operators.

Proposition 11.

Given two formal variables we have

(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)

The first four (resp. last four) formal identities correspond to actual converging series when (resp. ).

We give in Appendix A a combinatorial proof of these identities (note that it is sufficient to establish only one of them, the others follow by taking duals, inverses and conjugates by ). For algebraic derivations, see for instance the references given at the beginning of Section 3.

4.2. Constrained transfer matrices

The fermionic operators can be used to enumerate constrained dimer configurations. A first natural idea, already used in [OR03], consists in inserting some orthogonal projectors or (with various ’s) within the product of bosonic operators (22) forming the partition function, which has the effect of forcing black or white marbles to be present at given positions. However, this does not fully determine the positions of the dimers (there are ambiguities for the columns containing both horizontal and diagonal edges). Remarkably, for an arbitrary rail yard graph and an arbitrary finite set of edges, there is a suitable way of inserting fermionic operators which precisely forces each edge of to be covered by a dimer.

We first introduce convenient notations. Recall that writing for an edge implies that its endpoints and are such that is even and is odd. Any finite set of edges of a rail yard graph can be decomposed “column by column”, hence written in the form

(44)

where , and . Here (resp. ) is the number of edges of connecting vertices with abscissas and (resp.  and ), and is zero except for finitely many .

Theorem 12 (Constrained transfer matrix decomposition).

Let be an arbitrary finite subset of edges of the graph , which we decompose as in (44), and let () be the number of diagonal edges of with an endpoint of abscissa . The sum of the weights (1) of all pure dimer configurations containing reads

(45)