Local Statistics of Realizable Vertex Models

Local Statistics of Realizable Vertex Models

Zhongyang Li***Department of mathematics, Brown University, Providence, RI 02912, USA, zli@math.brown.edu
Abstract

We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.

1 Introduction

A vertex model is a graph where we associate to each vertex a signature . A local configuration at a vertex is a subset of incident edges of . A configuration of the graph is an edge subset of . The signature at a vertex is a function which assigns a nonnegative real number (weight) to each local configuration at . The partition function of the vertex model is the weighted sum of configurations , where the weight of a configuration is the product of weights of local configurations, obtained by restricting the configuration at each vertex. Dimers, loop models, and random tiling models are some special examples of vertex models.

Direct computations of the partition function of a general vertex model usually require exponential time. On the other hand, using the Fisher-Kasteleyn-Temperley method [4, 5], we can efficiently count the number of perfect matchings (dimer configurations) of a finite planar graph. The idea of generalized holographic reduction is to reduce a vertex model on a planar graph to a dimer model on another planar graph, essentially by a linear base change, see section 3. For an original version of the holographic reduction (Valiant’s Algorithm), see [16].

However, not all satisfying assignment problems can be reduced to a perfect matching problem (“realized”) using the holographic algorithm. We study the realizability problem of the generalized algorithm for vertex models on the hexagonal lattice and prove that the signature of realizable models form a submanifold of positive codimension of the manifold of all signatures, see Theorem 3.6. An example of realizable models is the 1-2 model, which is a signature on the honeycomb lattice, only one or two edges allowed to be present in each local configuration, see Figure 13, 14. The realizability problem of Valiant’s algorithm is studied by Cai [2], and the realizability problem of uniform 1-2 model (not-all-equal relation), a special 1-2 model which assigns all the configurations weight 1, under Valiant’s algorithm is studied by Schwartz and Bruck [14].

Realizable vertex models may be reduced to dimer models in more than one way, that is, using different bases. However, all the dimer models corresponding to the same vertex model are shown to be gauge equivalent, i.e. obtained from one another by a trivial reweighting.

One of the simplest vertex configuration models is a graph with the same signature at all vertices. Using the singular value decomposition, we prove that such models on a hexagonal lattice are realizable if and only if they are realizable under orthogonal base change. Moreover, the orthogonal realizability condition takes a very nice form; see section 3.2.

We compute the local statistics of realizable vertex models on a hexagonal lattice with the help of the generalized holographic reduction, i.e. for the natural probability measure, we compute the probabilities of given configurations at finitely many fixed vertices, which are proved to be computable by sums of finitely many Pfaffians, see Theorem 5.1 and Theorem 5.2.

The weak limit of probability measures of the vertex model on finite graphs are of considerable interest. Using an torus to approximate the infinite periodic graph, we give an explicit integral formula for the probability of a specific local configuration at a fixed vertex, see section 6. These results follow from a study of the zeros of the characteristic polynomial, or the spectral curve, on the unit torus . For a more general result about the intersection of the spectral curve with , see [13]. For example, using our method, we compute the probability that a dimer occurs and for uniform 1-2 model, and the probability that a configuration occurs at a vertex for critical 1-2 model, see Examples 7.2 and 7.3.

The main result of this paper can be stated in the following theorems

Theorem 1.1.

For a periodic, realizable, positive-weight vertex model on a hexagonal lattice with period , assume the corresponding Fisher graph has positive edge weights, then the free energy of G is

where is the quotient graph , is the characteristic polynomial.

Theorem 1.2.

Assume the periodic vertex model on hexagonal lattice is realizable to the dimer model on a Fisher graph with positive, periodic edge weights, and assume the entries of the corresponding base change matrices are nonzero. Let be the probability measure defined for configurations on toroidal hexagonal lattice . Moreover, for a configuration at a vertices

where

are local dimer configurations on the gadget of the Fisher graph corresponding to . is the set of vertices involved in the configuration .

Acknowledgments The author would like to thank Richard Kenyon for stimulating discussions. The author is also grateful to David Wilson, Béatrice de Tilière and Sunil Chhita for valuable comments.

2 Background

2.1 Vertex Models

Let denote the set of all binary sequences of length . A vertex model is a graph where we associate to each vertex a function

is called the signature of the vertex model at vertex . We give a linear ordering on the edges adjacent to , and we fix such an ordering around each vertex once and for all. This way the binary sequences of length are in one-to-one correspondence with the local configurations at . Each edge corresponds to a digit; if that edge is included in the configuration, the corresponding digit is 1, otherwise the corresponding digit is 0. Hence we can also consider as a column vector indexed by local configurations at :

Example 2.1 (signature of the vertex model at a vertex).

Assume we have a degree-2 vertex with signature

that means we give weights to the four different local configurations as in Figure 1:

Figure 1: Relation at a Vertex

Assume is a finite graph. We define a probability measure with sample space the set of all configurations, . The probability of a specific configuration is

(1)

The product is over all vertices. is the weight of the local configuration obtained by restricting to the vertex , and is a normalizing constant called the partition function for vertex models, defined to be

The sum is over all possible configurations of .

Now we consider a vertex model on a -periodic planar graph . By this we mean that is embedded in the plane so that translations in act by signature-preserving isomorphisms of . Examples of such graphs are the square and Fisher lattices, as shown in Figure 7. Let be the quotient of by the action of . It is a finite graph embedded into a torus. Let be the partition function of the vertex model on . The free energy of the infinite periodic vertex model is defined to be

2.2 Perfect Matching

For more information, see [6]. A perfect matching, or a dimer cover, of a graph is a collection of edges with the property that each vertex is incident to exactly one edge. A graph is bipartite if the vertices can be 2-colored, that is, colored black and white so that black vertices are adjacent only to white vertices and vice versa.

To a weighted finite graph , the weight is a function from the set of edges to positive real numbers. We define a probability measure, called the Boltzmann measure with sample space the set of dimer covers. Namely, for a dimer cover

where the product is over all edges present in , and is a normalizing constant called the partition function for dimer models, defined to be

the sum over all dimer configurations of .

If we change the weight function by multiplying the edge weights of all edges incident to a single vertex by the same constant, the probability measure defined above does not change. So we define two weight functions to be gauge equivalent if one can be obtained from the other by a sequence of such multiplications.

The key objects used to obtain explicit expressions for the dimer model are Kasteleyn matrices. They are weighted, oriented adjacency matrices of the graph defined as follows. A clockwise-odd orientation of is an orientation of the edges such that for each face (except the infinite face) an odd number of edges pointing along it when traversed clockwise. For a planar graph, such an orientation always exists [5]. The Kasteleyn matrix corresponding to such a graph is a skew-symmetric matrix defined by

It is known [4, 5, 15, 8] that for a planar graph with a clock-wise odd orientation, the partition function of dimers satisfies

Now let be a -periodic planar graph. Let be a quotient graph of , as defined before. Let be a path in the dual graph of winding once around the torus horizontally(vertically). Let be the set of edges crossed by . We give a crossing orientation for the toroidal graph as follows. We orient all the edges of except for those in . This is possible since no other edges are crossing. Then we orient the edges of as if did not exist. Again this is possible since is planar. To complete the orientation, we also orient the edges of as if did not exist.

For , let be the Kasteleyn matrix in which the weights of edges in are multiplied by , and those in are multiplied by . It is proved in [15] that the partition function of the graph is

Let be a subset of edges of . Kenyon [7] proved that the probability of these edges occurring in a dimer configuration of with respect to the Boltzmann measure is

where , and is the submatrix of whose lines and columns are indexed by .

The asymptotic behavior of when is large is an interesting subject. One important concept is the partition function per fundamental domain, which is defined to be

Let be a Kasteleyn matrix for the graph . Given any parameters , we construct a matrix as follows. Let , be the paths introduced as above. Multiply by if Pfaffian orientation on that edge is from to , otherwise multiply by , and similarly for on . Define the characteristic polynomial . The spectral curve is defined to be the locus .

Gauge equivalent dimer weights give the same spectral curve. That is because after Gauge transformation, the determinant multiplies by a nonzero constant, and the locus of does not change.

A formula for enlarging the fundamental domain is proved in [3, 8]. Let be the characteristic polynomial of with period , and be the characteristic polynomial of , then

2.3 Matchgates, Matchgrids

A matchgate is a planar local graph including a set of external vertices, i.e. vertices located along the boundary of the local graph. The external vertices are ordered anti-clockwise on the boundary. is called an odd(even) matchgate if it has an odd(even) number of vertices.

The signature of the matchgate is a vector indexed by subsets of external vertices, . For a subset , the entry of the signature at is the partition function of dimer configurations on a subgraph of the matchgate. The subgraph is obtained from the matchgate by removing all the external vertices in .

Example 2.2 (signature of a matchgate).

Assume we have a matchgate with external vertices 1, 2, 3, and edge weights as illustrated in the following figure:

Figure 2: Matchgate

then the signature of the matchgate is

A matchgrid is a weighted planar graph consisting of a collection of matchgates and connecting edges. Each connecting edge has weight 1 and joins an external vertex of a matchgate with an external vertex of another matchgate, so that every external vertex is incident to exactly one connecting edge.

3 Generalized Holographic Reduction

In this section we introduce a generalized holographic algorithm to compute the partition function of the vertex model on a finite planar graph in terms of the partition function for perfect matchings on a matchgrid. The idea is using a matchgate to replace each vertex, and perform a change of basis, such that after the base change process, the signature of a vertex becomes the signature of the corresponding matchgate. We describe the algorithm in detail as follows.

For a finite graph , we associate to each oriented edge a 2-dimensional vector space . To the edge with the reversed orientation, the associated vector space is the dual space, i.e. . Give a set basis for each , satisfying

Let be a vertex with incident edges , oriented away from . The signature of the vertex model at a vertex , , can be considered as an element in . Hence has representations under bases and as follows

(2)
(3)

where are the set of standard bases for each

. are binary sequences of length . From the definition of the signature of vertex models, obviously is the weight of the configuration at vertex .

We construct a matchgrid as follows. We replace each vertex by a matchgate , such that the number of external vertices of is the same as the degree of , and the edges of the vertex model graph become connecting edges joining different matchgates in the matchgrid . Examples of such replacements are illustrated in the following Figure.

Figure 3: degree-3 vertex and matchgate
Figure 4: degree-4 vertex and matchgate

If the signature of the matchgate satisfies

(4)

that is, the representation of under bases is the same as the representation of under bases , then we have the following theorem:

Theorem 3.1.

Under the above base change process, the partition function of the vertex model of is equal to the partition function of the dimer model of .

Proof.

There is a natural mapping from to induced by , where is the natural pairing from to . Note that in , each and appear exactly once. Since the representation of under bases is the same as the representation of under bases , we have

(5)

(5) follows from the fact that each is independent of bases as long as we choose the dual basis for the dual vector space. However, the left side of (5) is exactly the partition function of the dimer model of , while the right side of (5) is exactly the partition function of the vertex model of . ∎

Define the base change matrix at edge , . The base change matrix at vertex , , acting on by multiplication, is defined to be

In order for a vertex model problem to be reduced to dimer model problem, one sufficient condition is that at each vertex, the signature of the vertex model under the bases is the same as the signature of the matchgate under the standard bases. Namely,

(6)

(6) follows directly from (3) and (4).

Example 3.2.

Consider the graph in Figure 3 with standard dimer signature

Figure 5:

Define the base change matrix on edges to be

By definition, . Note that After the base change, we have the standard loop signature

For instance, after the base change, the dimer configuration with only -edge occupied becomes

which is the configuration , the loop configuration with -edge and -edge occupied.

However, since the number of vertices in a matchgate is either even or odd, at a vertex of degree , the signature of a matchgate must be a dimensional subspace of , with those entries being 0. These are the entries correspond to the partition function of dimer configurations on a subgraph of the matchgate with an odd number of vertices, see example 2.1. This is the parity constraint. As a result, by dimension count we can see that it is not possible to use holographic algorithm to reduce all vertex models into dimer models. To characterize the special class of vertex models applicable to holographic reduction, we introduce the following definition.

Definition 3.3.

A network of relations on a finite graph is realizable, if there exists a system of bases reducing the model to the set of perfect matchings of a matchgrid.

Definition 3.4.

A network of relations is bipartite realizable, if it is realizable and the corresponding matchgrid is a bipartite graph.

Remark. The above generalizes Valiant’s algorithm [16] in the sense that our basis can be different from edge to edge. As a consequence, our approach results in an enlargement of the dimension of realizable submanifold, which will be shown in the next section.

3.1 Realizability

We are interested in periodic vertex models on the honeycomb lattice with period . The quotient graph can be embedded on a torus . We classify all the edges into -type, -type and -type according to their direction, and assume -type and -type edges have the same direction as the two basic homology cycles and of torus, respectively, see Figure 6.

Figure 6: Periodic Honeycomb Lattice

Assume the vertex model is realizable, then each corresponding matchgate may have either an even or an odd number of vertices. By enlarging the fundamental domain, we can always assume that there are an even number of matchgates with an even number of vertices. Then by permuting rows of matrices on a finite number of edges, we can always have all the matchgates having an odd number of vertices. For example, assume we have a pair of adjacent matchgates with matrix on the connecting edge , ( here is actually the inverse of the base change matrix defined before ), as illustrated in Figure 7.

Figure 7: Permutation of Basis Vectors 1

Then if we assume obtained by permutating two basis vectors of , we actually interchange the roles of 0 and 1 at the corresponding digit of the binary sequences as indices of signatures of the matchgate. Namely, assume , and , where . Then for any binary sequence , the entry is the same as , according to equation (6). If originally we have an even matchgate at , by parity constraint, the entries of are zero. After the permutation of basis vectors, will have entries to be zero. Hence has to be an odd matchgate, as illustrate in Figure 8.

Figure 8: Permutation of Basis Vectors 2

By finitely many times of such permutations, we can move two even matchgates to adjacent positions. Then we permutate the basis on the connecting edge, we can decrease the number of even matchgates by 2. Since we have an even number of even matchgates in total, if we repeat the process, in the end, all matchgates will be odd.

Assumption 3.5.

From now on, we make the following assumption:

  • All entries of base change matrices are nonzero.

  • All entries of the matchgate signature are nonzero.

Since the honeycomb lattice is a bipartite graph, we can color all the vertices in black and white such that black vertices are adjacent only to white vertices and vice versa. Assume vertex signatures at black and white vertices are as follows:

(7)
(8)

where is the row and column index of white and black vertices. Assume , then entries of signatures are indexed from 1 to 8. Associate to an -edge, -edge and -edge incident to a white or black vertices, a basis

where for white vertices and for black vertices.

By the realizability equation (6), and the fact that all matchgates are odd, the 1st, 4th, 6th, 7th entries of the matchgate signatures are 0, so we have a system of 4 algebraic equations at each vertex . For example, at each black vertex, we have

(9)

and the fact that the 1st entry of is 0 gives the following equation

That the 4th, 6th, 7th entries of are 0 give three other similar equations. Similarly at each white vertex, we have

(10)

the same process gives a system of 4 equations at the white vertex.

Let ,, ,. Then the equations we get are linear with respect to , , , , which can be solved explicitly.

(11)
(12)
(13)
(14)

where

(15)

Since and , after clearing denominators and some reducing, for any , equations (11)-(14) are equivalent to

(16)
(17)
(18)
(19)

If we solve explicitly, a similar process yields

(20)
(21)

We get two equations per edge, one involving the a-variables, the other involving the b-variables. But a-variables and b-variables are actually the same thing for each single edge. From equation (16),(17), we can solve basis entries ; from equation (18),(19), we can solve basis entries . Finally from (11)-(14), we can solve and , the only constraint left will be , which is two polynomial equations with respect to relation signature at each a-edge. Together with Theorem 2 in appendix, we have the following theorem

Theorem 3.6.

Under assumption 3.5, the realizable signatures on the periodic honeycomb lattice form a dimensional submanifold of the dimensional manifold of all positive signatures; the bipartite realizable signatures on the periodic honeycomb lattice form a dimensional submanifold of the dimensional manifold of all positive signatures.

For the exact realizability condition, see the appendix.

Under assumption 3.5, the weight {111} is non-vanishing at each matchgate. Since the probability measure will not change if all entries of the signature at one vertex are multiplied by a constant, we can assume at each matchgate, the weight of is 1. Therefore, we have

Theorem 3.7.

A realizable vertex model on a finite hexagonal lattice can always be transformed to dimers on , a Fisher graph as shown in Figure 9, with the partition function of the vertex model of equal to the partition function of the dimer model of , up to multiplication of a constant.

Figure 9: Matchgrid with 3 Fundamental Domain

It is possible to construct a matchgrid with different weights to produce the same vertex model. Since the holographic reduction is an invertible process, by which we mean that because the base change matrices are nonsingular, we can always achieve the matchgate signature from the vertex signature and vice versa, there is an equivalence relation on dimer models producing the same vertex model.

Definition 3.8.

A vertex model is holographically equivalent to a dimer model on a matchgrid, if it can be reduced to the dimer model on the matchgrid using the holographic algorithm, in such a way that partition function of the vertex model corresponds to perfect matchings of the matchgrid. Two matchgrids are holographically equivalent, if they are holographically equivalent to the same vertex model.

Proposition 3.9.

Holographically equivalent matchgrids give rise to gauge equivalent dimer models. Therefore, they have the same probability measure.

Proof.

See the appendix. ∎

3.2 Orthogonal Realizability

Definition 3.10.

A vertex model is orthogonally realizable if it is realizable by an orthonormal base change matrix on each edge.

Consider a single vertex. To the incident edges of the vertex, associate an matrices . Without loss of generality, we can assume . In fact, if , we multiply the first row of by . The new signature of the matchgate will be multiplied by . This change does not violate the parity constraint.

Assume . Assume , then , each term of which is product of three of . Moreover, the eigenvalues are . Using trigonometric identities, each entry is a linear combination of and . If we further define

then if , we have

(22)
(23)
(24)
(25)
(26)

By (22), we have ,,,. If we define

Then the following theorem holds:

Theorem 3.11.

A vertex model on a periodic honeycomb lattice with period is orthogonally realizable if and only if its signatures satisfy the following system of equations

where , and the same relation for and . and are defined by (7) and .

It is trivial to verify these equations in any given situation.

Definition 3.12.

A vertex model on a periodic honeycomb lattice with period is positively orthogonally realizable if it is orthogonal realizable and for each vertex , there exists angles , such that

Proposition 3.13.

If a vertex model on a bi-periodic hexagonal lattice is orthogonally realizable, then the corresponding dimer configuration has positive edge weights.

Proof.

Under the assumption that the vertex model have nonnegative signature, we have