Combinatorics of Tripartite Boundary Connections
for Trees and Dimers
00footnotetext: 2000 Mathematics Subject Classification. 60c05, 82b20, 05c05, 05c50.
00footnotetext: Key words and phrases. Tree, grove, double-dimer model, Dirichlet-to-Neumann matrix, Pfaffian.
A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for “tripartite” pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the double-dimer model on bipartite planar graphs.
In a companion paper [KW06] we studied two probability models on finite planar graphs: groves and the double-dimer model.
Given a finite planar graph and a set of vertices on the outer face, referred to as nodes, a grove is a spanning forest in which every component tree contains at least one of the nodes. A grove defines a partition of the nodes: two nodes are in the same part if and only if they are in the same component tree of the grove. See Figure 1.
When the edges of the graph are weighted, one defines a probability measure on groves, where the probability of a grove is proportional to the product of its edge weights. We proved in [KW06] that the connection probabilities—the partition of nodes determined by a random grove—could be computed in terms of certain “boundary” measurements. Explicitly, one can think of the graph as a resistor network in which the edge weights are conductances. Suppose the nodes are numbered in counterclockwise order. The matrix, or Dirichlet-to-Neumann matrix111Our matrix is the negative of the Dirichlet-to-Neumann matrix of [CdV98]. (also known as the response matrix or discrete Hilbert transform), is then the function indexed by the nodes, with being the vector of net currents out of the nodes when is a vector of potentials applied to the nodes (and no current loss occurs at the internal vertices). For any partition of the nodes, the probability that a random grove has partition is
where is the partition which connects no nodes, and is a polynomial in the entries with integer coefficients (we think of it as a normalized probability, , hence the notation). In [KW06] we showed how the polynomials could be constructed explicitly as integer linear combinations of elementary polynomials.
For certain partitions , however, there is a simpler formula for : for example, Curtis, Ingerman, and Morrow [CIM98], and Fomin [Fom01], showed that for certain partitions , is a determinant of a submatrix of . We generalize these results in several ways.
Firstly, we give an interpretation (§ 8) of every minor of in terms of grove probabilities. This is analogous to the all-minors matrix-tree theorem [Cha82] [Che76, pg. 313 Ex. 4.12–4.16, pg. 295], except that the matrix entries are entries of the response matrix rather than edge weights, so in fact the all-minors matrix-tree theorem is a special case.
Secondly, we consider the case of tripartite partitions (see Figure 2), showing that the grove probabilities can be written as the Pfaffian of an antisymmetric matrix derived from the matrix. One motivation for studying tripartite partitions is the work of Carroll and Speyer [CS04] and Petersen and Speyer [PS05] on so-called Carroll-Speyer groves (Figure 7) which arose in their study of the cube recurrence. Our tripartite groves directly generalize theirs. See § 9.
A third motivation is the conductance reconstruction problem. Under what circumstances does the response matrix ( matrix), which is a function of boundary measurements, determine the conductances on the underlying graph? This question was studied in [CIM98, CdV98, CdVGV96]. Necessary and sufficient conditions are given in [CdVGV96] for two planar graphs on nodes to have the same response matrix. In [CdV98] it was shown which matrices arise as response matrices of planar graphs. Given a response matrix satisfying the necessary conditions, in § 7 we use the tripartite grove probabilities to give explicit formulas for the conductances on a standard graph whose response matrix is . This question was first solved in [CIM98], who gave an algorithm for recursively computing the conductances, and was studied further in [CM02, Rus03]. In contrast, our formulas are explicit.
1.2 Double-dimer model
A number of these results extend to another probability model, the double-dimer model on bipartite planar graphs, also discussed in [KW06].
Let be a finite bipartite graph222Bipartite means that the vertices can be colored black and white such that adjacent vertices have different colors. embedded in the plane with a set of distinguished vertices (referred to as nodes) which are on the outer face of and numbered in counterclockwise order. One can consider a multiset (a subset with multiplicities) of the edges of with the property that each vertex except the nodes is the endpoint of exactly two edges, and the nodes are endpoints of exactly one edge in the multiset. In other words, it is a subgraph of degree at the internal vertices, degree at the nodes, except for possibly having some doubled edges. Such a configuration is called a double-dimer configuration; it will connect the nodes in pairs.
If edges of are weighted with positive real weights, one defines a probability measure in which the probability of a configuration is a constant times the product of weights of its edges (and doubled edges are counted twice), times where is the number of loops (doubled edges do not count as loops).
We proved in [KW06] that the connection probabilities—the matching of nodes determined by a random configuration—could be computed in terms of certain boundary measurements.
Let be the weighted sum of all double-dimer configurations. Let be the subgraph of formed by deleting the nodes except the ones that are black and odd or white and even, and let be defined as was, but with nodes and included in if and only if they were not included in .
A dimer cover, or perfect matching, of a graph is a set of edges whose endpoints cover the vertices exactly once. When the graph has weighted edges, the weight of a dimer configuration is the product of its edge weights. Let and be the weighted sum of dimer configurations of an , respectively, and define and similarly but with the roles of black and white reversed. Each of these quantities can be computed via determinants, see [Kas67].
One can easily show that ; this is essentially equivalent to Ciucu’s graph factorization theorem [Ciu97]. (The two dimer configurations in Figure 3 are on the graphs and .) The variables that play the role of in groves are defined by
We showed in [KW06] that for each matching , the normalized probability that a random double-dimer configuration connects nodes in the matching , is an integer polynomial in the quantities .
In the present paper, we show in Theorem 6.1 that when is a tripartite pairing, that is, the nodes are divided into three consecutive intervals around the boundary and no node is paired with a node in the same interval, is a determinant of a matrix whose entries are the ’s or .
1.3 Conductance reconstruction
Recall that an electrical transformation of a resistor network is a local rearrangement of the type shown in Figure 4. These transformations do not change the response matrix of the graph. [CdVGV96] showed that a planar graph with nodes can be reduced, using electrical transformations, to a standard graph (shown in Figure 5 for up to ), or a minor of one of these graphs (obtained from by deletion/contraction of edges).
In particular the response matrix of any planar graph on nodes is the same as that for a minor of the standard graph (with certain conductances). [CdV98] computed which matrices occur as response matrices of a planar graph. [CIM98] showed how to reconstruct recursively the edge conductances of from the response matrix, and the reconstruction problem was also studied in [CM02] and [Rus03]. Here we give an explicit formula for the conductances as ratios of Pfaffians of matrices derived from the matrix and its inverse. These Pfaffians are irreducible polynomials in the matrix entries (Theorem 5.1), so this is in some sense the minimal expression for the conductances in terms of the .
Here we collect the relevant facts from [KW06].
We assume that the nodes are labeled through counterclockwise around the boundary of the graph . We denote a partition of the nodes by the sequences of connected nodes, for example denotes the partition consisting of the parts and , i.e., where nodes , , and are connected to each other but not to node . A partition is crossing if it contains four items such that and are in the same part, and are in the same part, and these two parts are different. A partition is planar if and only if it is non-crossing, that is, it can be represented by arranging the items in order on a circle, and placing a disjoint collection of connected sets in the disk such that items are in the same part of the partition when they are in the same connected set. For example is the only non-planar partition on nodes.
2.2 Bilinear form and projection matrix
Let be the vector space consisting of formal linear combinations of partitions of . Let be the subspace consisting of formal linear combinations of planar partitions.
On we define a bilinear form: if and are partitions, takes value or and is equal to if and only if the following two conditions are satisfied:
The number of parts of and add up to .
The transitive closure of the relation on the nodes defined by the union of and has a single equivalence class.
For example but (We write the subscript to distinguish this form from ones that arise in the double-dimer model in § 6.)
This form, restricted to the subspace , is essentially the “meander matrix”, see [KW06, DFGG97], and has non-zero determinant. Hence the bilinear form is non-degenerate on . We showed in [KW06], Proposition 2.6, that is the direct sum of and a subspace on which is identically zero. In other words, the rank of is the Catalan number , which is the dimension of . Projection to along the kernel associates to each partition a linear combination of planar partitions. The matrix of this projection is called . It has integer entries [KW06]. Observe that preserves the number of parts of a partition: each non-planar partition with parts projects to a linear combination of planar partitions with parts (this follows from condition 1 above).
The rows of the projection matrix determine the crossing probabilities, see Theorem 2.5 below. In this section we give tools for computing columns of .
We say two elements of are equivalent () if their difference is in , that is, their inner product with any partition is equal. We have, for example,
which is another way of saying that
This lemma, together with the following two equivalences, will allow us to write any partition as an equivalent sum of planar partitions. That is, it allows us to compute the columns of .
Suppose , is a partition of , and . Then
If is a partition of , we can insert into the part containing item to get a partition of .
Suppose , is a partition of , , and . Then
One more lemma is quite useful for computations.
Lemma 2.4 ([Kw06, Lemma 4.1]).
If a planar partition contains only singleton and doubleton parts, and is the partition obtained from by deleting all the singleton parts, then the rows of the matrices for and are equal, in the sense that they have the same non-zero entries (when the columns are matched accordingly by deleting the corresponding singletons).
2.4 Connection probabilities
For a partition on we define
where the sum is over those spanning forests of the complete graph on vertices for which trees of span the parts of .
This definition makes sense whether or not the partition is planar. For example, and .
Theorem 2.5 (Theorem 1.2 of [Kw06]).
3 Tripartite pairing partitions
Recall that a tripartite partition is defined by three circularly contiguous sets of nodes , , and , which represent the red nodes, green nodes, and blue nodes (a node may be split between two color classes), and the number of nodes of the different colors satisfy the triangle inequality. In this section we deal with tripartite partitions in which all the parts are either doubletons or singletons. (We deal with tripod partitions in the next section.) By Lemma 2.4 above, in fact additional singleton nodes could be inserted into the partition at arbitrary locations, and the -polynomial for the partition would remain unchanged. Thus we lose no generality in assuming that there are no singleton parts in the partition, so that it is a tripartite pairing partition. This assumption is equivalent to assuming that each node has only one color.
Let be the tripartite pairing partition defined by circularly contiguous sets of nodes , , and , where , , and satisfy the triangle inequality. Then
Here is the submatrix of whose columns are the red nodes and rows are the green nodes. Similarly for and . Also recall that the Pfaffian of an antisymmetric matrix is a square root of the determinant of , and is a polynomial in the matrix entries:
where the sum can be interpreted as a sum over pairings of , since any of the permutations associated with a pairing would give the same summand.
In Appendix B there is a corresponding formula for tripartite pairings in terms of the matrix of pairwise resistances between the nodes.
Observe that we may renumber the nodes while preserving their cyclic order, and the above Pfaffian remains unchanged: if we move the last row and column to the front, the sign of the Pfaffian changes, and then if we negate the (new) first row and column so that the entries above the diagonal are non-negative, the Pfaffian changes sign again.
As an illustration of the theorem, we have
Note that when one of the colors (say blue) is absent, the Pfaffian becomes a determinant (in which the order of the green vertices is reversed). This bipartite determinant special case was proved by Curtis, Ingerman, and Morrow [CIM98, Lemma 4.1] and Fomin [Fom01, Eqn. 4.4]. See § 8 for a (different) generalization of this determinant special case.
Proof of Theorem 3.1.
From Theorem 2.5 we are interested in computing the non-planar partitions (columns of ) for which .
When we project , if has singleton parts, its image must consist of planar partitions having those same singleton parts, by the lemmas above: all the transformations preserve the singleton parts. Since consists of only doubleton parts, because of the on the number of parts, is non-zero only when contains only doubleton parts. Thus in Lemma 2.1 we may use the abbreviated transformation rule
Notice that if we take any crossing pair of indices, and apply this rule to it, each of the two resulting partitions has fewer crossing pairs than the original partition, so repeated application of this rule is sufficient to express as a linear combination of planar partitions.
If a non-planar partition contains a monochromatic part, and we apply Rule (4) to it, then because the colors are contiguous, three of the above vertices are of the same color, so both of the resulting partitions contain a monochromatic part. When doing the transformations, once there is a monochromatic doubleton, there always will be one, and since contains no such monochromatic doubletons, we may restrict attention to columns with no monochromatic doubletons.
When applying Rule (4) since there are only three colors, some color must appear twice. In one of the resulting partitions there must be a monochromatic doubleton, and we may disregard this partition since it will contribute . This allows us to further abbreviate the uncrossing transformation rule:
and similarly for green and blue. Thus for any partition with only doubleton parts, none of which are monochromatic, we have , and otherwise .
If we consider the Pfaffian of the matrix
each monomial corresponds to a monomial in the -polynomial of , up to a possible sign change that may depend on the term.
Suppose that the nodes are numbered from to starting with the red ones, continuing with the green ones, and finishing with the blue ones. Let us draw the linear chord diagram corresponding to . Pick any chord, and move one of its endpoints to be adjacent to its partner, while maintaining their relative order. Because the chord diagram is non-crossing, when doing the move, an integer number of chords are traversed, so an even number of transpositions are performed. We can continue this process until the items in each part of the partition are adjacent and in sorted order, and the resulting permutation will have even sign. Thus in the above Pfaffian, the term corresponding to has positive sign, i.e., the same sign as the monomial in ’s -polynomial.
Next we consider other pairings , and show by induction on the number of transpositions required to transform into , that the sign of the monomial in ’s -polynomial equals the sign of the monomial in the Pfaffian. Suppose that we do a swap on to obtain a pairing closer to . In ’s polynomial, and have opposite sign. Next we compare their signs in the Pfaffian. In the parts in which the swap was performed, there is at least one duplicated color (possibly two duplicated colors). If we implement the swap by transposing the items of the same color, then the items in each part remain in sorted order, and the sign of the permutation has changed, so and have opposite signs in the Pfaffian.
Thus ’s -polynomial is the Pfaffian of the above matrix. ∎
4 Tripod partitions
In this section we show how to compute for tripod partitions , i.e., tripartite partitions in which one of the parts has size three. The three lower-left panels of Figure 2 and the left panels of Figure 6 and Figure 7 show some examples.
4.1 Via dual graph and inverse response matrix
For every tripod partition , the dual partition is also tripartite, and contains no part of size three. As a consequence, we can compute the probability when is a tripod in terms of a Pfaffian in the entries of the response matrix of the dual graph :
The last ratio in the right is known to be an minor of (see e.g., § 8); it remains to express the matrix in terms of .
Let be the node of the dual graph which is located between the nodes and of .
The entries of are related to the entries of as follows:
Here even though is not invertible, the vector is in the image of and is perpendicular to the kernel of , so the above expression is well defined.
4.2 Via Pfaffianiod
In § 4.1 we saw how to compute for tripartite partitions . It is clear that the formula given there is a rational function of the ’s, but from Theorem 2.5, we know that it is in fact a polynomial in the ’s. Here we give the polynomial.
We saw in § 3 that the Pfaffian was relevant to tripartite pairing partitions, and that this was in part because the Pfaffian is expressible as a sum over pairings. For tripod partitions (without singleton parts), the relevant matrix operator resembles a Pfaffian, except that it is expressible as a sum over near-pairings, where one of the parts has size , and the remaining parts have size . We call this operator the Pfaffianoid, and abbreviate it . Analogous to (2), the Pfaffianoid of an antisymmetric matrix is defined by
where the sum can (almost) be interpreted as a sum over near-pairings (one tripleton and rest doubletons) of , since for any permutation associated with the near-pairing , the summand only depends on the order of the items in the tripleton part.
The sum-over-pairings formula for the Pfaffian is fine as a definition, but there are more computationally efficient ways (such as Gaussian elimination) to compute the Pfaffian. Likewise, there are more efficient ways to compute the Pfaffianoid than the above sum-over-near-pairings formula. For example, we can write
where denotes the matrix with rows and columns , , and deleted. It is also possible to represent the Pfaffianoid as a double-sum of Pfaffians.
The tripod probabilities can written as a Pfaffianoid in the ’s as follows:
Let be the tripod partition without singletons defined by circularly contiguous sets of nodes , , and , where , , and satisfy the triangle inequality. Then
Unlike the situation for tripartite partitions, here we cannot appeal to Lemma 2.4 to eliminate singleton parts from a tripod partition, since Lemma 2.4 does not apply when there is a part with more than two nodes. However, any nodes in singleton parts of the partition can be split into two monochromatic nodes of different color, one of which is a leaf. The response matrix of the enlarged graph is essentially the same as the response matrix of the original graph, with some extra rows and columns for the leaves which are mostly zeroes. Theorem 4.2 may then be applied to this enlarged graph to compute for the original graph.
For any non-crossing partition , is an irreducible polynomial in the ’s.
By looking at the dual graph, it is a straightforward consequence of Theorem 5.1 that is an irreducible polynomial on the pairwise resistances. In contrast, for the double-dimer model, the polynomials sometimes factor (the first, second, and fourth examples in § 6 factor).
Proof of Theorem 5.1.
Suppose that factors into where and are polynomials in the ’s. Because and each is multilinear in the ’s, we see that no variable occurs in both polynomials and .
Suppose that for distinct vertices , the variables and both occur in , but occur in different factors, say occurs in while occurs in . Then the product contains monomials divisible by . If we consider one such monomial, then the connected components (with edges given by the indices of the variables of the monomial) define a partition for which and for which contains a part containing at least three distinct items , , and . Then contains , so also occurs in one of or , say (w.l.o.g.) that it occurs in . Because contains monomials divisible by , so does , and hence must contain monomials divisible by . But then would contain monomials divisible by , but contains no such monomials, a contradiction, so in fact and must occur in the same factor of .
If we consider the graph which has an edge for each variable of , we aim to show that the graph is connected except possibly for isolated vertices; it will then follow that is irreducible.
We say that two parts and of a non-crossing partition are mergeable if the partition is non-crossing. It suffices, to complete the proof, to show that if and are mergeable parts of , then contains for some and .
Suppose and are mergeable parts of that both have at least two items. When the items are listed in cyclic order, say that is the last item of before , is the first item of after , is the last item of before , and is the first item of after . Let be the partition formed from by swapping and . Let , and let and . Then and . Then
Each of the partitions on the right-hand side is non-crossing, so , so in particular occurs in .
Now suppose that contains a singleton part and another part containing at least three items , , , where , , and are the first, second, and last items of the part as viewed from item . Let and . Let be the partition
The second, third, fourth, fifth, and sixth terms on the RHS contribute nothing to because their restrictions to the intervals , , , , and respectively are planar and do not agree with . Thus , and hence occurs in .
Finally, if contains singleton parts but no parts with at least three items, then is formally identical to the polynomial where is the partition with the singleton parts removed from , and we have already shown above that the polynomial is irreducible. ∎
6 Tripartite pairings in the double-dimer model
In this section we prove a determinant formula for the tripartite pairing in the double-dimer model.
Suppose that the nodes are contiguously colored red, green, and blue (a color may occur zero times), and that is the (unique) planar pairing in which like colors are not paired together. Let denote the item that pairs with item . We have
|(this first example formula is essentially Theorems 2.1 and 2.3 of [Kuo04], see also [Kuo06] for a generalization different from the one considered here)|
Recall our theorem from [KW06], which shows how to compute the “” polynomials for the double-dimer model in terms of the “” polynomials for groves:
Theorem 6.2 (Kenyon-Wilson ’06).
If a partition contains only doubleton parts, then if we make the following substitutions to the grove partition polynomial :
then the result is the double-dimer pairing polynomial , when we interpret as a pairing.
Thus our Pfaffian formula for tripartite groves in terms of the ’s immediately gives a Pfaffian formula for the double-dimer model. For the double-dimer tripartite formula there are node parities as well as colors (recall that the graph is bipartite). Rather than take a Pfaffian of the full matrix, we can take the determinant of the odd/even submatrix, whose rows are indexed by red-even, green-even, and blue-even vertices, and whose columns are indexed by red-odd, green-odd, and blue-odd vertices. For example, when computing the probability
nodes , , and are red, and are green, and , , and are blue; the odd nodes are black, and the even ones are white. The -polynomial is
Next we do the substitution when is even, and reorder the rows and columns so that the odd nodes are listed first. Each time we swap a pair of rows and do the same swap on the corresponding pair of columns, the sign of the Pfaffian changes by . Since there are nodes the number of swaps is . If is congruent to or the sign does not change, and otherwise it does change. After the rows and columns have been sorted by their parity, the matrix has the form
where represents the odd nodes and the even nodes, and where the individual signs are if the odd node has smaller index than the even node, and otherwise. The Pfaffian of this matrix is just the determinant of the upper-right submatrix, times the sign of the permutation , which is . This sign cancels the above sign. In this example we get
Next we do the substitution. The entry of this matrix is . Each time that or are incremented or decremented by , the sign will flip, unless the sign also flips. After the substitution, the signs of the are staggered in a checkerboard pattern. If we then multiply every other row by and every other column by , the determinant is unchanged and all the signs are . In the example we get
There is then a global sign of where the sign of the pairing is the sign of sign of the permutation of the even elements when the parts are arranged in increasing order of their odd parts. In our example, the sign of is the sign of , which is . This global sign may be canceled by reordering the columns in this order, i.e., so that the pairing can be read in the indices along the diagonal of the matrix, which for our example is
7 Reconstruction on the “standard graphs”
Given a planar resistor network, can we determine (or “reconstruct”) the conductances on the edges from boundary measurements, that is, from the entries in the matrix?
While reconstruction is not possible in general, each planar graph is equivalent, through a sequence of electrical transformations, to a graph on which generically the conductances can be reconstructed. Let denote the standard graph on nodes, illustrated in Figure 5 for up to . Every circular planar graph with nodes is electrically equivalent to a minor of a standard graph .
Here we will use the Pfaffian formulas to give explicit formulas for reconstruction on standard graphs. For minors of standard graphs, the conductances can be computed by taking limits of the formulas for standard graphs.
Curtis, Ingerman and Morrow [CIM98] gave a recursive construction to compute conductances for standard graphs from the -matrix. Card and Muranaka [CM02] give another way. Russell [Rus03] shows how to recover the conductances, and shows that they are rational functions of -matrix entries. However the solution is sometimes given parametrically, as a solution to polynomial constraints, even when graph is recoverable.
For a vertex we define to be the tripod partition of the nodes indicated in Figure 6, with a single triple connection joining the nodes horizontally across from and the two nodes vertically located from (in the same column as ), and the remaining nodes joined in nested pairs between and , and , and and (with up to two singletons if and/or have an odd number of nodes between them).
Similarly, for a bounded face of define to be the tripartite partition of the nodes indicated in Figure 6. It has three nested sequences of pairwise connections (with two of the nested sequences going to the NE and SE, possibly terminating in singletons). We think of the unbounded face as containing many “external faces,” each consisting of a unit square which is adjacent to an edge of . For each of these external faces, we define in the same manner as for internal faces. For the external faces on the left of , the “left-going” nested sequence of is empty. For the other external faces , the partition is , independent of .
Observe that for the standard graphs , there is only one grove of type or of type . Let denote the conductance of edge in . Each and is a monomial in these conductances . To simplify notation we write and
Each conductance can be written in terms of the