## Combinatorics of Tripartite Boundary Connections for Trees and Dimers

### BibTeX

@MISC{Kenyon_combinatoricsof,

author = {Richard W. Kenyon and David B. Wilson},

title = {Combinatorics of Tripartite Boundary Connections for Trees and Dimers},

year = {}

}

### OpenURL

### Abstract

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. 1

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Citation Context ...rly of dimer configurations of GBW an GBW i,j , respectively, and define ZWB and ZWB i,j 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 Z DD = Z BW Z WB ; this is essentially equivalent to Ciucu’s graph factorization theorem [Ciu97]. (The two dimer configurations in Figure 3 are on the graphs G BW and G WB .... |

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Citation Context ...f L. We generalize these results in several ways. Firstly, we give an interpretation (§ 8) of every minor of L in terms of grove probabilities. This is analogous to the all-minors matrix-tree theorem =-=[Cha82]-=- [Che76, pg. 313 1 Our L matrix is the negative of the Dirichlet-to-Neumann matrix of [CdV98]. the electronic journal of combinatorics 16 (2009), #R112 2Ex. 4.12–4.16, pg. 295], except that the matri... |

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(Show Context)
Citation Context ...onstructed explicitly as integer linear combinations of elementary polynomials. For certain partitions π, however, there is a simpler formula for .... Pr(π): for example, Curtis, Ingerman, and Morrow =-=[CIM98]-=-, and Fomin [Fom01], showed that for certain partitions π, .... Pr(π) is a determinant of a submatrix of L. We generalize these results in several ways. Firstly, we give an interpretation (§ 8) of eve... |

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Citation Context ...r example, ̂Pr(σ) = det[1i, j colored differentlyXi,j] i=1,3,...,2n−1 j=σ1,σ3,...,σ2n−1 . 3 2 4 1 ̂Pr( 1 4 |3 2 ) = ∣ X1,4 0 0 X3,2∣ (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) 4 3 2 5 6 1 ̂Pr( 1 2 | 3 6 | 5 4 ) = ∣ X1,2 0 X1,4 0 X3,6 0 X5,2 0 X5,4 ∣ the electronic journal of combinatorics 16 (20... |

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Citation Context ...ly as integer linear combinations of elementary polynomials. For certain partitions π, however, there is a simpler formula for .... Pr(π): for example, Curtis, Ingerman, and Morrow [CIM98], and Fomin =-=[Fom01]-=-, showed that for certain partitions π, .... Pr(π) is a determinant of a submatrix of L. We generalize these results in several ways. Firstly, we give an interpretation (§ 8) of every minor of L in te... |

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Citation Context ... every minor of L in terms of grove probabilities. This is analogous to the all-minors matrix-tree theorem [Cha82] [Che76, pg. 313 1 Our L matrix is the negative of the Dirichlet-to-Neumann matrix of =-=[CdV98]-=-. the electronic journal of combinatorics 16 (2009), #R112 2Ex. 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-min... |

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Citation Context ...hich 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 n 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 L satisfying the necessa... |

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Citation Context ...grove probabilities .... Pr(π) can be written as the Pfaffian of an antisymmetric matrix derived from the L 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 moti... |

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(Show Context)
Citation Context ...can be written as the Pfaffian of an antisymmetric matrix derived from the L 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 recon... |

2 |
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(Show Context)
Citation Context ...es occur as response matrices of a planar graph. [CIM98] showed how to reconstruct recursively the edge conductances of Σn 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 L matrix and its inverse. These Pfaffians are irreducible polynomials in the mat... |