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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Boundary partitions in trees and dimers

Authors: Richard W. Kenyon and David B. Wilson
Journal: Trans. Amer. Math. Soc. 363 (2011), 1325-1364
MSC (2010): Primary 60C05, 82B20, 05C05, 05C50
Published electronically: October 25, 2010
MathSciNet review: 2737268
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Abstract: Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure; i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and, as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have a direct application to connection probabilities for multiple-strand $ \operatorname{SLE}_2$, $ \operatorname{SLE}_8$, and $ \operatorname{SLE}_4$.

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Additional Information

Richard W. Kenyon
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912

David B. Wilson
Affiliation: Microsoft Research, Redmond, Washington 98052

Keywords: Tree, grove, double-dimer model, Gaussian free-field, Dirichlet-to-Neumann matrix, meander, SLE
Received by editor(s): March 26, 2008
Received by editor(s) in revised form: November 25, 2008
Published electronically: October 25, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.