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16
Cluster algebras: Notes for the CDM03 conference
"... Abstract. This is an expanded version of the notes of our lectures ..."
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Cited by 17 (6 self)
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Abstract. This is an expanded version of the notes of our lectures
A periodicity theorem for the octahedron recurrence
 J. Algebraic Combin
"... In this paper we investigate a variant of the octahedron recurrence of RobbinsRumsey [8] called the bounded octahedron recurrence. It was first described by Kamnitzer and the author in [4], where it was used to relate the commutativity isomorphism for gl(n)crystals with the Schützenberger involuti ..."
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Cited by 9 (0 self)
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In this paper we investigate a variant of the octahedron recurrence of RobbinsRumsey [8] called the bounded octahedron recurrence. It was first described by Kamnitzer and the author in [4], where it was used to relate the commutativity isomorphism for gl(n)crystals with the Schützenberger involution on Young tableaux.
RIMS1659 Plücker environments, wiring and tiling diagrams, and weakly separated setsystems By
, 2009
"... and weakly separated setsystems ..."
The Electrical Response Matrix of a Regular 2ngon
, 2008
"... Consider a unitresistive plate in the shape of a regular polygon with 2n sides, in which evennumbered sides are wired to electrodes and oddnumbered sides are insulated. The response matrix, or DirichlettoNeumann map, allows one to compute the currents flowing through the electrodes when they ar ..."
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Consider a unitresistive plate in the shape of a regular polygon with 2n sides, in which evennumbered sides are wired to electrodes and oddnumbered sides are insulated. The response matrix, or DirichlettoNeumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2ngon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests. 1
Boundary Partitions in Trees and Dimers
, 2006
"... We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as fu ..."
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We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition σ, Pr[grove has type σ] / Pr[grove is a tree] is a dyadiccoefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type σ] / Pr[grove has maximal number of trees] is an integercoefficient polynomial in the entries of the DirichlettoNeumann matrix. We give analogous integercoefficient polynomial formulas for the pairings of chains in the doubledimer model. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the doubledimer model. These partition probabilities are relevant to multichordal SLE2, SLE4, and SLE8. 1
Boundary Partitions in Trees and Dimers
, 2007
"... We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as fu ..."
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We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition σ, Pr[grove has type σ] / Pr[grove is a tree] is a dyadiccoefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type σ] / Pr[grove has maximal number of trees] is an integercoefficient polynomial in the entries of the DirichlettoNeumann matrix. We give analogous integercoefficient polynomial formulas for the pairings of chains in the doubledimer model. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the doubledimer model. These partition probabilities are relevant to multichordal SLE2, SLE4, and SLE8. 1
Boundary Partitions in Trees and Dimers
, 2008
"... 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 p ..."
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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 DirichlettoNeumann 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 integercoefficient polynomials in the matrix entries, where the coefficients have a natural combinatorial interpretation. A similar phenomenon holds in the socalled doubledimer 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 doubledimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiplestrand SLE2, SLE8, and SLE4.