The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary

Abstract

We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.

4 Pages to appear in the Physical Review E (2006)

References in corpus (5)

• Theory of resistor networks: The two-point resistance
• Theory of impedance networks: The two-point impedance and LC resonances
• Dimer statistics on the Möbius strip and the Klein bottle
• Packing dimers on $(2p+1)\times (2q+1)$ lattices
• Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries

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• Entropy of polydisperse chains: solution on the Husimi lattice
• Vacancy localization in the square dimer model
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• A Pfaffian formula for monomer-dimer partition functions
• Grassmannian representation of the two-dimensional monomer-dimer model
• Exact Solution of a Monomer-Dimer Problem: A Single Boundary Monomer on a Non-Bipartite Lattice
• Jamming and Tiling in Fragmentation of Rectangles
• A Statistical Model of Current Loops and Magnetic Monopoles
• Asymptotic behavior of the entropy of chains placed on stripes
• Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices
• State matrix recursion method and monomer--dimer problem
• The monopole-dimer model on Cartesian products of plane graphs
• A Pfaffian formula for the monomer-dimer model on surface graphs
• Errata and Addenda to Mathematical Constants