The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
F. Y. Wu
TL;DR
This paper revisits the dimer-monomer problem on a rectangular lattice, providing a Pfaffian-based derivation of the Tzeng-Wu solution for a boundary monomer and clarifying its mathematical structure.
Contribution
It offers a new Pfaffian approach to derive and interpret the Tzeng-Wu solution for a single boundary monomer in the dimer-monomer problem.
Findings
Re-derivation of the Tzeng-Wu solution using Pfaffians
Identification of the solution as eigenvalues of the Kasteleyn matrix
Clarification of the mathematical structure of the solution
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.
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