A Pfaffian formula for monomer-dimer partition functions

Abstract

We consider the monomer-dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the vertices and edges of the graph.

Added reference

References in corpus (5)

• The scaling limit of the energy correlations in non integrable Ising models
• The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
• Boundary monomers in the dimer model
• Exact Solution of a Monomer-Dimer Problem: A Single Boundary Monomer on a Non-Bipartite Lattice
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Cited by in corpus (7)

• Jamming and Tiling in Fragmentation of Rectangles
• Pfaffian Correlation Functions of Planar Dimer Covers
• Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases
• Two populations mean-field monomer-dimer model
• A Pfaffian formula for the monomer-dimer model on surface graphs
• Equivalence of Stabilizer and Shannon Rényi Entropies: Exact Results for Quantum Critical Chains
• Squareness for the Monopole-Dimer model