Isomorphism of dimer configurations and spanning trees on finite square lattices

Abstract

One-to-one mappings of the close-packed dimer configurations on a finite square lattice with free boundaries ℒ onto the spanning trees of a related graph (or two-graph) 𝒢 are found. The graph (two-graph) 𝒢 can be constructed from ℒ by: (1) deleting all the vertices of ℒ with arbitrarily fixed parity of the row and column numbers; (2) suppressing all the vertices of degree 2 except those of degree 2 in ℒ; (3) merging all the vertices of degree 1 into a single vertex g. The matrix Kirchhoff theorem reduces the enumeration problem for the spanning trees on 𝒢 to the eigenvalue problem for the discrete Laplacian on the square lattice ℒ′=𝒢 \ g with mixed Dirichlet–Neumann boundary conditions in at least one direction. That fact explains some of the unusual finite-size properties of the dimer model.