Hideyuki Narumi

Proof of the generalized expressions for the number of perfect matchings of polycube graphs

The number of perfect rnatchings for the linear 2 × 2 ×n cubic lattice was analytically derived by diagonalizing the skew—symmetric 4n × 4n determinant, whose non—zero off—diagonal elements are either ±1 or ±i (pure imaginary number). The basic formulation invoking the matrix manipulation follows that of Kasteleyn, but the result obtained in this paper is the first example of the analytical solution for a special case of the three-dimensional Ising model.

Generalized expression for the numbers of perfect matching of cylindrical m×n graphs

The analytical expression for the numbers of perfect matching for the cylindrical m×n lattice graphs was derived by diagonalizing the skew‐symmetric matrix after Kasteleyn but with a little modification. The results are compared with the recursion relations obtained earlier by the present authors.

Generalized expression of the perfect matching number for 2×3×n lattices

A closed form of the perfect matching number for the cubic lattice of 2×3×n was obtained in terms of the linear combination of determinants, which was derived from its recursive expression. This is the first analytic solution for dimer statistics on three‐dimensional, cubic‐cell lattices. It is potentially applicable to the analytic solution of the Ising model for the general l×m×n lattice.

Expressions for the perfect matching numbers of cubicl x m x n lattices and their asymptotic values

A general expression of the perfect matching number for thel x m x n cubic lattice was conjectured and examined for infinitely large systems. The asymptotic value of the square of the perfect matching number was calculated by numerical integration. The present treatment will give a key to obtain the true analytic solution of the perfect matching numbers for the 3-dimensional lattices.