Dimer Statistics and Phase Transitions

After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics of non-overlapping dimers, each occupying two neighboring sites of a lattice graph. It is shown that the configurational partition function of this system can be expressed in terms of a Pfaffian, and hence calculated explicitly, if the lattice graph is planar and if the dimers occupy all lattice sites. By the examples of the quadratic and the hexagonal lattice, it is found that the dimer system may show a phase transition similar to that of a two-dimensional Ising model, or one of a different nature, or no transition at all, depending on the activities of various classes of bonds. The Ising problem is then shown to be equivalent to a generalized dimer problem, and a rederivation, of Onsager's expression for the Ising partition function of a rectangular lattice graph is sketched on the basis of this equivalence.