We investigate Kasteleyn’s method of computing partition functions of Onsager’s lattice, who discovered that the partition function of the two-dimensional Ising model with H=0 was related to a combinatorial problem about dimers. We are interested in the combinatorial problem of a two-dimensional quadratic lattice covered completely with dimers, i.e., in terms of graph theory, we look for the number of ‘perfect matchings’ of the lattice. We also investigate the relation between this problem and another combinatorial problem connected with the Ising model of cooperative phenomena.

The one-dimensional Ising model with only nearest neighbor interactions is one of the simplest models to solve exactly in statistical mechanics. It can be solved exactly by various methods, including the transfer matrix, generating function, and induction. However, when second or third-nearest-neighbor interactions are taken into account, the problem becomes significantly more complicated. In higher dimensions, this method is more useful than other methods for finding exact analytical solutions. We present a recursive method for solving nearest-neighbor interactions exactly. We’ll follow the paper “Recursive Method in One-Dimensional Ising Model” by E. Marchi and J. Vila.