@MISC{08dimersand, author = {}, title = {Dimers and the Ising model}, year = {2008} }

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Abstract

We present a connection between the ground state of the Ising Model and the dimer problem. We find the generating function for dimers as the appropriate limit of the free energy per spin for the Ising Model. 1. Dimers. Dimers are objects connecting two neighboring sites of the lattice. We consider a lattice which can be covered with dimers in the way that each site of the lattice belongs to exactly one dimer. How to calculate the number of such coverings for a given lattice was shown in 1961 by Kastelayn, Fisher and Temporley in 1961 [1-3]. The connection between the Ising model on the plane lattice and the dimer problem is well known. In 1961, Fisher [4] gave a general rule connecting the partition function for the Ising model with a generating function for configurations of dimers on an appropriately constructed lattice. For the Ising model on the honeycomb, for instance, the dimers on 3-12 lattice are considered. Moreover, the Ising modelâ€”as well