Configurational studies of the Potts models

Abstract

Eigenvalues and eigenvectors are derived for the q-state two-level model introduced by Potts in 1952, and it is shown that for any net the partition function depends only on the topology and not on any two-degree vertices. This enables a simple method to be used calculating the partition functions of standard star topologies. The second q-orientation model introduced by Potts (termed the planar Potts model) is discussed, and it is shown that the same property holds. Partition functions for certain star topologies are derived for this model. By considering the asymptotic form of the coefficients in high-temperature expansions of the partition function, estimates are obtained of the critical temperatures of these models in terms of the geometrical properties of self-avoiding walks.