In this thesis results on the Partition function ZG(T, q) for the q–state Potts–Model on finite polygonal lattices G are presented.
These are polynomials in a ≡ e− J and q. The first step is to calculate all the coefficients of ZG(a, q) using a transfer matrix method.
The only points of non–analyticity are the zeros of the partition function; in the thermodynamic limit the complex temperature zeros form a continuous curve B via coalescence. This is the locus where the free energy is non–analytic. The zeros of Z(a, q) in the complex q–plane for finite a and in the complex a–plane for integer and non–integer values of q are plotted.
For a = 0 the partition function reduces to the chromatic polynomial PG(q) of the graph and the zeros are called chromatic zeros.
The behavior of those zeros as a increases from zero is investigated.
As for complex temperature the zeros in q form a continuous curve in the thermodynamic limit. This is the locus where the limiting function WG(q) = limn→∞ PG(q)1/n is non–analytic. WG(q) is the ground state degeneracy and is connected to the ground state entropy via S0(G, q) = kB ln(WG(q)).
Thus the characteristics of the zeros of q and a for finite lattices help to understand the properties of the model in the thermodynamic limit.

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