On Complex Zeros of the q-Potts Partition Function for a Self-dual Family of Graphs

Research paper by J.-M. Billiot, F. Corset, E. Fontenas

Indexed on: 04 May '10Published on: 04 May '10Published in: Journal of Statistical Physics


This paper deals with the location of the complex zeros of q-Potts partition function for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane. Thus, in these regions, the analyticity of the free energy per site can be derived easily. Next, some examples of graphs with their Tutte polynomial having few eigenvalues are given. The case of the cycle with an edge having a high order of multiplicity is presented in detail. In particular, we show that the well known conjecture of Chen et al. is false in the finite case. Furthermore we obtain a sequence of self-dual graphs for which the unit circle does not belong to the accumulation sets of the zeros.