This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied by D. Andr{e}n and K. Markstr{o}m in 2009. We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well-known to be closely related to the Potts model in the absence of an external field. We show that Z(G;x,y,z) is #P-hard to evaluate at all points in $mathbb{Q}^3$, except those in an exception set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by H. Dell, T. Husfeldt and M. Wahl{e}n in 2010 in order to study the complexity of the Tutte polynomial. In analogy to their results, we give a dichotomy theorem stating that evaluations of Z(G;t,y) either take exponential time in the number of vertices of $G$ to compute, or can be done in polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.
تحميل البحث