Complexity of Ising Polynomials

Abstract. 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 [14]. Z(G; x, y, z) generalizes a bivariate polynomial Z(G; t, y), which was studied in by D. Andrén and K. Markström in [1]. 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 Q, except those in an exceptional 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. Wahlén in [6] in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETH 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.