Multi-point correlations and dual boundary conditions

As we have seen in Chapter 2, the partition function of the Ising model on a planar graph G with positive boundary conditions is proportional to the even subgraph generating function ZG⇤(x), where G⇤ is the weak dual of G and x is the weight vector defined by the low-temperature expansion. Similarly, the partition function of the model with free boundary conditions is proportional to ZG(x), where x is the high-temperature weight vector. This is the famous Kramers–Wannier duality of the planar Ising model. We used it together with Theorem 2.10 to express the free energy density and the correlation functions in terms of signed loops. Then, by analyzing the asymptotic growth rate of the signed loops, we were able to show analyticity of the free energy density and describe the behaviour of the correlation functions for off-critical temperatures. However, the low-temperature loop expansions work only for 2 ( c,1) and the high-temperature loop expansions are valid only for 2 (0, c). This is because for