New results on the domain of analyticity of the free energy for the Ising model

We investigate the analyticity of the free energy of the Ising model in the presence of a non-zero external magnetic field, at high temperature, and at low temperature. Using the Fernandez–Procacci convergence criterion for cluster expansions, together with generating-function techniques and graph-theoretical methods, we derive improved convergence conditions in all three regimes. In particular, the generating-function approach yields sharper estimates for polymers and contours in the strong-field and low-temperature regimes, while a new high-temperature expansion based on Veblen’s theorem provides a substantially larger analyticity region than the classical results in the literature.