Positivity constraints for the Ising ferromagnetic model

The consequences of the Lee and Yang representation (see Phys. Rev., vol.87, p.404, 410 (1952)) for a ferromagnetic Ising model are reviewed. Bounds for the Lee and Yang angle are given and the critical magnetic behaviour is analysed. A main result is that identities for the Mayer-Yvon coefficients are deduced, which lead to the expansion of the magnetization with respect to the variable (1-tanh 2 beta H). This expansion appears to exhibit remarkable positivity properties and the corresponding coefficients are tabulated. A central role is played by the Bethe lattice which is completely analysed, since its coefficients seem to give upper bounds for the more realistic regular lattices.