Bounds on the Correlations and Analyticity Properties of Ferromagnetic Ising Spin Systems*

We consider a ferromagnetic Ising spin system isomorphic to a lattice gas with attractive interactions. Using the Fortuin, Kasteleyn and Ginibre (FKG) inequalities we derive bounds on the decay of correlations between two widely separated sets of particles in terms of the decay of the pair correlation. This leads to bounds on the derivatives of various orders of the free energy with respect to the magnetic field h, and reciprocal temperature β. In particular, if the pair correlation has an upper bound (uniform in the size of the system) which decays exponentially with distance in some neighborhood of (/?', h') then the thermodynamic free energy density ψ(β, h) and all the correlation functions are infinitely differentiable at (/?', h'). We then show that when only pair interactions are present it is sufficient to obtain such a bound only at h = 0 (and only in the infinite volume limit) for systems with suitable boundary conditions. This is the case in the two dimensional square lattice with nearest neighbor interactions for 0^β<β0, where β^ 1 is the Onsager temperature at which ψ(β,h = 0) has a singularity. For β > β0, dψ(β, h)/dh is discontinuous at h = 0, i.e. β0 = βc, where β~ 1 is the temperature below which there is spontaneous magnetization.