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Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(-2mHß) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsns(x1, ...,xs; ß,z) are analytic in the two variables ß andH if the reciprocal temperature ß>0 and the complex numberH is not a limit point of zeros of the grand partition function ?, and (ii) the Ursell functionsus(x1, ...,xs; ß,z) tend to 0 as ?s=Maxi, j |xi-xj| ? 8 if ß>0 and ReH?0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value ?, then |us|C exp [(-2 ßm |ReH|+e) ?s/?] where e is any small positive number andC is independent of ?s. One consequence of the result (i) is that a phase transition can occur as ß is varied at constantH only ifH is a limit point of zeros of ? (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant ß.

Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems