Analytic and Clustering Properties of Thermodynamic Functions and Distribution Functions for Classical Lattice and Continuum Systems*

Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic field H (or the corresponding lattice gas model with fugacity z = const. exp(—2 m H β) where m 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 functions ne(x19 . . .,xs; β,z) are analytic in the two variables β and H if the reciprocal temperature β > 0 and the complex number H is not a limit point of zeros of the grand partition function Ξ, and (ii) the Ursell functions us(x19 . . ., xg; β, z) tend to 0 as Δs ΞΞ maxίf _, |acf — Xj\ -> oo if β > 0 and ReiϊΓφO; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value λ, then \us\ < C exp[(—2βm \ReH\ + ε) AJλ] where ε is any small positive number and C is independent of Δ8. One consequence of the result (i) is that a phase transition can occur as β is varied at constant H only if H is a limit point of zeros of Ξ (which can happen only if Reiϊ = 0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition when II is varied at constant β. For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided β > 0 and the complex fugacity z is less than the radius of convergence of the Mayer z expansion for the continuum gas, however, nt and us must be replaced by their values integrated over small volumes surrounding each of the points x2, . . ., xs. It is shown that the pressure p is analytic in both β and z, if it is analytic in z at fixed β over a suitable range of values of β and z, and further that, except for continuum systems without hard cores, φ. ns and us have convergent Maclaurin expansions in β for small enough z.