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Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems. (English) Zbl 0162.59002

Summary: 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 \(z=\text{const.}\, \exp(-2mH\beta)\) 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 \(n_s (\mathbf x_1, \dots,\mathbf x_s; \beta,z)\) are analytic in the two variables \(\beta\) and \(H\) if the reciprocal temperature \(\beta>0\) and the complex number \(H\) is not a limit point of zeros of the grand partition function \(\Xi\), and (ii) the Ursell functions \(u_s (\mathbf x_1, \dots,\mathbf x_s; \beta,z)\) tend to 0 as \(\Delta_s \equiv \max_{i, j} |\mathbf x_i -\mathbf x_j | \to\infty\) if \(\beta>0\) and \(\text{Re}\, H\neq 0\); in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value lambda, then \[ |u_s |C \exp [(-2 \beta m |\text{Re}\,H|+\varepsilon) \Delta_s /\lambda] \] where \(\varepsilon\) is any small positive number and \(C\) is independent of \(\Delta_s\). One consequence of the result (i) is that a phase transition can occur as \(\beta\) is varied at constant \(H\) only if \(H\) is a limit point of zeros of \(\Xi\) (which can happen only if \(\text{Re}\,H=0\)); this supplements Lee and Yang’s result that the same condition is necessary for a phase transition when \(H\) is varied at constant \(\beta\).
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 \(\beta>0\) and the complex fugacity \(z\) is less than the radius of convergence of the Mayer \(z\) expansion; for the continuum gas, however, \(n_s\) and \(u_s\) must be replaced by their values integrated over small volumes surrounding each of the points \(\mathbf x_2,\dots, \mathbf x_s\).
It is shown that the pressure \(p\) is analytic in both \(\beta\) and \(z\), if it is analytic in \(z\) at fixed \(\beta\) over a suitable range of values of \(\beta\) and \(z\), and further that, except for continuum systems without hard cores, \(p,n_s\) and \(u_s\) have convergent Maclaurin expansions in \(\beta\) for small enough \(z\).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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