Mixing properties in lattice systems

Abstract

We study mixing or spatial cluster properties and some of their consequences in classical lattice systems, in particular complete regularity and the weaker notion of strong mixing. Introducing the notion of reflection positivity as a generalization ofT-positivity of [1], we construct a generalized transfer matrixP and relate complete regularity to a spectral gap inP. It is shown that all reflection invariant Ising systems with n.n. and ferromagnetic n.n.n. interaction satisfy reflection positivity. For Ising ferromagnets with reflection positivity, exponential decay of the truncated 2-point function implies complete regularity. In particular, the 2-dimensional spin-1/2 Ising model is completely regular, except at the critical point. This complements a result of [2] that strong mixing fails at the critical point of this model and in this case verifies the suggestion of Jona-Lasinio [3] that critical behaviour should be linked with failure of strong mixing. We then show that strong mixing imposes severe restrictions on the possible form of limits of block spins. Strong mixing in each direction allows onlyindependent Gaussians as non-zero limit if the 2-point function exists; strong mixing in a single direction only will allow infinitely divisible distributions.