This is the fin1 of three papers on the Glauber evolution of king spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range. y-', of the interaction while time is kepi finite: ye prove that in this limit the magnetization density converges to the solution of a deterministic, nonlinear, nonlocal evolution equation. We also show that the long time behaviour of this equation describes correctly the evolution of the spin system till times which diverge as y + 0 but are small in units logy-'. In this time regime we can give a "ry precise description of the evolution and a sharp characterhion of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of extema! magnetic fields and below the critical temperature, on a suitable macroscopic limit an interface between two stable phases moves by mean curvature. All ihe proofs are consequek of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.

We study one-dimensional Ising spin systems with ferromagnetic, long-range interaction decaying as n −2+α , α ∈ ( 1 2 , ln 3 ln 2 − 1), in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with IP = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures.

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In this paper we address the questions of perfectly sampling a Gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations. We solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures. The algorithm works for general Gibbsian interaction under requirements on the tails of the interaction. As a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure.