Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics

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Abstract
This is the first of three papers on the Glauber evolution of Ising spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range, gamma -1, of the interaction while time is kept finite: we 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 gamma to 0 but are small in units log gamma -1. In this time regime we can give a very precise description of the evolution and a sharp characterization of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of external magnetic fields and below the critical temperature, on a suitable macroscopic limit, an interface between two stable phases moves by mean curvature. All the proofs are consequence of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

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