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Approach to equilibrium of Glauber dynamics in the one phase region. I: The attractive case. (English) Zbl 0793.60110

Summary: Various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed. In particular some finite size conditions are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamics in arbitrary large volumes. It is shown that Dobrushin-Shlosman’s theory of complete analyticity and its dynamical counterpart due to D. W. Stroock and B. Zegarlinski [ibid. 149, No. 1, 175–193 (1992; Zbl 0758.60070)] cannot be applied, in general, to the whole one phase region since it requires mixing properties for regions of arbitrary shape.
An alternative approach, based on previous ideas of the second author [J. Stat. Phys. 50, No. 5-6, 1179–1200 (1988; Zbl 1084.82523)] and the second author and P. Picco [ibid. 59, No. 1, 221–256 (1990; Zbl 1083.82509)], is developed, which allows to establish results on rapid approach to equilibrium deeply inside the one phase region. In particular, in the ferromagnetic case, we considerably improve some previous results by R. Holley [Particle systems, random media and large deviations, Proc. Conf., Bowdoin Coll. 1984, Contemp. Math. 41, 215–234 (1985; Zbl 0577.60099)] and M. Aizenman and R. Holley [Percolation theory and ergodic theory of infinite particle systems, Proc. Workshop IMA, Minneapolis/Minn. 1984/85, IMA Vol. Math. Appl. 8, 1–11 (1987; Zbl 0621.60118)]. Our results are optimal in the sense that, for example, they show for the first time fast convergence of the dynamics for any temperature above the critical one for the \(d\)-dimensional Ising model with or without an external field.
In part II (see below) we extensively consider the general case (not necessarily attractive) and we develop a new method, based on renormalization group ideas and on an assumption of strong mixing in a finite cube, to prove hypercontractivity of the Markov semigroup of the Glauber dynamics.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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