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Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. (English) Zbl 0621.60118
Percolation theory and ergodic theory of infinite particle systems, Proc. Workshop IMA, Minneapolis/Minn. 1984/85, IMA Vol. Math. Appl. 8, 1-11 (1987).
[For the entire collection see Zbl 0615.00015.]
We show that, under the conditions of the Dobrushin Shlosman theorem for uniqueness of the Gibbs state, the reversible stochastic Ising model converges to equilibrium exponentially fast on the \(L^ 2\) space of that Gibbs state. For stochastic Ising models with attractive interactions and under conditions which are somewhat stronger than Dobrushin’s, we prove that the semi-group of the stochastic Ising model converges to equilibrium exponentially fast in the uniform norm. We also give a new, much shorter, proof of a theorem which says that if the semi-group of an attractive spin flip system converges to equilibrium faster than \(1/t^ d\) where d is the dimension of the underlying lattice, then the convergence must be exponentially fast.
Reviewer: Abstract

60K35 Interacting random processes; statistical mechanics type models; percolation theory