## One dimensional stochastic Ising models with small migration.(English)Zbl 0739.60099

A stochastic Ising model in $$\mathbb{Z}$$ is considered. That is a process $$\eta_ t$$ on $$\{-1,1\}^ \mathbb{Z}$$, where a component $$\eta(x)$$ changes its value at rate $$C(x,\eta)=\exp(-\sum_{F\ni x}I_ F\prod_{y\in F}\eta(y))$$. The paper deals with the case where these rates are translation invariant, depend only on a finite number of components and $$C(x,\eta)$$ is decreasing on $$\eta(x)=1$$ and increasing on $$\eta(x)=-1$$. Such a system is ergodic, that is as time tends to infinity, $${\mathcal L}(\eta_ t)$$ ($${\mathcal L}=$$law) converges weakly to a unique and reversible equilibrium state, which is identified as the Gibbs measure for the potential $$\{I_ F; F\leq \mathbb{Z}\}$$.
The question of the present article is to study what happens, if we add a transition (stirring) $$\eta\to \eta_{x,y}$$ with $$\eta_{x,y}(u)$$ given by $$\eta(y)$$ for $$u=x,\;\eta(x)$$ for $$u=y$$, and $$\eta(u)$$ otherwise. This new transition occurs at a rate $$\varepsilon p(x,y)$$ where $$p(x,y)$$ is a symmetric homogeneous transition kernel with finite range. Denote the resulting semigroup with $$T^ \varepsilon_ t$$ and let $$f$$ be a function depending on only finitely many coordinates. Then there exist constants $$A_ f$$, $$\lambda>0$$, $$\varepsilon_ 0>0$$ such that for all $$\xi,\eta$$, $$| T^ \varepsilon_ t(f)(\eta)-T^ \varepsilon_ tf(\xi)|\leq A_ fe^{-\lambda t}$$ whenever $$\varepsilon<\varepsilon_ 0$$. In particular, the system remains ergodic under the perturbation by the stirring as long as the strength $$\varepsilon$$ of that perturbation remains below a certain value $$\varepsilon_ 0$$. The proof uses first of all a theorem of D. Holley establishing such a property for the stochastic Ising model, i.e. $$\varepsilon=0$$ and secondly coupling arguments.

### MSC:

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