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

Neuhauser, Claudia

Ann. Probab. 18, No. 4, 1539-1546 (1990).

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.

Reviewer: A.Greven (Göttingen)

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

interacting particle systems; exponential convergence; stochastic Ising model; Gibbs measure; perturbation; coupling arguments