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Escape from the unstable equilibrium in a random process with infinitely many interacting particles. (English) Zbl 0652.60110
Summary: We consider a one-dimensional version of the model introduced by the first author, P. A. Ferrari and J. L. Lebowitz, ibid. 44, 589-644 (1986; Zbl 0629.60107). At each site of Z there is a particle with spin $$\pm 1$$. Particles move according to the stirring process and spins change according to the Glauber dynamics. In the hydrodynamical limit, with the stirring process suitably speeded up, the local magnetic density $$m_ l(r)$$ is proven in the paper cited above to satisfy the reaction-diffusion equation $(*)\quad \partial_ tm_ t(r)=2^{- 1}\partial^ 2_ rm_ t(r)-V'(m_ t)$ $$V(m)=-2^{-1}\alpha m^ 2+4^{-1}\beta m^ 4$$, $$\alpha$$ and $$\beta >0$$, $$\alpha$$ and $$\beta$$ being determined by the parameters of the Glauber dynamics.
In the present paper we consider an initial state with zero magnetization, $$m_ 0(r)=0$$. We then prove that at long times, before taking the hydrodynamical limit, the evolution departs from that predicted by (*) and that the microscopic state becomes a nontrivial mixture of states with different magnetizations.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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