A coupling of infinite particle systems. II (Q1296407)

The paper is presented as a continuation of the paper of the second author [ibid. 35, No.~1, 43-52 (1995; Zbl 0840.60097)]. In the first section interacting particle systems on \(D^{\mathbb{Z}}\) with \(D\) finite are studied for which the flip rates are of finite range and are uniformly bounded. As consequences of results of the mentioned paper and of the paper of the first author [Trans. Am. Math. Soc. 318, No.~2, 601-614 (1990; Zbl 0698.60089)] it is shown that, in the case that the process is translation invariant, extremal elements of the set of measures that are both stationary and translation invariant are ergodic with respect to translations on \(\mathbb{Z}\). Under the same assumptions extremal stationary measures that are translation invariant are mixing with respect to translations on \(\mathbb{Z}\). Also, if \(\mu\) is an extremal invariant distribution, then \(E^{\mu}f(\xi_0)g(\xi_t)-\langle\mu,f\rangle\langle\mu,g\rangle\to 0\) for \(t\to\infty\), \(f, g\in L^2(D^{\mathbb Z},\mu)\). The second section is devoted to generalizations of one-dimensional voter models on \(\{0,1\}^{\mathbb Z}\). Under two possible variants of assumptions on the finite range and translation invariant flip rates the convergence of \(P^{\xi_0}(\xi_t(x)\neq\xi_t(y))\) to zero for \(t\to\infty\) is shown whenever \(x,y\) are two distinct integers and \(\xi_0\in\{0,1\}^{\mathbb Z}\) is an arbitrary initial value. This proves a conjecture of R. Durrett on the voter model. This conclusion was known for the voter model if the initial condition \(\xi_0\) was replaced by a translation invariant initial distribution.