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A complete convergence theorem for attractive reversible nearest particle systems. (English) Zbl 0891.60092

A class of one-dimensional nearest particle systems (NPS) on the state space \(\{0, 1\}^{Z}\) is studied. Such systems are defined as random processes being continuous Markov chains on the countable state space of subsets \(\gamma\) on \(Z\). The process is called finite if the function \(\gamma (x): Z \to \{0, 1\}\) is equal to unity only for a finite set of lattice sites \(x\in Z\) at all times \(t \geq 0\). Analogously, it is called infinite if this function is nonzero for an infinite set of sites at all \(t \geq 0\). Correctness of these notions is guaranteed by process dynamics. Namely, each chain is determined by flip rates \(c(x, \gamma)\) being equal to unity if \(\gamma (x) = 1\) and to \(\beta (l_x, r_x)\) if \(\gamma (x) = 0\), where \(l_x = x - \sup\{y < x: \gamma (y) =1\}\) and \(r_x = \inf\{y > x: \gamma (y) = 1\} - x\). They say that each finite process survives if there is some finite initial configuration \(\gamma_0\) so that trap state 0 is not hit a.s.. An infinite process is said to survive if there exists a nontrivial invariant measure, i.e. a probability measure \(\nu\) on \(\{0, 1\}^Z\) so that for all continuous \(f\) and all positive \(t\), \(\int f(\omega)\nu(d\omega) = \int P_t (\omega)\nu (d\omega) \), where \(P_t\) is the semigroup for the process. So-called attractive and reversible processes are considered. For these processes the flip-rate determining function \(\beta (\cdot, \cdot)\) is of the form \[ \beta (l, r) = \beta (l) \beta (r)/\beta (l+r),\qquad \beta (\infty, l) = \beta (l,\infty) = \beta (l) \] for some strictly positive function \(\beta : Z \to R\), satisfying \(\sum_n\beta (n) < \infty\). For these processes, two existence theorems are proved:
Theorem A. A finite reversible nearest particle system with flip rates determined by \(\beta\) survives if and only if \(\sum_n \beta(n) > 1\).
Theorem B. For attractive, reversible NPS satisfying \(\sum_n \beta (n, n) < \infty\), and either \(\sum_n \beta(n) > 1\) or \(\sum_n \beta(n) = 1\) and \(\sum_n n\beta(n) < \infty\), the renewal measure Ren\((\beta)\) is the unique non-trivial, translation invariant, stationary probability measure. Here, Ren\((\beta)\) is the measure on \(\{0,1\}^Z\) for which 1’s are distributed according to the stationary renewal process corresponding to the probability law on the integers, \(\{\beta (n)\theta^n\}, \theta \in (0, 1]\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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