Steif, Jeffrey E. The threshold voter automaton at a critical point. (English) Zbl 0814.60095 Ann. Probab. 22, No. 3, 1121-1139 (1994). Summary: We consider the threshold voter automaton in one dimension with threshold \(\tau > n/2\), where \(n\) is the number of neighbors and where we start from a product measure with density \({1/2}\). It has recently been shown that there is a critical value \(\theta_ c \approx 0.6469076\), so that if \(\tau = \theta n\) with \(\theta > \theta_ c\) and \(n\) is large, then most sites never flip, while for \(\theta \in (1/2, \theta_ c)\) and \(n\) large, there is a limiting state consisting mostly of large regions of points of the same type. Using a supercritical branching process, we show that the behavior at \(\theta_ c\) differs from both the \(\theta > \theta_ c\) regime and the \(\theta < \theta_ c\) regime and that, in some sense, there is a discontinuity both from the left and from the right at this critical value. Cited in 1 Document MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F10 Large deviations Keywords:cellular automata; critical value; Chen-Stein method; branching processes; threshold voter automaton PDFBibTeX XMLCite \textit{J. E. Steif}, Ann. Probab. 22, No. 3, 1121--1139 (1994; Zbl 0814.60095) Full Text: DOI