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The threshold voter automaton at a critical point. (English) Zbl 0814.60095

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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations
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