We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly $1-\Theta(\frac{1}{n})$ as conjectured by Cameron \cite{CamConj} for the most interesting binary alphabet case. Finally, we describe a deterministic algorithm which decides whether a given random automaton is synchronizing in linear expected time.

中文翻译：

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关于可同步的概率

我们证明了具有 $n$ 个状态和任何固定的非单一字母表的随机自动机以高概率同步。此外，我们还证明了收敛率正好是 $1-\Theta(\frac{1}{n})$，正如 Cameron \cite{CamConj} 对最有趣的二进制字母情况所推测的那样。最后，我们描述了一种确定性算法，该算法决定给定的随机自动机是否在线性预期时间内同步。