一个具有两种遍历性相变的扰动元胞自动机
A perturbed cellular automaton with two phase transitions for the ergodicity
AI总结：
本文利用Gács结果构造了一个具有两种遍历性相变的元胞自动机，证明其在小噪声速率下遍历、中等速率下非遍历、接近1的速率下再次遍历。
摘要：
正速率猜想指出，具有严格正转移速率的一维概率元胞自动机（PCA）必定是遍历的。该猜想已被Gács反驳，其反例是一个在速率足够小的均匀随机噪声下非遍历的元胞自动机。对于所有已知反例，非遍历性都是在足够小的速率下被证明的。反之，所有元胞自动机在足够高速率的噪声下都是遍历的。目前尚未发现其他类型的遍历性相变，而已知反例在中等噪声速率下的行为也未知。我们给出一个具有两种相变的元胞自动机例子。利用Gács的结果作为黑箱，我们构造了一个元胞自动机，它在小噪声速率下是遍历的，在稍高速率下是非遍历的，而在接近1的速率下再次变为遍历的。
英文摘要：
The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using Gács's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.
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