Synchronization of finite-state pulse-coupled oscillators on ℤ
arXiv: Probability(2017)
摘要
We study class of cellular automata called κ-color firefly cellular automata, which were introduced recently by the first author. At each discrete time t, each vertex in a graph has a state in {0, …, κ-1}, and a special state b(κ) = ⌊κ-1/2⌋ is designated as the `blinking' state. At step t, simultaneously for all vertices, the state of a vertex increments from k to k+1κ unless k>b(κ) and at least one of its neighbors is in the state b(κ). We study this system on ℤ where the initial configuration is drawn from a uniform product measure on {0, …, κ-1}^ℤ. We show that the system clusters with high probability for all κ≥ 3, and give upper and lower bounds on the rate. In the special case of κ=3, we obtain sharp results using a combination of functional central limit theorem for Markov chains and generating function method. Our proof relies on a classic idea of relating the limiting behavior with an accompanying interacting particle system, as well as another monotone comparison process that we introduce in this paper.
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