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- W2570250091 abstract "We study class of cellular automata called $kappa$-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 ${{0, ldots, kappa-1}}$, and a special state $b(kappa) = lfloorfrac{kappa-1}{2}rfloor$ is designated as the `blinking' state. At step $t$, simultaneously for all vertices, the state of a vertex increments from $k$ to $k+1mod kappa$ unless $k>b(kappa)$ and at least one of its neighbors is in the state $b(kappa)$. We study this system on $mathbb{Z}$ where the initial configuration is drawn from a uniform product measure on ${0, ldots, kappa-1}^mathbb{Z}$. We show that the system clusters with high probability for all $kappage 3$, and give upper and lower bounds on the rate. In the special case of $kappa=3$, 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." @default.
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- W2570250091 date "2017-01-02" @default.
- W2570250091 modified "2023-09-27" @default.
- W2570250091 title "Synchronization of finite-state pulse-coupled oscillators on $mathbb{Z}$" @default.
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