A signal-recovery system: asymptotic properties, and construction of an infinite-volume process.

*(English)*Zbl 1058.60093Summary: We consider a linear sequence of ‘nodes’, each of which can be in state 0 (‘off’) or 1 (‘on’). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are ‘on’. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present results for finite systems and use some of these results to construct an infinite-volume process (with signals ‘coming from infinity’), which has some peculiar properties. This construction is related to a question by D. Aldous and we hope that it sheds some light on, and stimulates further investigation of, that question.

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C43 | Time-dependent percolation in statistical mechanics |

92D25 | Population dynamics (general) |

##### Keywords:

on-off sequence; long-range interactions; infinite-volume dynamics; 1-D time-dependent percolation
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\textit{J. van den Berg} and \textit{B. Tóth}, Stochastic Processes Appl. 96, No. 2, 177--190 (2001; Zbl 1058.60093)

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##### References:

[1] | Aldous, D., The percolation process on a tree where infinite clusters are frozen, Math. proc. Cambridge philos. soc., 128, 465-477, (2000) · Zbl 0961.60096 |

[2] | Benjamini, I., Schramm, O. 1999. Private communication (via D. Aldous). |

[3] | Flajolet, P.; Sedgewick, R., Mellin transforms and asymptotics: finite differences and Rice’s integrals, Theoret. comput. sci., 144, 101-124, (1995) · Zbl 0869.68056 |

[4] | Járai, A., 1999. Private communication. |

[5] | Lukács, A., 1999. Private communication. |

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