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On the meeting of random walks on random DFA

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  • Quattropani, Matteo
  • Sau, Federico

Abstract

We consider two random walks evolving synchronously on a random out-regular graph of n vertices with bounded out-degree r≥2, also known as a random Deterministic Finite Automaton (DFA). We show that, with high probability with respect to the generation of the graph, the meeting time of the two walks is stochastically dominated by a geometric random variable of rate (1+o(1))n−1, uniformly over their starting locations. Further, we prove that this upper bound is typically tight, i.e., it is also a lower bound when the locations of the two walks are selected uniformly at random. Our work takes inspiration from a recent conjecture by Fish and Reyzin (2017) in the context of computational learning, the connection with which is discussed.

Suggested Citation

  • Quattropani, Matteo & Sau, Federico, 2023. "On the meeting of random walks on random DFA," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:spapps:v:166:y:2023:i:c:s0304414923001898
    DOI: 10.1016/j.spa.2023.104225
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    References listed on IDEAS

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    1. Aldous, David J., 1982. "Markov chains with almost exponential hitting times," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 305-310, September.
    2. Caputo, Pietro & Quattropani, Matteo, 2021. "Mixing time trichotomy in regenerating dynamic digraphs," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 222-251.
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