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Speeding up Markov chains with deterministic jumps

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Abstract

We show that the convergence of finite state space Markov chains to stationarity can often be considerably speeded up by alternating every step of the chain with a deterministic move. Under fairly general conditions, we show that not only do such schemes exist, they are numerous.

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Acknowledgements

We thank Shirshendu Ganguly for bringing the paper [24] to our attention, which was crucial for the proof of Theorem 2.2. We thank Kannan Soundararajan, Perla Sousi, Ron Graham, Fan Chung, Charles Bordenave, Jimmy He, Huy Pham, and especially Jonathan Hermon for many insightful comments and references. Lastly, we thank the referee for a number of useful comments.

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Correspondence to Sourav Chatterjee.

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In memory of Harry Kesten.

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Sourav Chatterjee’s research was partially supported by NSF Grant DMS-1855484. Persi Diaconis’s research was partially supported by NSF Grant DMS-1954042. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Chatterjee, S., Diaconis, P. Speeding up Markov chains with deterministic jumps. Probab. Theory Relat. Fields 178, 1193–1214 (2020). https://doi.org/10.1007/s00440-020-01006-4

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  • DOI: https://doi.org/10.1007/s00440-020-01006-4

Keywords

  • Markov chain
  • Mixing time
  • Spectral gap
  • Cheeger constant

Mathematics Subject Classification

  • 60J10
  • 60J22