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Random walks supported on random points ofZ/nZ
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  • Published: June 1994

Random walks supported on random points ofZ/nZ

  • Martin Hildebrand1 

Probability Theory and Related Fields volume 100, pages 191–203 (1994)Cite this article

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Summary

This paper considers random walks on the integers modn supported onk points and asks how long does it take for these walks to get close to uniformly distributed. Ifk is a constant, Greenhalgh showed that at least some constant timesn 2/(k−1) steps are necessary to make the distance of the random walk from the uniform distribution small; here we show that ifn is prime, some constant timesn 2/(k−1) steps suffice to make this distance small for almost all choices ofk points. The proof uses the Upper Bound Lemma of Diaconis and Shahshahani and some averaging techniques. This paper also explores some cases wherek varies withn. In particular, ifk=⌊(logn)a⌋, we find different kinds of results for different values ofa, and these results disprove a conjecture of Aldous and Diaconis.

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References

  • [AD] Aldous, D., Diaconis, P.: Shuffling Cards and Stopping Times. (Techn. Rep. No. 231) Department of Statistics, Stanford University 1985

  • [AR] Alon, N., Roichman, Y.: Random Cayley Graphs Expanders. Random Struct. Algorithms5, 271–284 (1994)

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  • [Do] Dou, C.: Studies of Random Walks on Groups and Random Graphs. Ph.D. thesis. Department of Mathematics, Massachusetts Institute of Technology 1992

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  • [Gr] Greenhalgh, A.: Random walks on groups with subgroup invariance properties. Ph.D. thesis. Department of Mathematics, Stanford University 1989

  • [Gr2] Greenhalgh, A.: On a Model for Random Random-Walks on Finite Groups. (Preprint 1990)

  • [Hi] Hildebrand, M.: Rates of Convergence of Some Random Processes on Finite Groups. Ph.D. thesis. Department of Mathematics, Harvard University 1990

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Author information

Authors and Affiliations

  1. Institute for Mathematics and its Applications, University of Minnesota, 55455-0436, Minneapolis, MN, USA

    Martin Hildebrand

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  1. Martin Hildebrand
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Additional information

Research Supported in Part by a Rackham Faculty Fellowship at the University of Michigan

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Cite this article

Hildebrand, M. Random walks supported on random points ofZ/nZ . Probab. Th. Rel. Fields 100, 191–203 (1994). https://doi.org/10.1007/BF01199265

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  • Received: 13 November 1993

  • Revised: 21 March 1994

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199265

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Mathematics Subject Classification (1991)

  • 60B15
  • 60J15
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