Probability Theory and Related Fields

, Volume 108, Issue 4, pp 441–457 | Cite as

Random random walks on ℤ2d

  • David Bruce Wilson


We consider random walks on classes of graphs defined on the d-dimensional binary cube ℤ2 d by placing edges on n randomly chosen parallel classes of vectors. The mixing time of a graph is the number of steps of a random walk before the walk forgets where it started, and reaches a random location. In this paper we resolve a question of Diaconis by finding exact expressions for this mixing time that hold for all n>d and almost all choices of vector classes. This result improves a number of previous bounds. Our method, which has application to similar problems on other Abelian groups, uses the concept of a universal hash function, from computer science.

Key words and phrases. Random walk hypercube mixing time threshold. 
Mathematics Subject Classification (1991): Primary 60J15; Secondary 60B15 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • David Bruce Wilson
    • 1
  1. 1.University of California, 387 Soda Hall Berkeley, CA 94720-1776, USA e-mail:

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