Summary.
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.
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Wilson, D. Random random walks on ℤ2 d . Probab Theory Relat Fields 108, 441–457 (1997). https://doi.org/10.1007/s004400050116
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DOI: https://doi.org/10.1007/s004400050116
- Key words and phrases. Random walk
- hypercube
- mixing time
- threshold.
- Mathematics Subject Classification (1991): Primary 60J15; Secondary 60B15