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