Random Random Walks on Z2d

David Bruce Wilson

We consider random walks on classes of graphs defined on the d-dimensional binary cube Z2d 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.

Probability Theory and Related Fields, 108(4):441--457, 1997.

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