Random random walks on Z2d

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.